{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T11:29:09Z","timestamp":1768735749692,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T00:00:00Z","timestamp":1668988800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006701","name":"Umm Al-Qura University","doi-asserted-by":"publisher","award":["22UQU4310396DSR35"],"award-info":[{"award-number":["22UQU4310396DSR35"]}],"id":[{"id":"10.13039\/501100006701","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann\u2013Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.<\/jats:p>","DOI":"10.3390\/sym14112462","type":"journal-article","created":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T03:09:30Z","timestamp":1669000170000},"page":"2462","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A New Numerical Approach for Variable-Order Time-Fractional Modified Subdiffusion Equation via Riemann\u2013Liouville Fractional Derivative"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9516-1485","authenticated-orcid":false,"given":"Dowlath","family":"Fathima","sequence":"first","affiliation":[{"name":"Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Jeddah 23442, Saudi Arabia"}]},{"given":"Muhammad","family":"Naeem","sequence":"additional","affiliation":[{"name":"Department of Mathematics of Applied Sciences, Umm-Al-Qura University, Makkah 21955, Saudi Arabia"}]},{"given":"Umair","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1787-5036","authenticated-orcid":false,"given":"Abdul Hamid","family":"Ganie","sequence":"additional","affiliation":[{"name":"Basic Sciences Department, College of Science and Theoretical Studies, Saudi Electronic University, Abha 61421, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5215-9617","authenticated-orcid":false,"given":"Farah Aini","family":"Abdullah","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang 11800, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.cnsns.2016.12.022","article-title":"A compact fnite difference scheme for variable order subdiffusion equation","volume":"48","author":"Cao","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4586","DOI":"10.1016\/j.physa.2009.07.024","article-title":"Variable-order fractional differential operators in anomalous diffusion modeling","volume":"388","author":"Sun","year":"2009","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.cma.2019.02.035","article-title":"A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation","volume":"350","author":"Shekari","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1137\/090771715","article-title":"Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation","volume":"32","author":"Chen","year":"2010","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1186\/s13662-018-1544-8","article-title":"Finite difference scheme for multi-term variable-order fractional diffusion equation","volume":"2018","author":"Xu","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","first-page":"13","article-title":"A simultaneous inversion problem for the variable-order time fractional differential equation with variable coefficient","volume":"2019","author":"Wang","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s11071-014-1854-7","article-title":"Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation","volume":"80","author":"Bhrawy","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jcp.2014.08.015","article-title":"Second-order approximations for variable order fractional derivatives: Algorithms and applications","volume":"293","author":"Zhao","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_10","first-page":"10861","article-title":"Numerical techniques for the variable order time fractional diffusion equation","volume":"218","author":"Shen","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"2347","DOI":"10.1016\/j.aej.2020.02.028","article-title":"An approximate approach for the generalized variable-order fractional pantograph equation","volume":"59","author":"Avazzadeh","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.jcp.2014.12.001","article-title":"Fractional spectral collocation methods for linear and nonlinear variable order FPDEs","volume":"293","author":"Zayernouri","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_14","unstructured":"Ali, U. (2019). Numerical Solutions for Two Dimensional Time-Fractional Differential Sub-Diffusion Equation. [Ph.D. Thesis, University Sains Malaysia]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"417942","DOI":"10.1155\/2012\/417942","article-title":"Numerical solutions of a variable-order fractional financial system","volume":"2012","author":"Ma","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_16","unstructured":"Katsikadelis, J.T. (2018). Numerical solution of variable order fractional differential equations. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"112","DOI":"10.11121\/ijocta.01.2017.00368","article-title":"On solutions of variable-order fractional differential equations","volume":"7","author":"Baleanu","year":"2017","journal-title":"Int. J. Optim. Control Theor. Appl. (IJOCTA)"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.aml.2016.08.018","article-title":"A numerical solution for variable order fractional functional differential equation","volume":"64","author":"Jia","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s40324-018-0173-1","article-title":"A numerical solution of variable order fractional functional differential equation based on the shifted Legendre polynomials","volume":"76","author":"Dehghan","year":"2019","journal-title":"SeMA J."