{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T07:34:29Z","timestamp":1774683269134,"version":"3.50.1"},"reference-count":360,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T00:00:00Z","timestamp":1610668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["MOMS 1761423"],"award-info":[{"award-number":["MOMS 1761423"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DCSD 1825837"],"award-info":[{"award-number":["DCSD 1825837"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["D19AP00052"],"award-info":[{"award-number":["D19AP00052"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.<\/jats:p>","DOI":"10.3390\/e23010110","type":"journal-article","created":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T12:44:05Z","timestamp":1610714645000},"page":"110","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":85,"title":["Applications of Distributed-Order Fractional Operators: A Review"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3832-6500","authenticated-orcid":false,"given":"Wei","family":"Ding","sequence":"first","affiliation":[{"name":"Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8975-130X","authenticated-orcid":false,"given":"Sansit","family":"Patnaik","sequence":"additional","affiliation":[{"name":"Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8320-6543","authenticated-orcid":false,"given":"Sai","family":"Sidhardh","sequence":"additional","affiliation":[{"name":"Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA"}]},{"given":"Fabio","family":"Semperlotti","sequence":"additional","affiliation":[{"name":"Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,15]]},"reference":[{"key":"ref_1","first-page":"301","article-title":"Letter from Hanover, Germany, to GFA L\u2019Hopital, 30 September 1695","volume":"2","author":"Leibniz","year":"1849","journal-title":"Math. Schriften"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"De Oliveira, E.C., and Tenreiro Machado, J.A. (2014). A review of definitions for fractional derivatives and integral. Math. Probl. Eng., 2014.","DOI":"10.1155\/2014\/238459"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Erd\u00e9lyi, A. (1940). On fractional integration and its application to the theory of Hankel transforms. Q. J. Math., 293\u2013303.","DOI":"10.1093\/qmath\/os-11.1.293"},{"key":"ref_4","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1122\/1.549724","article-title":"A theoretical basis for the application of fractional calculus to viscoelasticity","volume":"27","author":"Bagley","year":"1983","journal-title":"J. Rheol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1115\/1.3167615","article-title":"On the appearance of the fractional derivative in the behavior of real materials","volume":"51","author":"Torvik","year":"1984","journal-title":"J. Appl. Mech. Trans. ASME"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1016\/j.jsv.2004.09.019","article-title":"Statistical origins of fractional derivatives in viscoelasticity","volume":"284","author":"Chatterjee","year":"2005","journal-title":"J. Sound Vib."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s42102-019-00007-9","article-title":"A fractional approach to non-Newtonian blood rheology in capillary vessels","volume":"1","author":"Alotta","year":"2019","journal-title":"J. Peridyn. Nonlocal Model."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1016\/0960-0779(95)00125-5","article-title":"Fractional relaxation-oscillation and fractional diffusion-wave phenomena","volume":"7","author":"Mainardi","year":"1996","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gritsenko, D., and Paoli, R. (2020). Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions. Appl. Sci., 10.","DOI":"10.3390\/app10249093"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gritsenko, D., and Paoli, R. (2020). Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: Parametric Study. Appl. Sci., 10.","DOI":"10.3390\/app10249080"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Failla, G., and Zingales, M. (2020). Advanced Materials Modelling Via Fractional Calculus: Challenges and Perspectives, Royal Society.","DOI":"10.1098\/rsta.2020.0050"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.mechrescom.2006.05.001","article-title":"Non-local continuum mechanics and fractional calculus","volume":"33","author":"Lazopoulos","year":"2006","journal-title":"Mech. Res. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10659-011-9346-1","article-title":"A fractional model of continuum mechanics","volume":"107","author":"Drapaca","year":"2012","journal-title":"J. Elast."},{"key":"ref_15","first-page":"20120433","article-title":"The mechanically based non-local elasticity: An overview of main results and future challenges","volume":"371","author":"Failla","year":"2013","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1080\/01495739.2014.885332","article-title":"Thermoelasticity in the framework of the fractional continuum mechanics","volume":"37","author":"Sumelka","year":"2014","journal-title":"J. Therm. Stress."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2551","DOI":"10.1007\/s11012-014-0044-5","article-title":"Nonlocal elasticity: An approach based on fractional calculus","volume":"49","author":"Carpinteri","year":"2014","journal-title":"Meccanica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1016\/j.ijsolstr.2020.05.034","article-title":"A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity","volume":"202","author":"Patnaik","year":"2020","journal-title":"Int. J. Solids Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"103529","DOI":"10.1016\/j.ijnonlinmec.2020.103529","article-title":"Geometrically nonlinear response of a fractional-order nonlocal model of elasticity","volume":"125","author":"Sidhardh","year":"2020","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Patnaik, S., Sidhardh, S., and Semperlotti, F. (2020). Fractional-order models for the static and dynamic analysis of nonlocal plates. Commun. Nonlinear Sci. Numer. Simul., 105601.","DOI":"10.1016\/j.cnsns.2020.105601"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"105710","DOI":"10.1016\/j.ijmecsci.2020.105710","article-title":"Geometrically nonlinear analysis of nonlocal plates using fractional calculus","volume":"179","author":"Patnaik","year":"2020","journal-title":"Int. J. Mech. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sidhardh, S., Patnaik, S., and Semperlotti, F. (2020). Fractional-Order Structural Stability: Formulation and Application to the Critical Load of Slender Structures. arXiv.","DOI":"10.1016\/j.ijmecsci.2021.106443"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Sidhardh, S., Patnaik, S., and Semperlotti, F. (2020). Analysis of the Post-Buckling Response of Nonlocal Plates via Fractional Order Continuum Theory. J. Appl. Mech., 1\u201322.","DOI":"10.1115\/1.4049224"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"20200200","DOI":"10.1098\/rspa.2020.0200","article-title":"A generalized fractional-order elastodynamic theory for non-local attenuating media","volume":"476","author":"Patnaik","year":"2020","journal-title":"Proc. R. Soc. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2668","DOI":"10.1016\/j.sigpro.2006.02.015","article-title":"Application of fractional calculus to ultrasonic wave propagation in human cancellous bone","volume":"86","author":"Sebaa","year":"2006","journal-title":"Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1121\/1.3377056","article-title":"Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian","volume":"127","author":"Treeby","year":"2010","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.cnsns.2015.06.014","article-title":"Modeling and simulation of the fractional space-time diffusion equation","volume":"30","author":"Baleanu","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1016\/j.physa.2018.05.137","article-title":"Analysis of reaction\u2013diffusion system via a new fractional derivative with non-singular kernel","volume":"509","author":"Saad","year":"2018","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"115035","DOI":"10.1016\/j.jsv.2019.115035","article-title":"Application of fractional order operators to the simulation of ducts with acoustic black hole terminations","volume":"465","author":"Hollkamp","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"203101","DOI":"10.1063\/5.0004605","article-title":"Scattering cross sections of acoustic nonlocal inclusions: A fractional dynamic approach","volume":"127","author":"Buonocore","year":"2020","journal-title":"J. Appl. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(94)00055-7","article-title":"Fractal physiology for physicists: L\u00e9vy statistics","volume":"246","author":"West","year":"1994","journal-title":"Phys. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0960-0779(00)00238-1","article-title":"A fractional calculus approach to the description of stress and strain localization in fractal media","volume":"13","author":"Carpinteri","year":"2002","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","first-page":"2521","article-title":"Fractal solids, product measures and fractional wave equations","volume":"465","author":"Li","year":"2009","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"12","DOI":"10.3389\/fphys.2010.00012","article-title":"Fractal physiology and the fractional calculus: A perspective","volume":"1","author":"West","year":"2010","journal-title":"Front. Physiol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"20190288","DOI":"10.1098\/rsta.2019.0288","article-title":"Thermo-poromechanics of fractal media","volume":"378","author":"Li","year":"2020","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Sheng, H., Chen, Y., and Qiu, T. (2011). Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4471-2233-3"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.sigpro.2010.08.003","article-title":"On the fractional signals and systems","volume":"91","author":"Magin","year":"2011","journal-title":"Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Lazarevi\u0107, M.P., Mandi\u0107, P.D., and Ostoji\u0107, S. (2020). Further results on advanced robust iterative learning control and modeling of robotic systems. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci., 0954406220965996.","DOI":"10.1177\/0954406220965996"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Oziablo, P., Mozyrska, D., and Wyrwas, M. (2020). Discrete-Time Fractional, Variable-Order PID Controller for a Plant with Delay. Entropy, 22.","DOI":"10.3390\/e22070771"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"20190498","DOI":"10.1098\/rspa.2019.0498","article-title":"Applications of variable-order fractional operators: A review","volume":"476","author":"Patnaik","year":"2020","journal-title":"Proc. R. Soc. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1515\/fca-2019-0003","article-title":"A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications","volume":"22","author":"Sun","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Caputo, M., and Fabrizio, M. (2017). The kernel of the distributed order fractional derivatives with an application to complex materials. Fractal Fract., 1.","DOI":"10.3390\/fractalfract1010013"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"58","DOI":"10.3389\/fphy.2018.00058","article-title":"Towards multifractional calculus","volume":"6","author":"Calcagni","year":"2018","journal-title":"Front. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1016586905654","article-title":"Variable order and distributed order fractional operators","volume":"29","author":"Lorenzo","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_46","first-page":"383","article-title":"Linear models of dissipation whose Q is almost frequency independent","volume":"19","author":"Caputo","year":"1966","journal-title":"Ann. Geophys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_48","unstructured":"Caputo, M. (1969). Elasticita e Dissipazione, Zanichelli."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02826009","article-title":"Mean fractional-order-derivatives differential equations and filters","volume":"41","author":"Caputo","year":"1995","journal-title":"Annali dell\u2019Universita di Ferrara"},{"key":"ref_50","first-page":"865","article-title":"On the existence of the order domain and the solution of distributed order equations\u2014Part I","volume":"2","author":"Bagley","year":"2000","journal-title":"Int. J. Appl. Math."},{"key":"ref_51","first-page":"965","article-title":"On the existence of the order domain and the solution of distributed order equations\u2014Part II","volume":"2","author":"Bagley","year":"2000","journal-title":"Int. J. Appl. Math."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Garrappa, R., Kaslik, E., and Popolizio, M. (2019). Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial. Mathematics, 7.","DOI":"10.3390\/math7050407"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Atanackovi\u0107, T.M., Pilipovi\u0107, S., Stankovi\u0107, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics, Wiley Online Library.","DOI":"10.1002\/9781118577530"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Sandev, T., and Tomovski, \u017d. (2019). Fractional Equations and Models: Theory and Applications, Springer.","DOI":"10.1007\/978-3-030-29614-8"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.1016\/j.camwa.2012.01.053","article-title":"Distributed order equations as boundary value problems","volume":"64","author":"Ford","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.cam.2008.07.018","article-title":"Numerical analysis for distributed-order differential equations","volume":"225","author":"Diethelm","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1023\/B:DIEQ.0000028722.41328.21","article-title":"On the theory of the continual integro-differentiation operator","volume":"40","author":"Pskhu","year":"2004","journal-title":"Differ. Equ."},{"key":"ref_58","unstructured":"Pskhu, A. (2005). Partial Differential Equations of Fractional Order, Nauka."},{"key":"ref_59","first-page":"449","article-title":"Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations: Part one","volume":"245","author":"Umarov","year":"2005","journal-title":"J. Anal. Its Appl."},{"key":"ref_60","first-page":"243","article-title":"Numerical solution of linear multi-term initial value problems of fractional order","volume":"6","author":"Diethelm","year":"2004","journal-title":"J. Comput. Anal. Appl"},{"key":"ref_61","unstructured":"Diethelm, K., and Ford, N.J. (2005). Numerical Solution Methods for Distributed Order Differential Equations, Institute of Mathematics & Informatics, Bulgarian Academy of Sciences."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/j.jmaa.2006.05.038","article-title":"On a nonlinear distributed order fractional differential equation","volume":"328","author":"Oparnica","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Van Bockstal, K. (2020). Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order). Mathematics, 8.","DOI":"10.3390\/math8081283"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Noroozi, H., Ansari, A., and Dahaghin, M.S. (2012). Existence results for the distributed order fractional hybrid differential equations. Abstr. Appl. Anal., 2012.","DOI":"10.1155\/2012\/163648"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1016\/j.camwa.2011.03.041","article-title":"Theory of fractional hybrid differential equations","volume":"62","author":"Zhao","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_66","first-page":"55","article-title":"Basic results on distributed order fractional hybrid differential equations with linear perturbations","volume":"2","author":"Noroozi","year":"2014","journal-title":"J. Math. Model."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1080\/10652460802568069","article-title":"Distributional framework for solving fractional differential equations","volume":"20","author":"Atanackovic","year":"2009","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"4101","DOI":"10.1016\/j.na.2010.01.042","article-title":"Semilinear ordinary differential equation coupled with distributed order fractional differential equation","volume":"72","author":"Atanackovic","year":"2010","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Fedorov, V.E. (2020). Generators of analytic resolving families for distributed order equations and perturbations. Mathematics, 8.","DOI":"10.3390\/math8081306"},{"key":"ref_70","first-page":"3","article-title":"Analytic study on linear systems of distributed order fractional differential equations","volume":"67","author":"Refahi","year":"2012","journal-title":"Le Matematiche"},{"key":"ref_71","first-page":"409","article-title":"Boundary value problems for the generalized time-fractional diffusion equation of distributed order","volume":"12","author":"Luchko","year":"2009","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1016\/j.jmaa.2010.08.048","article-title":"Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation","volume":"374","author":"Luchko","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1016\/j.jmaa.2008.04.065","article-title":"Boundary value problems for multi-term fractional differential equations","volume":"345","author":"Bhalekar","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.amc.2014.11.073","article-title":"Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients","volume":"257","author":"Li","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"1766","DOI":"10.1016\/j.camwa.2009.08.015","article-title":"Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation","volume":"59","author":"Luchko","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_76","first-page":"207","article-title":"An operational method for solving fractional differential equations with the Caputo derivatives","volume":"24","author":"Luchko","year":"1999","journal-title":"Acta Math. Vietnam"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.jmaa.2011.12.055","article-title":"Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain","volume":"389","author":"Jiang","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1080\/10652469.2015.1039224","article-title":"Completely monotone functions and some classes of fractional evolution equations","volume":"26","author":"Bazhlekova","year":"2015","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"050008","DOI":"10.1063\/1.5082107","article-title":"Complete monotonicity of the relaxation moduli of distributed-order fractional Zener model","volume":"Volume 2048","author":"Bazhlekova","year":"2018","journal-title":"AIP Conference Proceedings"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1016\/j.nonrwa.2011.08.028","article-title":"Fractional relaxation equations of distributed order","volume":"13","year":"2012","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1177\/1077546307077468","article-title":"The two forms of fractional relaxation of distributed order","volume":"13","author":"Mainardi","year":"2007","journal-title":"J. Vib. Control"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Lorenzo, C.F., and Hartley, T.T. (2007). Initialization, conceptualization, and application in the generalized (fractional) calculus. Crit. Rev. Biomed. Eng., 35.","DOI":"10.1615\/CritRevBiomedEng.v35.i6.10"},{"key":"ref_83","first-page":"531","article-title":"Numerical solution methods for distributed order differential equations","volume":"4","author":"Diethelm","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.apm.2019.01.013","article-title":"Legendre wavelets approach for numerical solutions of distributed order fractional differential equations","volume":"70","author":"Yuttanan","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"112739","DOI":"10.1016\/j.cam.2020.112739","article-title":"Crank\u2014Nicolson\/Galerkin spectral method for solving two-dimensional time-space distributed-order weakly singular integro-partial differential equation","volume":"374","author":"Abbaszadeh","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Abbaszadeh, M., and Dehghan, M. (2019). Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection\u2013diffusion equation. Eng. Comput., 1\u201317.","DOI":"10.1007\/s00366-019-00861-7"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1016\/j.apnum.2019.11.010","article-title":"Space-time finite element method for the distributed-order time fractional reaction diffusion equations","volume":"152","author":"Bu","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"3476","DOI":"10.1002\/mma.4839","article-title":"A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation","volume":"41","author":"Dehghan","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"1929","DOI":"10.1016\/j.apm.2015.09.035","article-title":"Deterministic analysis of distributed order systems using operational matrix","volume":"40","author":"Duong","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Fakhar-Izadi, F. (2020). Fully Petrov\u2014Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation. Eng. Comput., 1\u201310.","DOI":"10.1007\/s00366-020-00968-2"},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Hafez, R.M., Zaky, M.A., and Abdelkawy, M.A. (2020). Jacobi Spectral Galerkin method for Distributed-Order Fractional Rayleigh-Stokes problem for a Generalized Second Grade Fluid. Front. Phys., 7.","DOI":"10.3389\/fphy.2019.00240"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"A1003","DOI":"10.1137\/16M1073121","article-title":"Petrov\u2013Galerkin and spectral collocation methods for distributed order differential equations","volume":"39","author":"Kharazmi","year":"2017","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Jibenja, N., Yuttanan, B., and Razzaghi, M. (2018). An Efficient Method for Numerical Solutions of Distributed-Order Fractional Differential Equations. J. Comput. Nonlinear Dyn., 13.","DOI":"10.1115\/1.4040951"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.apnum.2016.11.001","article-title":"Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method","volume":"114","author":"Morgado","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.jcp.2016.01.041","article-title":"Numerical solution of distributed order fractional differential equations by hybrid functions","volume":"315","author":"Mashayekhi","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.apnum.2019.03.005","article-title":"Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation","volume":"142","author":"Xu","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_97","unstructured":"Pourbabaee, M., and Saadatmandi, A. (2020). Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations. Comput. Methods Differ. Equ."