},{"key":"ref_20","first-page":"62","article-title":"Numerical methods for solving a two-dimensional variable-order modified diffusion equation","volume":"225","author":"Chen","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ali, U., Sohail, M., and Abdullah, F.A. (2020). An Efficient Numerical Scheme for Variable-Order Fractional Sub-diffusion Equation. Symmetry, 12.","DOI":"10.3390\/sym12091437"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"8829017","DOI":"10.1155\/2020\/8829017","article-title":"New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations","volume":"2020","author":"Ahmad","year":"2020","journal-title":"Complexity"},{"key":"ref_23","first-page":"31","article-title":"On the numerical solutions of the variable order fractional heat equation","volume":"2","author":"Sweilam","year":"2011","journal-title":"Stud. Nonlinear Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112908","DOI":"10.1016\/j.cam.2020.112908","article-title":"Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method","volume":"377","author":"Babaei","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2647","DOI":"10.1007\/s10444-019-09690-0","article-title":"Analysis and numerical solution of a nonlinear variable-order fractional differential equation","volume":"45","author":"Wang","year":"2019","journal-title":"Adv. Comput. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"020058","DOI":"10.1063\/5.0111641","article-title":"October. A new computational procedure for the solution of the time-fractional advection problem","volume":"2451","author":"Kaur","year":"2022","journal-title":"AIP Conf. Proc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6196","DOI":"10.1016\/j.apm.2015.01.065","article-title":"Eigenvalue approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity","volume":"39","author":"Abbas","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Alzahrani, F., Hobiny, A., Abbas, I., and Marin, M. (2020). An eigenvalues approach for a two-dimensional porous medium based upon weak, normal and strong thermal conductivities. Symmetry, 12.","DOI":"10.3390\/sym12050848"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Ahmad, H., Thounthong, P., Chu, Y.M., and Cesarano, C. (2020). Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry, 12.","DOI":"10.3390\/sym12071195"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"060021","DOI":"10.1063\/1.5136453","article-title":"December. Modified implicit difference method for one-dimensional fractional wave equation","volume":"2184","author":"Ali","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2827","DOI":"10.1016\/j.aej.2020.06.029","article-title":"Numerical simulation of simulate an anomalous solute transport model via local meshless method","volume":"59","author":"Ahmad","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Mahmood, A., Md Basir, M.F., Ali, U., Mohd Kasihmuddin, M.S., and Mansor, M. (2019). Numerical Solutions of Heat Transfer for Magnetohydrodynamic Jeffery-Hamel Flow Using Spectral Homotopy Analysis Method. Processes, 7.","DOI":"10.3390\/pr7090626"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1140\/epjp\/s13360-020-00784-z","article-title":"Numerical study of integer-order hyperbolic telegraph model arising in physical and related sciences","volume":"135","author":"Ahmad","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"347","DOI":"10.2298\/TSCI200225210S","article-title":"Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method","volume":"25","author":"Srivastava","year":"2020","journal-title":"Therm. Sci."},{"key":"ref_35","first-page":"18","article-title":"Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation","volume":"2017","author":"Ali","year":"2017","journal-title":"J. Interpolat. Approx. Sci. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1186\/s13662-017-1192-4","article-title":"Modified implicit fractional difference scheme for 2D modified anomalous fractional sub-diffusion equation","volume":"2017","author":"Ali","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1016\/j.camwa.2006.02.001","article-title":"Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results","volume":"51","author":"Jumarie","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Ali, U., Sohail, M., Usman, M., Abdullah, F.A., Khan, I., and Nisar, K.S. (2020). Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation. Symmetry, 12.","DOI":"10.3390\/sym12050691"},{"key":"ref_39","first-page":"1","article-title":"Solving time-fractional differential diffusion equation by theta-method","volume":"2","author":"Aslefallah","year":"2014","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1090\/S0025-5718-1985-0790648-7","article-title":"A general equivalence theorem in the theory of discretization methods","volume":"45","author":"Palencia","year":"1985","journal-title":"Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2462\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:22:34Z","timestamp":1760145754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2462"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,21]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112462"],"URL":"https:\/\/doi.org\/10.3390\/sym14112462","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,21]]}}}