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"2460","DOI":"10.1016\/j.camwa.2018.08.042","article-title":"A Crank\u2014Nicolson ADI Galerkin\u2014Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation","volume":"76","author":"Zhang","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_99","doi-asserted-by":"crossref","unstructured":"Zaky, M., Doha, E., and Tenreiro Machado, J. (2018). A spectral numerical method for solving distributed-order fractional initial value problems. J. Comput. Nonlinear Dyn., 13.","DOI":"10.1115\/1.4041030"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1007\/s11071-017-4038-4","article-title":"A Legendre collocation method for distributed-order fractional optimal control problems","volume":"91","author":"Zaky","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1515\/ijnsns-2018-0111","article-title":"A collocation method based on Jacobi and fractional order Jacobi basis functions for multi-dimensional distributed-order diffusion equations","volume":"19","author":"Abdelkawy","year":"2018","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s40314-019-0845-1","article-title":"Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction\u2013diffusion equations","volume":"38","author":"Abdelkawy","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.camwa.2019.07.008","article-title":"Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations","volume":"79","author":"Zaky","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2019.05.023","article-title":"An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets","volume":"145","author":"Rahimkhani","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-020-1070-7","article-title":"Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order","volume":"39","author":"Abdelkawy","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s11075-016-0201-0","article-title":"An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate","volume":"75","author":"Abbaszadeh","year":"2017","journal-title":"Numer. Algorithms"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1186\/s13662-018-1817-2","article-title":"Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term","volume":"2018","author":"Hu","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.jcp.2016.03.044","article-title":"Finite difference\/spectral approximations for the distributed order time fractional reaction\u2013diffusion equation on an unbounded domain","volume":"315","author":"Chen","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.camwa.2020.04.019","article-title":"A novel finite element method for the distributed-order time fractional Cable equation in two dimensions","volume":"80","author":"Gao","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.jcp.2015.05.047","article-title":"Some high-order difference schemes for the distributed-order differential equations","volume":"298","author":"Gao","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1007\/s10915-015-0064-x","article-title":"Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations","volume":"66","author":"Gao","year":"2016","journal-title":"J. Sci. Comput."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1007\/s11075-018-0476-4","article-title":"Two temporal second-order H1-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations","volume":"79","author":"Li","year":"2018","journal-title":"Numer. Algorithms"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.jcp.2013.11.013","article-title":"Numerical solution of distributed order fractional differential equations","volume":"259","author":"Katsikadelis","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.enganabound.2018.08.007","article-title":"An RBF based meshless method for the distributed order time fractional advection\u2013diffusion equation","volume":"96","author":"Liu","year":"2018","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1007\/s10915-017-0360-8","article-title":"Finite difference\/finite element methods for distributed-order time fractional diffusion equations","volume":"72","author":"Bu","year":"2017","journal-title":"J. Sci. Comput."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1016\/j.camwa.2015.02.023","article-title":"Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations","volume":"69","author":"Gao","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1007\/s11075-016-0167-y","article-title":"Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations","volume":"74","author":"Gao","year":"2017","journal-title":"Numer. Algorithms"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1080\/00207160.2019.1608968","article-title":"Galerkin\u2014Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation","volume":"97","author":"Fei","year":"2020","journal-title":"Int. J. Comput. Math."},{"key":"ref_119","first-page":"848","article-title":"Numerical Solution of Caputo-Fabrizio Time Fractional Distributed Order Reaction-diffusion Equation via Quasi Wavelet based Numerical Method","volume":"6","author":"Kumar","year":"2020","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"131","DOI":"10.4208\/eajam.020615.030216a","article-title":"Lubich second-order methods for distributed-order time-fractional differential equations with smooth solutions","volume":"6","author":"Du","year":"2016","journal-title":"East Asian J. Appl. Math."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.cam.2016.02.039","article-title":"On a class of non-linear delay distributed order fractional diffusion equations","volume":"318","author":"Pimenov","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1109\/JAS.2017.7510646","article-title":"Numerical solution of the distributed-order fractional Bagley-Torvik equation","volume":"6","author":"Aminikhah","year":"2017","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.aml.2017.10.005","article-title":"A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain","volume":"77","author":"Fan","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_124","doi-asserted-by":"crossref","unstructured":"Ford, N.J., Morgado, M.L., and Rebelo, M. (2014, January 23\u201325). A numerical method for the distributed order time-fractional diffusion equation. Proceedings of the ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967389"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.aml.2018.06.005","article-title":"Finite difference\/spectral-Galerkin method for a two-dimensional distributed-order time\u2013space fractional reaction\u2013diffusion equation","volume":"85","author":"Guo","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s11075-015-0051-1","article-title":"An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation","volume":"72","author":"Hu","year":"2016","journal-title":"Numer. Algorithms"},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1016\/j.camwa.2017.05.017","article-title":"A novel finite volume method for the Riesz space distributed-order diffusion equation","volume":"74","author":"Li","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1016\/j.apm.2017.01.065","article-title":"A novel finite volume method for the Riesz space distributed-order advection\u2013diffusion equation","volume":"46","author":"Li","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.cam.2014.07.029","article-title":"Numerical approximation of distributed order reaction\u2013diffusion equations","volume":"275","author":"Morgado","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.apnum.2018.09.019","article-title":"Numerical approach for a class of distributed order time fractional partial differential equations","volume":"136","author":"Moghaddam","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1093\/imamat\/hxu015","article-title":"Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains","volume":"80","author":"Ye","year":"2015","journal-title":"IMA J. Appl. Math."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1016\/j.jcp.2015.06.025","article-title":"Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains","volume":"298","author":"Ye","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_133","doi-asserted-by":"crossref","unstructured":"Wang, X., Liu, F., and Chen, X. (2015). Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations. Adv. Math. Phys., 2015.","DOI":"10.1155\/2015\/590435"},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1016\/j.camwa.2020.06.017","article-title":"Finite difference\/spectral methods for the two-dimensional distributed-order time-fractional cable equation","volume":"80","author":"Zheng","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1007\/s41980-018-0191-x","article-title":"Fractional backward differential formulas for the distributed-order differential equation with time delay","volume":"45","author":"Heris","year":"2019","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s40324-019-00192-z","article-title":"Analysis and numerical methods for the Riesz space distributed-order advection-diffusion equation with time delay","volume":"76","author":"Javidi","year":"2019","journal-title":"SEMA J."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11071-018-4063-y","article-title":"Local discontinuous Galerkin method for distributed-order time and space-fractional convection\u2013diffusion and Schr\u00f6dinger-type equations","volume":"92","author":"Aboelenen","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1007\/s11075-016-0223-7","article-title":"Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations","volume":"75","author":"Liao","year":"2017","journal-title":"Numer. Algorithms"},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s11075-018-0606-z","article-title":"Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation","volume":"82","author":"Li","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"112589","DOI":"10.1016\/j.cam.2019.112589","article-title":"A Galerkin finite element method for the modified distributed-order anomalous sub-diffusion equation","volume":"368","author":"Li","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.apnum.2018.04.013","article-title":"A block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation","volume":"131","author":"Li","year":"2018","journal-title":"Appl. Numer. Math."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1186\/s13662-018-1655-2","article-title":"A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients","volume":"2018","author":"Jian","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.aml.2019.04.030","article-title":"A numerical method for distributed order time fractional diffusion equation with weakly singular solutions","volume":"96","author":"Ren","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_144","doi-asserted-by":"crossref","unstructured":"Li, C., and Zeng, F. (2015). Numerical Methods for Fractional Calculus, CRC Press.","DOI":"10.1201\/b18503"},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1016\/j.cma.2004.06.006","article-title":"Algorithms for the fractional calculus: A selection of numerical methods","volume":"194","author":"Diethelm","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"1230014","DOI":"10.1142\/S0218127412300145","article-title":"Finite difference methods for fractional differential equations","volume":"22","author":"Li","year":"2012","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1007\/s11071-015-2087-0","article-title":"A review of operational matrices and spectral techniques for fractional calculus","volume":"81","author":"Bhrawy","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_148","doi-asserted-by":"crossref","unstructured":"Guo, S., Mei, L., Zhang, Z., Li, C., Li, M., and Wang, Y. (2020). A linearized finite difference\/spectral-Galerkin scheme for three-dimensional distributed-order time\u2013space fractional nonlinear reaction\u2013diffusion-wave equation: Numerical simulations of Gordon-type solitons. Comput. Phys. Commun., 107144.","DOI":"10.1016\/j.cpc.2020.107144"},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"1813","DOI":"10.1007\/s00366-019-00797-y","article-title":"High-order continuous Galerkin methods for multi-dimensional advection\u2013reaction\u2013diffusion problems","volume":"36","author":"Hafez","year":"2019","journal-title":"Eng. Comput."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s40314-020-1102-3","article-title":"Implicit Runge\u2013Kutta and spectral Galerkin methods for Riesz space fractional\/distributed-order diffusion equation","volume":"39","author":"Zhao","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_151","unstructured":"Samiee, M., Kharazmi, E., Zayernouri, M., and Meerschaert, M.M. (2018). Petrov-Galerkin method for fully distributed-order fractional partial differential equations. arXiv."},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1007\/s11071-017-3525-y","article-title":"Numerical simulation of multi-dimensional distributed-order generalized Schr\u00f6dinger equations","volume":"89","author":"Bhrawy","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1007\/s40314-019-0922-5","article-title":"On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems","volume":"38","author":"Zaky","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.amc.2019.05.030","article-title":"A novel Legendre operational matrix for distributed order fractional differential equations","volume":"361","author":"Pourbabaee","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"2364","DOI":"10.1016\/j.camwa.2011.07.024","article-title":"A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order","volume":"62","author":"Doha","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"5662","DOI":"10.1016\/j.apm.2011.05.011","article-title":"Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations","volume":"35","author":"Doha","year":"2011","journal-title":"Appl. Math. Model."},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1016\/j.jcp.2014.10.060","article-title":"A method based on the Jacobi tau approximation for solving multi-term time\u2013space fractional partial differential equations","volume":"281","author":"Bhrawy","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_158","doi-asserted-by":"crossref","first-page":"3525","DOI":"10.1007\/s40314-017-0530-1","article-title":"A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations","volume":"37","author":"Zaky","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_159","doi-asserted-by":"crossref","first-page":"8903","DOI":"10.1016\/j.apm.2013.04.019","article-title":"Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations","volume":"37","author":"Atabakzadeh","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_160","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.amc.2017.11.047","article-title":"Modified methods for solving two classes of distributed order linear fractional differential equations","volume":"323","author":"Semary","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_161","first-page":"193","article-title":"Simulating the solution of the distributed order fractional differential equations by block-pulse wavelets","volume":"79","author":"Mashoof","year":"2017","journal-title":"UPB Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_162","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1080\/00207160.2017.1421949","article-title":"Fractional pseudo-spectral methods for distributed-order fractional PDEs","volume":"95","author":"Kharazmi","year":"2018","journal-title":"Int. J. Comput. Math."},{"key":"ref_163","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.aml.2015.10.009","article-title":"A numerical method for solving distributed order diffusion equations","volume":"53","author":"Li","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_164","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.physleta.2007.06.016","article-title":"Numerical studies for a multi-order fractional differential equation","volume":"371","author":"Sweilam","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_165","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1016\/j.camwa.2009.06.020","article-title":"Homotopy analysis method for solving multi-term linear and nonlinear diffusion\u2013wave equations of fractional order","volume":"59","author":"Jafari","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_166","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.1016\/j.camwa.2011.03.066","article-title":"An algorithm for solving multi-term diffusion-wave equations of fractional order","volume":"62","author":"Jafari","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_167","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1016\/j.asej.2016.03.007","article-title":"Approximate analytical solutions of distributed order fractional Riccati differential equation","volume":"9","author":"Aminikhah","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_168","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1007\/s11071-020-05488-8","article-title":"Application of variable-and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators","volume":"100","author":"Patnaik","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_169","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.amc.2006.11.129","article-title":"Solving a multi-order fractional differential equation using Adomian decomposition","volume":"189","author":"Jafari","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_170","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.amc.2008.01.027","article-title":"Solving multi-term linear and non-linear diffusion\u2013wave equations of fractional order by Adomian decomposition method","volume":"202","author":"Bhalekar","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_171","first-page":"14","article-title":"One solution of multi-term fractional differential equations by Adomian decomposition method","volume":"3","author":"Sadeghinia","year":"2015","journal-title":"Int. J. Sci. Innov. Math. Res."},{"key":"ref_172","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/16M1089320","article-title":"Numerical analysis of nonlinear subdiffusion equations","volume":"56","author":"Jin","year":"2018","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_173","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.cma.2018.12.011","article-title":"Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview","volume":"346","author":"Jin","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_174","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1137\/16M1082329","article-title":"Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation","volume":"55","author":"Stynes","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_175","doi-asserted-by":"crossref","first-page":"C464","DOI":"10.21914\/anziamj.v55i0.7888","article-title":"A numerical investigation of the time distributed-order diffusion model","volume":"55","author":"Hu","year":"2013","journal-title":"ANZIAM J."},{"key":"ref_176","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.amc.2015.06.045","article-title":"Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation","volume":"268","author":"Alikhanov","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_177","doi-asserted-by":"crossref","first-page":"070002","DOI":"10.1063\/1.4965348","article-title":"Introducing graded meshes in the numerical approximation of distributed-order diffusion equations","volume":"Volume 1776","author":"Morgado","year":"2016","journal-title":"AIP Conference Proceedings"},{"key":"ref_178","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.aml.2018.08.024","article-title":"Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation","volume":"88","author":"Abbaszadeh","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_179","doi-asserted-by":"crossref","first-page":"4906","DOI":"10.1002\/mma.4938","article-title":"An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations","volume":"41","author":"Li","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_180","doi-asserted-by":"crossref","first-page":"1502","DOI":"10.1007\/s10915-018-0672-3","article-title":"WSGD-OSC scheme for two-dimensional distributed order fractional reaction\u2013diffusion equation","volume":"76","author":"Yang","year":"2018","journal-title":"J. Sci. Comput."},{"key":"ref_181","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.apnum.2018.03.005","article-title":"New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order","volume":"129","author":"Ran","year":"2018","journal-title":"Appl. Numer. Math."},{"key":"ref_182","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","article-title":"A fully discrete difference scheme for a diffusion-wave system","volume":"56","author":"Sun","year":"2006","journal-title":"Appl. Numer. Math."},{"key":"ref_183","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/s13662-020-2514-5","article-title":"Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo\u2013Fabrizio fractional derivative","volume":"2020","author":"Qiao","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_184","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.apnum.2020.07.020","article-title":"A POD-based reduced-order Crank-Nicolson\/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation","volume":"158","author":"Abbaszadeh","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_185","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","article-title":"The Galerkin finite element method for a multi-term time-fractional diffusion equation","volume":"281","author":"Jin","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_186","doi-asserted-by":"crossref","first-page":"1540001","DOI":"10.1142\/S1793962315400012","article-title":"Finite element multigrid method for multi-term time fractional advection diffusion equations","volume":"6","author":"Bu","year":"2015","journal-title":"Int. J. Model. Simul. Sci. Comput."},{"key":"ref_187","doi-asserted-by":"crossref","first-page":"8810","DOI":"10.1016\/j.apm.2016.05.039","article-title":"Analytical solution and nonconforming finite element approximation for the 2D multi-term fractional subdiffusion equation","volume":"40","author":"Zhao","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_188","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1515\/fca-2016-0005","article-title":"Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data","volume":"19","author":"Jin","year":"2016","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_189","doi-asserted-by":"crossref","unstructured":"Hou, Y., Wen, C., Li, H., Liu, Y., Fang, Z., and Yang, Y. (2020). Some Second-Order \u03c3 Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation. Mathematics, 8.","DOI":"10.3390\/math8020187"},{"key":"ref_190","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s12190-018-1182-z","article-title":"Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order","volume":"59","author":"Wei","year":"2019","journal-title":"J. Appl. Math. Comput."},{"key":"ref_191","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10915-018-0694-x","article-title":"The unstructured mesh finite element method for the two-dimensional multi-term time\u2013space fractional diffusion-wave equation on an irregular convex domain","volume":"77","author":"Fan","year":"2018","journal-title":"J. Sci. Comput."},{"key":"ref_192","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1016\/j.camwa.2019.01.007","article-title":"Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time\u2013space fractional Bloch\u2013Torrey equations on irregular convex domains","volume":"78","author":"Liu","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_193","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1016\/j.apm.2019.04.023","article-title":"An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain","volume":"73","author":"Shi","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_194","unstructured":"Yin, B., Liu, Y., Li, H., and Zhang, Z. (2017). Approximation methods for the distributed order calculus using the convolution quadrature. Discret. Contin. Dyn. Syst. B, 22."},{"key":"ref_195","first-page":"359","article-title":"Matrix approach to discrete fractional calculus","volume":"3","author":"Podlubny","year":"2000","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_196","doi-asserted-by":"crossref","first-page":"3137","DOI":"10.1016\/j.jcp.2009.01.014","article-title":"Matrix approach to discrete fractional calculus II: Partial fractional differential equations","volume":"228","author":"Podlubny","year":"2009","journal-title":"J. Comput. Phys."},{"key":"ref_197","doi-asserted-by":"crossref","first-page":"20120153","DOI":"10.1098\/rsta.2012.0153","article-title":"Matrix approach to discrete fractional calculus III: Non-equidistant grids, variable step length and distributed orders","volume":"371","author":"Podlubny","year":"2013","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_198","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/S0096-3003(03)00739-2","article-title":"Multi-order fractional differential equations and their numerical solution","volume":"154","author":"Diethelm","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_199","doi-asserted-by":"crossref","first-page":"9","DOI":"10.2478\/s13540-013-0002-2","article-title":"Numerical methods for solving the multi-term time-fractional wave-diffusion equation","volume":"16","author":"Liu","year":"2013","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_200","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/j.amc.2013.11.015","article-title":"Maximum principle and numerical method for the multi-term time\u2013space Riesz\u2014Caputo fractional differential equations","volume":"227","author":"Ye","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_201","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.apnum.2019.08.019","article-title":"An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions","volume":"147","author":"Kazmi","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_202","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1007\/s10915-019-00979-2","article-title":"An efficient finite volume method for nonlinear distributed-order space-fractional diffusion equations in three space dimensions","volume":"80","author":"Zheng","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_203","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1016\/j.camwa.2017.09.003","article-title":"A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains","volume":"75","author":"Jia","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_204","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1007\/s11227-014-1123-z","article-title":"An efficient parallel solution for Caputo fractional reaction\u2013diffusion equation","volume":"68","author":"Gong","year":"2014","journal-title":"J. Supercomput."},{"key":"ref_205","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1515\/jnma-2014-0016","article-title":"A parallel Crank\u2013Nicolson finite difference method for time-fractional parabolic equation","volume":"22","author":"Sweilam","year":"2014","journal-title":"J. Numer. Math."},{"key":"ref_206","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.jcp.2018.08.034","article-title":"Parallel algorithms for nonlinear time\u2013space fractional parabolic PDEs","volume":"375","author":"Biala","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_207","doi-asserted-by":"crossref","unstructured":"Liu, J., Gong, C., Bao, W., Tang, G., and Jiang, Y. (2014). Solving the Caputo fractional reaction-diffusion equation on GPU. Discret. Dyn. Nat. Soc., 2014.","DOI":"10.1155\/2014\/820162"},{"key":"ref_208","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.cam.2019.05.019","article-title":"A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time\u2013space fractional diffusion equation","volume":"362","author":"Zhao","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_209","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Gu, X., Li, M., and Jian, H. (2020). Preconditioners for all-at-once system from the fractional mobile\/immobile advection\u2013diffusion model. J. Appl. Math. Comput., 1\u201323.","DOI":"10.1007\/s12190-020-01410-y"},{"key":"ref_210","doi-asserted-by":"crossref","unstructured":"Li, Y., and Chen, Y. (2011, January 28\u201331). Theory and implementation of distributed-order element networks. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA.","DOI":"10.1115\/DETC2011-48063"},{"key":"ref_211","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/S0378-4371(02)01377-8","article-title":"Multifractality of cerebral blood flow","volume":"318","author":"West","year":"2003","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_212","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1103\/PhysRevLett.58.1100","article-title":"L\u00e9vy dynamics of enhanced diffusion: Application to turbulence","volume":"58","author":"Shlesinger","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_213","doi-asserted-by":"crossref","first-page":"105992","DOI":"10.1016\/j.ijmecsci.2020.105992","article-title":"Towards a unified approach to nonlocal elasticity via fractional-order mechanics","volume":"189","author":"Patnaik","year":"2020","journal-title":"Int. J. Mech. Sci."},{"key":"ref_214","doi-asserted-by":"crossref","first-page":"051112","DOI":"10.1103\/PhysRevE.74.051112","article-title":"Fractional diffusion interpretation of simulated single-file systems in microporous materials","volume":"74","author":"Demontis","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_215","doi-asserted-by":"crossref","first-page":"031135","DOI":"10.1103\/PhysRevE.78.031135","article-title":"L\u00e9vy flights in nonhomogeneous media: Distributed-order fractional equation approach","volume":"78","author":"Srokowski","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_216","doi-asserted-by":"crossref","first-page":"2313","DOI":"10.1016\/j.amc.2012.07.053","article-title":"Fractional Schr\u00f6dinger equation with noninteger dimensions","volume":"219","author":"Martins","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_217","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physleta.2013.10.038","article-title":"Langevin equation for a free particle driven by power law type of noises","volume":"378","author":"Sandev","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_218","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1063\/1.1745400","article-title":"A method of analyzing experimental results obtained from elasto-viscous bodies","volume":"7","author":"Gemant","year":"1936","journal-title":"Physics"},{"key":"ref_219","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1080\/14786443808562036","article-title":"On fractional differentials","volume":"25","author":"Gemant","year":"1938","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_220","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1016\/S0165-1684(03)00182-8","article-title":"Fractional-order system identification based on continuous order-distributions","volume":"83","author":"Hartley","year":"2003","journal-title":"Signal Process."},{"key":"ref_221","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/BF01171449","article-title":"A generalized model for the uniaxial isothermal deformation of a viscoelastic body","volume":"159","author":"Atanackovic","year":"2002","journal-title":"Acta Mech."},{"key":"ref_222","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.crme.2003.08.003","article-title":"On a distributed derivative model of a viscoelastic body","volume":"331","author":"Atanackovic","year":"2003","journal-title":"Comptes Rendus Mec."},{"key":"ref_223","doi-asserted-by":"crossref","unstructured":"Atanackovi\u0107, T.M., Konjik, S., Oparnica, L., and Zorica, D. (2011). Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. Abstr. Appl. Anal., 2011.","DOI":"10.1155\/2011\/975694"},{"key":"ref_224","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF02820620","article-title":"Linear models of dissipation in anelastic solids","volume":"1","author":"Caputo","year":"1971","journal-title":"La Rivista Del Nuovo Cimento (1971\u20131977)"},{"key":"ref_225","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s00161-010-0177-2","article-title":"Distributed-order fractional wave equation on a finite domain: Creep and forced oscillations of a rod","volume":"23","author":"Atanackovic","year":"2011","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_226","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.ijengsci.2010.11.004","article-title":"Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod","volume":"49","author":"Atanackovic","year":"2011","journal-title":"Int. J. Eng. Sci."},{"key":"ref_227","doi-asserted-by":"crossref","first-page":"6703","DOI":"10.1088\/0305-4470\/38\/30\/006","article-title":"On a fractional distributed-order oscillator","volume":"38","author":"Atanackovic","year":"2005","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_228","doi-asserted-by":"crossref","first-page":"1750040","DOI":"10.1142\/S1793962317500404","article-title":"Mechanical response and simulation for constitutive equations with distributed order derivatives","volume":"8","author":"Duan","year":"2017","journal-title":"Int. J. Model. Simul. Sci. Comput."},{"key":"ref_229","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1006\/jmaa.2001.7816","article-title":"Dynamics of a rod made of generalized Kelvin\u2013Voigt visco-elastic material","volume":"268","author":"Stankovic","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_230","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0020-7225(00)00025-2","article-title":"A new method for solving dynamic problems of fractional derivative viscoelasticity","volume":"39","author":"Rossikhin","year":"2001","journal-title":"Int. J. Eng. Sci."},{"key":"ref_231","doi-asserted-by":"crossref","first-page":"26","DOI":"10.2478\/s13540-013-0003-1","article-title":"On a fractional Zener elastic wave equation","volume":"16","author":"Holm","year":"2013","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_232","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s001610100056","article-title":"A modified Zener model of a viscoelastic body","volume":"14","author":"Atanackovic","year":"2002","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_233","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.jmaa.2009.10.043","article-title":"Waves in fractional Zener type viscoelastic media","volume":"365","author":"Konjik","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_234","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s00033-019-1097-z","article-title":"Distributed-order fractional constitutive stress\u2013strain relation in wave propagation modeling","volume":"70","author":"Konjik","year":"2019","journal-title":"Zeitschrift f\u00fcr angewandte Mathematik und Physik"},{"key":"ref_235","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1002\/1521-4001(200106)81:6<363::AID-ZAMM363>3.0.CO;2-9","article-title":"Analysis of dynamic behaviour of viscoelastic rods whose rheological models contain fractional derivatives of two different orders","volume":"81","author":"Rossikhin","year":"2001","journal-title":"ZAMM-J. Appl. Math. Mech."},{"key":"ref_236","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1006\/jsvi.1996.0406","article-title":"Analysis of four-parameter fractional derivative model of real solid materials","volume":"195","author":"Pritz","year":"1996","journal-title":"J. Sound Vib."},{"key":"ref_237","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1016\/S0022-460X(02)01530-4","article-title":"Five-parameter fractional derivative model for polymeric damping materials","volume":"265","author":"Pritz","year":"2003","journal-title":"J. Sound Vib."},{"key":"ref_238","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/j.dental.2015.05.009","article-title":"Viscoelastic properties of uncured resin composites: Dynamic oscillatory shear test and fractional derivative model","volume":"31","author":"Petrovic","year":"2015","journal-title":"Dent. Mater."},{"key":"ref_239","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.cnsns.2014.12.011","article-title":"Multi-objective optimization of distributed-order fractional damping","volume":"24","author":"Naranjani","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_240","doi-asserted-by":"crossref","unstructured":"Jokar, M., Patnaik, S., and Semperlotti, F. (2020). Variable-Order Approach to Nonlocal Elasticity: Theoretical Formulation and Order Identification via Deep Learning Techniques. arXiv.","DOI":"10.1007\/s00466-021-02093-3"},{"key":"ref_241","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1023\/A:1015506420053","article-title":"Dynamical stability of viscoelastic column with fractional derivative constitutive relation","volume":"22","author":"Li","year":"2001","journal-title":"Appl. Math. Mech."},{"key":"ref_242","first-page":"115","article-title":"Stability and creep of a fractional derivative order viscoelastic rod","volume":"25","year":"2000","journal-title":"Bulletin (Acad\u00e9mie serbe des sciences et des arts. Classe des sciences math\u00e9matiques et naturelles. Sciences math\u00e9matiques)"},{"key":"ref_243","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1002\/1521-4001(200206)82:6<377::AID-ZAMM377>3.0.CO;2-M","article-title":"Dynamics of a viscoelastic rod of fractional derivative type","volume":"82","author":"Atanackovic","year":"2002","journal-title":"ZAMM-J. Appl. Math. Mech."},{"key":"ref_244","first-page":"501","article-title":"On a model of a viscoelastic rod","volume":"4","author":"Stankovic","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_245","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1177\/1081286504036219","article-title":"On a viscoelastic rod with constitutive equation containing fractional derivatives of two different orders","volume":"9","author":"Stankovic","year":"2004","journal-title":"Math. Mech. Solids"},{"key":"ref_246","doi-asserted-by":"crossref","first-page":"D4016003","DOI":"10.1061\/(ASCE)EM.1943-7889.0001090","article-title":"Dynamic stability of axially loaded nonlocal rod on generalized Pasternak foundation","volume":"143","author":"Zorica","year":"2017","journal-title":"J. Eng. Mech."},{"key":"ref_247","unstructured":"Varghaei, P., Kharazmi, E., Suzuki, J.L., and Zayernouri, M. (2019). Vibration analysis of geometrically nonlinear and fractional viscoelastic cantilever beams. arXiv."},{"key":"ref_248","doi-asserted-by":"crossref","unstructured":"Duan, J., and Chen, L. (2019). Oscillatory shear flow between two parallel plates for viscoelastic constitutive model of distributed-order derivative. Int. J. Numer. Methods Heat Fluid Flow.","DOI":"10.1108\/HFF-05-2019-0424"},{"key":"ref_249","doi-asserted-by":"crossref","first-page":"R161","DOI":"10.1088\/0305-4470\/37\/31\/R01","article-title":"The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics","volume":"37","author":"Metzler","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_250","doi-asserted-by":"crossref","first-page":"5551","DOI":"10.1088\/1751-8113\/40\/21\/007","article-title":"Anomalous diffusion without scale invariance","volume":"40","author":"Hanyga","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_251","doi-asserted-by":"crossref","first-page":"52","DOI":"10.3389\/fphy.2017.00052","article-title":"The role of fractional time-derivative operators on anomalous diffusion","volume":"5","author":"Tateishi","year":"2017","journal-title":"Front. Phys."},{"key":"ref_252","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.jmaa.2010.12.056","article-title":"Distributed-order fractional diffusions on bounded domains","volume":"379","author":"Meerschaert","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_253","doi-asserted-by":"crossref","first-page":"042117","DOI":"10.1103\/PhysRevE.92.042117","article-title":"Distributed-order diffusion equations and multifractality: Models and solutions","volume":"92","author":"Sandev","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_254","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1103\/PhysRevLett.81.729","article-title":"Indication of a universal persistence law governing atmospheric variability","volume":"81","author":"Bunde","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_255","doi-asserted-by":"crossref","first-page":"2146","DOI":"10.1029\/2003GL018099","article-title":"Nonlinearity and multifractality of climate change in the past 420,000 years","volume":"30","author":"Ashkenazy","year":"2003","journal-title":"Geophys. Res. Lett."},{"key":"ref_256","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jhydrol.2012.07.033","article-title":"Distributed-order infiltration, absorption and water exchange in mobile and immobile zones of swelling soils","volume":"468","author":"Su","year":"2012","journal-title":"J. Hydrol."},{"key":"ref_257","doi-asserted-by":"crossref","first-page":"113118","DOI":"10.1016\/j.cma.2020.113118","article-title":"A variably distributed-order time-fractional diffusion equation: Analysis and approximation","volume":"367","author":"Yang","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_258","doi-asserted-by":"crossref","first-page":"016103","DOI":"10.1103\/PhysRevE.74.016103","article-title":"Wavelet versus detrended fluctuation analysis of multifractal structures","volume":"74","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_259","doi-asserted-by":"crossref","first-page":"3244","DOI":"10.1103\/PhysRevLett.80.3244","article-title":"Multiscale velocity correlations in turbulence","volume":"80","author":"Benzi","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_260","doi-asserted-by":"crossref","first-page":"1314","DOI":"10.1016\/j.chaos.2012.07.001","article-title":"Random-time processes governed by differential equations of fractional distributed order","volume":"45","author":"Beghin","year":"2012","journal-title":"Chaos Solitons Fractals"},{"key":"ref_261","doi-asserted-by":"crossref","first-page":"046129","DOI":"10.1103\/PhysRevE.66.046129","article-title":"Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations","volume":"66","author":"Chechkin","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_262","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.physa.2003.12.044","article-title":"Fractional diffusion equation for a power-law-truncated L\u00e9vy process","volume":"336","author":"Sokolov","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_263","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.chaos.2017.05.001","article-title":"Beyond monofractional kinetics","volume":"102","author":"Sandev","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_264","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1051\/mmnp\/201611302","article-title":"Comb model with slow and ultraslow diffusion","volume":"11","author":"Sandev","year":"2016","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_265","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1137\/1127028","article-title":"The limiting behavior of a one-dimensional random walk in a random medium","volume":"27","author":"Sinai","year":"1983","journal-title":"Theory Probab. Its Appl."},{"key":"ref_266","doi-asserted-by":"crossref","first-page":"R2390","DOI":"10.1103\/PhysRevE.56.R2390","article-title":"Dynamics of a polyampholyte hooked around an obstacle","volume":"56","author":"Schiessel","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_267","doi-asserted-by":"crossref","first-page":"114101","DOI":"10.1103\/PhysRevLett.87.114101","article-title":"Anomalous diffusion and dynamical localization in polygonal billiards","volume":"87","author":"Prosen","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_268","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1103\/PhysRevE.59.1465","article-title":"Anomalous diffusion in aperiodic environments","volume":"59","author":"Turban","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_269","doi-asserted-by":"crossref","first-page":"5998","DOI":"10.1103\/PhysRevLett.84.5998","article-title":"Strong anomaly in diffusion generated by iterated maps","volume":"84","author":"Klafter","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_270","first-page":"259","article-title":"Distributed order time fractional diffusion equation","volume":"6","author":"Chechkin","year":"2003","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_271","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1209\/epl\/i2003-00539-0","article-title":"Fractional Fokker-Planck equation for ultraslow kinetics","volume":"63","author":"Chechkin","year":"2003","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_272","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.amc.2006.08.126","article-title":"Some aspects of fractional diffusion equations of single and distributed order","volume":"187","author":"Mainardi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_273","doi-asserted-by":"crossref","unstructured":"Mainardi, F., Mura, A., Pagnini, G., and Gorenflo, R. (2007). Sub-diffusion equations of fractional order and their fundamental solutions. Mathematical Methods in Engineering, Springer.","DOI":"10.1007\/978-1-4020-5678-9_3"},{"key":"ref_274","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1177\/1077546307087452","article-title":"Time-fractional diffusion of distributed order","volume":"14","author":"Mainardi","year":"2008","journal-title":"J. Vib. Control"},{"key":"ref_275","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s12190-008-0084-x","article-title":"Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order","volume":"28","author":"Shen","year":"2008","journal-title":"J. Appl. Math. Comput."},{"key":"ref_276","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.physa.2010.10.012","article-title":"Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case","volume":"390","author":"Saxena","year":"2011","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_277","doi-asserted-by":"crossref","unstructured":"Gin\u00e9, E., Koltchinskii, V., Li, W., and Zinn, J. (2006). Random walk models associated with distributed fractional order differential equations. High Dimensional Probability, Institute of Mathematical.","DOI":"10.1214\/lnms\/1196284095"},{"key":"ref_278","unstructured":"Sokolov, I.M., Chechkin, A.V., and Klafter, J. (2004). Distributed-order fractional kinetics. arXiv."},{"key":"ref_279","doi-asserted-by":"crossref","first-page":"021111","DOI":"10.1103\/PhysRevE.78.021111","article-title":"Generalized fractional diffusion equations for accelerating subdiffusion and truncated L\u00e9vy flights","volume":"78","author":"Chechkin","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_280","doi-asserted-by":"crossref","unstructured":"Klafter, J., Lim, S.C., and Metzler, R. (2012). Natural and modified forms of distributed-order fractional diffusion equations. Fractional Dynamics: Recent Advances, World Scientific.","DOI":"10.1142\/9789814340595"},{"key":"ref_281","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.physa.2005.12.012","article-title":"Solution of a modified fractional diffusion equation","volume":"367","author":"Langlands","year":"2006","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_282","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1016\/j.spa.2006.01.006","article-title":"Stochastic model for ultraslow diffusion","volume":"116","author":"Meerschaert","year":"2006","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_283","doi-asserted-by":"crossref","first-page":"56","DOI":"10.2478\/s13540-011-0005-9","article-title":"Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations","volume":"14","author":"Hahn","year":"2011","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_284","doi-asserted-by":"crossref","first-page":"031136","DOI":"10.1103\/PhysRevE.83.031136","article-title":"Fractional Langevin equations of distributed order","volume":"83","author":"Eab","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_285","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1007\/s10959-010-0289-4","article-title":"SDEs driven by a time-changed L\u00e9vy process and their associated time-fractional order pseudo-differential equations","volume":"25","author":"Hahn","year":"2012","journal-title":"J. Theor. Probab."},{"key":"ref_286","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1016\/j.jmaa.2015.05.024","article-title":"L\u00e9vy mixing related to distributed order calculus, subordinators and slow diffusions","volume":"430","author":"Toaldo","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_287","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.physa.2018.12.005","article-title":"On the time-fractional Cattaneo equation of distributed order","volume":"518","author":"Awad","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_288","doi-asserted-by":"crossref","first-page":"1600","DOI":"10.1214\/EJP.v16-920","article-title":"The fractional Poisson process and the inverse stable subordinator","volume":"16","author":"Meerschaert","year":"2011","journal-title":"Electron. J. Probab."},{"key":"ref_289","doi-asserted-by":"crossref","first-page":"184005","DOI":"10.1088\/1751-8121\/aa651e","article-title":"Fractional diffusion equation with distributed-order material derivative. Stochastic foundations","volume":"50","author":"Magdziarz","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_290","doi-asserted-by":"crossref","first-page":"032110","DOI":"10.1103\/PhysRevE.87.032110","article-title":"Accelerating subdiffusions governed by multiple-order time-fractional diffusion equations: Stochastic representation by a subordinated Brownian motion and computer simulations","volume":"87","author":"Mydlarczyk","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_291","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1142\/S0218348X04002410","article-title":"Distributed order fractional sub-diffusion","volume":"12","author":"Naber","year":"2004","journal-title":"Fractals"},{"key":"ref_292","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.cam.2006.10.014","article-title":"The role of the Fox\u2013Wright functions in fractional sub-diffusion of distributed order","volume":"207","author":"Mainardi","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_293","doi-asserted-by":"crossref","first-page":"1114","DOI":"10.2478\/s13540-014-0217-x","article-title":"Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations","volume":"17","author":"Li","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_294","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1080\/00207160.2017.1366465","article-title":"Distributed-order wave equations with composite time fractional derivative","volume":"95","author":"Tomovski","year":"2018","journal-title":"Int. J. Comput. Math."},{"key":"ref_295","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jmaa.2007.08.024","article-title":"Distributed order calculus and equations of ultraslow diffusion","volume":"340","author":"Kochubei","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_296","doi-asserted-by":"crossref","first-page":"014012","DOI":"10.1088\/0031-8949\/2009\/T136\/014012","article-title":"Existence and calculation of the solution to the time distributed order diffusion equation","volume":"2009","author":"Atanackovic","year":"2009","journal-title":"Phys. Scr."},{"key":"ref_297","unstructured":"Li, Z., and Yamamoto, M. (2013). Initial-boundary value problems for linear diffusion equation with multiple time-fractional derivatives. arXiv."},{"key":"ref_298","doi-asserted-by":"crossref","first-page":"297","DOI":"10.2478\/s13540-013-0019-6","article-title":"Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density","volume":"16","author":"Gorenflo","year":"2013","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_299","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.camwa.2016.07.009","article-title":"Generalized distributed order diffusion equations with composite time fractional derivative","volume":"73","author":"Sandev","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_300","first-page":"95","article-title":"Initial-boundary value problem for distributed order time-fractional diffusion equations","volume":"115","author":"Li","year":"2019","journal-title":"Asymptot. Anal."},{"key":"ref_301","first-page":"123","article-title":"Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications","volume":"36","author":"Luchko","year":"2016","journal-title":"Analysis"},{"key":"ref_302","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2478\/s13540-011-0008-6","article-title":"Maximum principle and its application for the time-fractional diffusion equations","volume":"14","author":"Luchko","year":"2011","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_303","doi-asserted-by":"crossref","first-page":"035008","DOI":"10.1088\/1361-6420\/aa573e","article-title":"Fractional diffusion: Recovering the distributed fractional derivative from overposed data","volume":"33","author":"Rundell","year":"2017","journal-title":"Inverse Probl."},{"key":"ref_304","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1016\/j.camwa.2016.06.030","article-title":"Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem","volume":"73","author":"Li","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_305","doi-asserted-by":"crossref","unstructured":"Ruan, Z., and Wang, Z. (2020). A backward problem for distributed order diffusion equation: Uniqueness and numerical solution. Inverse Probl. Sci. Eng.","DOI":"10.1080\/17415977.2020.1795152"},{"key":"ref_306","doi-asserted-by":"crossref","first-page":"112564","DOI":"10.1016\/j.cam.2019.112564","article-title":"Uniqueness in the inversion of distributed orders in ultraslow diffusion equations","volume":"369","author":"Li","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_307","doi-asserted-by":"crossref","first-page":"055008","DOI":"10.1088\/1361-6420\/ab762c","article-title":"An inverse source problem for distributed order time-fractional diffusion equation","volume":"36","author":"Sun","year":"2020","journal-title":"Inverse Probl."},{"key":"ref_308","doi-asserted-by":"crossref","unstructured":"Sibatov, R.T. (2019). Anomalous grain boundary diffusion: Fractional calculus approach. Adv. Math. Phys., 2019.","DOI":"10.1155\/2019\/8017363"},{"key":"ref_309","doi-asserted-by":"crossref","first-page":"114704","DOI":"10.1063\/1.3637944","article-title":"Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell","volume":"135","author":"Santoro","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_310","doi-asserted-by":"crossref","first-page":"8773","DOI":"10.1021\/jp211097m","article-title":"Fractional diffusion equation and the electrical impedance: Experimental evidence in liquid-crystalline cells","volume":"116","author":"Ciuchi","year":"2012","journal-title":"J. Phys. Chem. C"},{"key":"ref_311","doi-asserted-by":"crossref","first-page":"2849","DOI":"10.1016\/S1452-3981(23)14355-0","article-title":"Anomalous diffusion and electrical response of ionic solutions","volume":"8","author":"Lenzi","year":"2013","journal-title":"Int. J. Electrochem. Sci."},{"key":"ref_312","first-page":"1","article-title":"A fractional diffusion random laser","volume":"9","author":"Chen","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_313","doi-asserted-by":"crossref","unstructured":"Kitsyuk, E.P., Sibatov, R.T., and Svetukhin, V.V. (2020). Memory Effect and Fractional Differential Dynamics in Planar Microsupercapacitors Based on Multiwalled Carbon Nanotube Arrays. Energies, 13.","DOI":"10.3390\/en13010213"},{"key":"ref_314","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0022-5193(81)90109-0","article-title":"A model for generating aspects of zebra and other mammalian coat patterns","volume":"93","author":"Bard","year":"1981","journal-title":"J. Theor. Biol."},{"key":"ref_315","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1098\/rstb.1981.0155","article-title":"On pattern formation mechanisms for Lepidopteran wing patterns and mammalian coat markings","volume":"295","author":"Murray","year":"1981","journal-title":"Philos. Trans. R. Soc. Lond. B Biol. Sci."},{"key":"ref_316","doi-asserted-by":"crossref","first-page":"083519","DOI":"10.1063\/1.4891922","article-title":"Distributed order reaction-diffusion systems associated with Caputo derivatives","volume":"55","author":"Saxena","year":"2014","journal-title":"J. Math. Phys."},{"key":"ref_317","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.physa.2016.03.020","article-title":"Fractional diffusion equations coupled by reaction terms","volume":"458","author":"Lenzi","year":"2016","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_318","doi-asserted-by":"crossref","first-page":"120","DOI":"10.3390\/axioms4020120","article-title":"Computational solutions of distributed order reaction-diffusion systems associated with Riemann-Liouville derivatives","volume":"4","author":"Saxena","year":"2015","journal-title":"Axioms"},{"key":"ref_319","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1007\/s10559-013-9522-3","article-title":"Mathematical modeling of the dynamics of anomalous migration fields within the framework of the model of distributed order","volume":"49","author":"Bulavatsky","year":"2013","journal-title":"Cybern. Syst. Anal."},{"key":"ref_320","doi-asserted-by":"crossref","unstructured":"Yin, M., Ma, R., Zhang, Y., Wei, S., Tick, G.R., Wang, J., Sun, Z., Sun, H., and Zheng, C. (2020). A distributed-order time fractional derivative model for simulating bimodal sub-diffusion in heterogeneous media. J. Hydrol., 125504.","DOI":"10.1016\/j.jhydrol.2020.125504"},{"key":"ref_321","doi-asserted-by":"crossref","first-page":"124515","DOI":"10.1016\/j.jhydrol.2019.124515","article-title":"Super-diffusion affected by hydrofacies mean length and source geometry in alluvial settings","volume":"582","author":"Yin","year":"2020","journal-title":"J. Hydrol."},{"key":"ref_322","doi-asserted-by":"crossref","first-page":"1262","DOI":"10.1016\/j.jhydrol.2015.09.033","article-title":"The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests","volume":"529","author":"Su","year":"2015","journal-title":"J. Hydrol."},{"key":"ref_323","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.cnsns.2018.10.010","article-title":"Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media","volume":"70","author":"Liang","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_324","unstructured":"Caputo, M. (2003). Diffusion with space memory modelled with distributed order space fractional differential equations. Ann. Geophys."},{"key":"ref_325","doi-asserted-by":"crossref","first-page":"5319","DOI":"10.1088\/1751-8113\/40\/20\/006","article-title":"A diffusion wave equation with two fractional derivatives of different order","volume":"40","author":"Atanackovic","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_326","first-page":"1869","article-title":"Time distributed-order diffusion-wave equation. I. Volterra-type equation","volume":"465","author":"Atanackovic","year":"2009","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_327","doi-asserted-by":"crossref","first-page":"015201","DOI":"10.1088\/1751-8121\/aaefa3","article-title":"Generalized diffusion-wave equation with memory kernel","volume":"52","author":"Sandev","year":"2018","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_328","first-page":"1893","article-title":"Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations","volume":"465","author":"Atanackovic","year":"2009","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_329","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1177\/1077546310368697","article-title":"Wave simulation in dissipative media described by distributed-order fractional time derivatives","volume":"17","author":"Caputo","year":"2011","journal-title":"J. Vib. Control"},{"key":"ref_330","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1140\/epjp\/s13360-019-00006-1","article-title":"Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control","volume":"135","author":"Mahmoud","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_331","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/j.isatra.2012.12.004","article-title":"Design, implementation and application of distributed order PI control","volume":"52","author":"Zhou","year":"2013","journal-title":"ISA Trans."},{"key":"ref_332","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1515\/fca-2019-0085","article-title":"Robust stability analysis of LTI systems with fractional degree generalized frequency variables","volume":"22","author":"Wang","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_333","doi-asserted-by":"crossref","first-page":"200","DOI":"10.3182\/20060719-3-PT-4902.00036","article-title":"Fractional-order system identification using complex order-distributions","volume":"39","author":"Adams","year":"2006","journal-title":"IFAC Proc. Vol."},{"key":"ref_334","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.cnsns.2017.04.026","article-title":"On the formulation and numerical simulation of distributed-order fractional optimal control problems","volume":"52","author":"Zaky","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_335","doi-asserted-by":"crossref","unstructured":"Li, Y., Sheng, H., and Chen, Y. (2010, January 15\u201317). On distributed order low-pass filter. Proceedings of the 2010 IEEE\/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, China.","DOI":"10.1109\/MESA.2010.5552095"},{"key":"ref_336","doi-asserted-by":"crossref","unstructured":"Jakovljevi\u0107, B.B., Rapai\u0107, M.R., Jelici\u0107, Z.D., and Sekara, T.B. (2014, January 23\u201325). Optimization of distributed order fractional PID controller under constraints on robustness and sensitivity to measurement noise. Proceedings of the ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy.","DOI":"10.1109\/ICFDA.2014.6967406"},{"key":"ref_337","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.aeue.2017.05.036","article-title":"On the distributed order PID controller","volume":"79","year":"2017","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_338","doi-asserted-by":"crossref","unstructured":"Jakovljevi\u0107, B., Lino, P., and Maione, G. (2019, January 25\u201328). Fractional and Distributed Order PID Controllers for PMSM Drives. Proceedings of the 2019 18th European Control Conference (ECC), Naples, Italy.","DOI":"10.23919\/ECC.2019.8796163"},{"key":"ref_339","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1016\/j.sigpro.2010.10.005","article-title":"On distributed order integrator\/differentiator","volume":"91","author":"Li","year":"2011","journal-title":"Signal Process."},{"key":"ref_340","doi-asserted-by":"crossref","unstructured":"Li, Y., and Chen, Y.Q. (2012, January 8\u201310). Theory and implementation of weighted distributed order integrator. Proceedings of the 2012 IEEE\/ASME 8th IEEE\/ASME International Conference on Mechatronic and Embedded Systems and Applications, Suzhou, China.","DOI":"10.1109\/MESA.2012.6275548"},{"key":"ref_341","doi-asserted-by":"crossref","unstructured":"Najafi, H.S., Sheikhani, A.R., and Ansari, A. (2011). Stability analysis of distributed order fractional differential equations. Abstr. Appl. Anal., 2011.","DOI":"10.1155\/2011\/175323"},{"key":"ref_342","doi-asserted-by":"crossref","unstructured":"Jiao, Z., Chen, Y., and Podlubny, I. (2012). Distributed Order Dynamic Systems, Modeling, Analysis and Simulation, Springer.","DOI":"10.1007\/978-1-4471-2852-6"},{"key":"ref_343","doi-asserted-by":"crossref","first-page":"640","DOI":"10.1002\/asjc.578","article-title":"Stability analysis of linear time-invariant distributed-order systems","volume":"15","author":"Jiao","year":"2013","journal-title":"Asian J. Control"},{"key":"ref_344","doi-asserted-by":"crossref","unstructured":"Rivero, M., Rogosin, S.V., Tenreiro Machado, J.A., and Trujillo, J.J. (2013). Stability of fractional order systems. Math. Probl. Eng., 2013.","DOI":"10.1155\/2013\/356215"},{"key":"ref_345","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1002\/asjc.1780","article-title":"Algebraic conditions for stability analysis of linear time-invariant distributed order dynamic systems: A Lagrange inversion theorem approach","volume":"21","author":"Taghavian","year":"2019","journal-title":"Asian J. Control"},{"key":"ref_346","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1177\/1077546313481049","article-title":"Fractional\/distributed-order systems and irrational transfer functions with monotonic step responses","volume":"20","author":"Tavazoei","year":"2014","journal-title":"J. Vib. Control"},{"key":"ref_347","doi-asserted-by":"crossref","first-page":"121010","DOI":"10.1115\/1.4037268","article-title":"Robust Stability Analysis of Distributed-Order Linear Time-Invariant Systems With Uncertain Order Weight Functions and Uncertain Dynamic Matrices","volume":"139","author":"Taghavian","year":"2017","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_348","doi-asserted-by":"crossref","first-page":"1733","DOI":"10.1080\/00207721.2020.1773959","article-title":"Properties of the stability boundary in linear distributed-order systems","volume":"51","author":"Majma","year":"2020","journal-title":"Int. J. Syst. Sci."},{"key":"ref_349","doi-asserted-by":"crossref","first-page":"2881","DOI":"10.1016\/j.jfranklin.2013.03.005","article-title":"Stabilization and passification of distributed-order fractional linear systems using methods of preservation","volume":"350","year":"2013","journal-title":"J. Frankl. Inst."},{"key":"ref_350","unstructured":"Li, Y., and Chen, Y. (2014, January 23\u201325). Lyapunov stability of fractional-order nonlinear systems: A distributed-order approach. Proceedings of the ICFDA\u201914 International Conference on Fractional Differentiation and Its Applications 2014, Catania, Italy."},{"key":"ref_351","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1515\/fca-2017-0048","article-title":"Stability analysis of linear distributed order fractional systems with distributed delays","volume":"20","author":"Boyadzhiev","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_352","doi-asserted-by":"crossref","unstructured":"He, B., Chen, Y., and Kou, C. (2017, January 6\u20139). On the Controllability of Distributed-Order Fractional Systems With Distributed Delays. Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Cleveland, OH, USA.","DOI":"10.1115\/DETC2017-67685"},{"key":"ref_353","doi-asserted-by":"crossref","unstructured":"Aminikhah, H., Refahi Sheikhani, A., and Rezazadeh, H. (2013). Stability analysis of distributed order fractional Chen system. Sci. World J., 2013.","DOI":"10.1155\/2013\/645080"},{"key":"ref_354","first-page":"207","article-title":"Stability analysis of linear distributed order system with multiple time delays","volume":"77","author":"Aminikhah","year":"2015","journal-title":"UPB Sci. Bull."},{"key":"ref_355","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.cnsns.2017.01.020","article-title":"Asymptotic stability of distributed order nonlinear dynamical systems","volume":"48","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_356","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1080\/00207721.2017.1412535","article-title":"Stability analysis of distributed-order nonlinear dynamic systems","volume":"49","author":"Taghavian","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_357","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-Anaya, G., Quezada-T\u00e9llez, L., and Franco-P\u00e9rez, L. (2020). Stability analysis of distributed order of Hilfer nonlinear systems. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.7017"},{"key":"ref_358","doi-asserted-by":"crossref","unstructured":"Nava-Antonio, G., Fernandez-Anaya, G., Hernandez-Martinez, E., Jamous-Galante, J., Ferreira-Vazquez, E., and Flores-Godoy, J. (2017, January 8\u201310). Consensus of multi-agent systems with distributed fractional order dynamics. Proceedings of the 2017 International Workshop on Complex Systems and Networks (IWCSN), Doha, Qatar.","DOI":"10.1109\/IWCSN.2017.8276526"},{"key":"ref_359","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s11071-019-04979-7","article-title":"Generalized Wright stability for distributed fractional-order nonlinear dynamical systems and their synchronization","volume":"97","author":"Mahmoud","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_360","doi-asserted-by":"crossref","unstructured":"Al Themairi, A., and Farghaly, A. (2020). The Dynamics Behavior of Coupled Generalized van der Pol Oscillator with Distributed Order. Math. Probl. Eng., 2020.","DOI":"10.1155\/2020\/5670652"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/110\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:11:38Z","timestamp":1760159498000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,15]]},"references-count":360,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010110"],"URL":"https:\/\/doi.org\/10.3390\/e23010110","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,15]]}}}