{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:11:01Z","timestamp":1760235061239,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,25]],"date-time":"2021-07-25T00:00:00Z","timestamp":1627171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-61-KNOW-030"],"award-info":[{"award-number":["KMUTNB-61-KNOW-030"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach\u2019s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann\u2013Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.<\/jats:p>","DOI":"10.3390\/sym13081341","type":"journal-article","created":{"date-parts":[[2021,7,25]],"date-time":"2021-07-25T22:07:00Z","timestamp":1627250820000},"page":"1341","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms"],"prefix":"10.3390","volume":"13","author":[{"given":"Surang","family":"Sitho","sequence":"first","affiliation":[{"name":"Department of Social and Applied Sciences, College of Industrial Technology, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-1800","authenticated-orcid":false,"given":"Sina","family":"Etemad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9969-3999","authenticated-orcid":false,"given":"Brahim","family":"Tellab","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Mathematics, Kasdi Merbah University, B.P. 511, Ouargla 30000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-3539","authenticated-orcid":false,"given":"Jessada","family":"Tariboon","sequence":"additional","affiliation":[{"name":"Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3233\/FI-2017-1490","article-title":"Monotone convergence of extended iterative methods and fractional calculus with applications","volume":"151","author":"Anastassiou","year":"2017","journal-title":"Fundam. Inform."},{"key":"ref_2","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of the Fractional Differential Equations, Elsevier."},{"key":"ref_3","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.chaos.2017.04.025","article-title":"New aspects of the adaptive synchronization and hyperchaos suppression of a financial model","volume":"99","author":"Jajarmi","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1007\/s10957-017-1186-0","article-title":"A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel","volume":"175","author":"Baleanu","year":"2017","journal-title":"J. Optim. Theory Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/s13662-020-2527-0","article-title":"Analyzing transient response of the parallel RCL circuit by using the Caputo-Fabrizio fractional derivative","volume":"2020","author":"Alizadeh","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"021013","DOI":"10.1115\/1.4038444","article-title":"An efficient nonstandard finite difference scheme for a class of fractional chaotic systems","volume":"13","author":"Hajipour","year":"2017","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1140\/epjp\/i2018-11950-y","article-title":"Cancer treatment model with the Caputo-Fabrizio fractional derivative","volume":"133","author":"Dokuyucu","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_9","first-page":"504","article-title":"A fractional epidemiological model for computer viruses pertaining to a new fractional derivative","volume":"316","author":"Singh","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1016\/j.chaos.2016.08.003","article-title":"Resonant functional problems of fractional order","volume":"91","author":"Kosmatov","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","first-page":"104","article-title":"Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions","volume":"327","author":"Lu","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109867","DOI":"10.1016\/j.chaos.2020.109867","article-title":"On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative","volume":"135","author":"Abdo","year":"2020","journal-title":"Chaos Solitons Fractal"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/s13661-020-01361-0","article-title":"A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1186\/s13662-020-02937-x","article-title":"A mathematical analysis of a system of Caputo-Fabrizio fractional differential equations for the anthrax disease model in animals","volume":"2020","author":"Rezapour","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1186\/s13662-020-02544-w","article-title":"Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1186\/s13662-020-03044-7","article-title":"A new mathematical model for Zika virus transmission","volume":"2020","author":"Rezapour","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"013128","DOI":"10.1063\/1.5079644","article-title":"Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative","volume":"29","author":"Khan","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"110668","DOI":"10.1016\/j.chaos.2021.110668","article-title":"A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control","volume":"144","author":"Mohammadi","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1186\/s13661-020-01468-4","article-title":"On a new structure of the pantograph inclusion problem in the Caputo conformable setting","volume":"2020","author":"Thabet","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_20","unstructured":"Adiguzel, R.S., Aksoy, U., Karapinar, E., and Erhan, I.M. (2020). On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci."},{"key":"ref_21","first-page":"1","article-title":"Solution of fractional differential equations via coupled fixed point","volume":"286","author":"Afshari","year":"2015","journal-title":"Electron. J. Diff. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ahmad, B., Alsaedi, A., Salem, S., and Ntouyas, S.K. (2019). Fractional differential equation involving mixed nonlinearities with nonlocal multi-point and Riemann-Stieltjes integral-multi-strip conditions. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020034"},{"key":"ref_23","first-page":"125","article-title":"A study of nonlocal integro-multi-point boundary value problems of sequential fractional integro-differential inclusions","volume":"25","author":"Ahmad","year":"2018","journal-title":"Dyn. Contin. Disc. Impuls. Syst. Ser. A Math. Anal."},{"key":"ref_24","first-page":"167","article-title":"\u03c8-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory","volume":"3","author":"Baitiche","year":"2020","journal-title":"Results Nonlinear Anal."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Etemad, S., and Rezapour, S. (2020). On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J.","DOI":"10.1186\/s13662-020-02713-x"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1186\/s13662-020-02736-4","article-title":"On the existence of solutions for fractional boundary value problems on the ethane graph","volume":"2020","author":"Etemad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Boucenna, D., Boulfoul, A., Chidouh, A., Ben Makhlouf, A., and Tellab, B. (2021). Some results for initial value problem of nonlinear fractional equation in Sobolev space. J. Appl. Math. Comput.","DOI":"10.1007\/s12190-021-01500-5"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Boulfoul, A., Tellab, B., Abdellouahab, N., and Zennir, K. (2020). Existence and uniqueness results for initial value problem of nonlinear fractional integro-differential equation on an unbounded domain in a weighted Banach space. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.6957"},{"key":"ref_29","first-page":"49","article-title":"Hilfer-Hadamard fractional differential equations; Existence and attractivity","volume":"5","author":"Abbas","year":"2021","journal-title":"Adv. Theory Nonlinear Anal. Appl."},{"key":"ref_30","first-page":"391","article-title":"Numerical solution for a class of linear system of fractional differential equations by the Haar wavelet method and the convergence analysis","volume":"97","author":"Chen","year":"2014","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.apnum.2020.10.019","article-title":"A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method","volume":"160","author":"Jong","year":"2021","journal-title":"Appl. Num. Math."},{"key":"ref_32","first-page":"102","article-title":"CAS Picard method for fractional nonlinear differential equation","volume":"307","author":"Saeed","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11071-017-3870-x","article-title":"A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses","volume":"91","author":"Kumar","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s40096-019-0284-6","article-title":"Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method","volume":"13","author":"Veeresha","year":"2019","journal-title":"Math. Sci."},{"key":"ref_35","first-page":"329","article-title":"Numerical solution for the variable order linear cable equation with Bernstein polynomials","volume":"238","author":"Chen","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1186\/s13662-017-1091-8","article-title":"On solutions of fractional Riccati differential equations","volume":"2017","author":"Sakar","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_37","first-page":"562140","article-title":"Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions","volume":"13","author":"Yin","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_38","first-page":"7","article-title":"Application of variational iteration method to nonlinear differential equations of fractional order","volume":"7","author":"Odibat","year":"2016","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1515\/phys-2016-0028","article-title":"Numerical solutions of multi-order fractional differential equations by Boubaker polynomials","volume":"14","author":"Bolandtalat","year":"2016","journal-title":"Open Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.cnsns.2015.10.020","article-title":"On the convergence of a new reliable algorithm for solving multi-order fractional differential equations","volume":"34","author":"Hesameddini","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"061025","DOI":"10.1115\/1.4029785","article-title":"On a numerical approach to solve multi order fractional differential equations with boundary initial conditions","volume":"10","author":"Firoozjaee","year":"2015","journal-title":"J. Comput. Nonlinear Dynam."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s11071-017-3654-3","article-title":"Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations","volume":"90","author":"Dabiri","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1002\/mma.6864","article-title":"Existence of solutions to fractional differential equation with fractional integral type boundary conditions","volume":"44","author":"Ali","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1186\/s13662-016-0847-x","article-title":"Existence and uniqueness results for nonlocal integral boundary value problems for fractional differential equations","volume":"2016","author":"Liu","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1515\/fca-2018-0038","article-title":"Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions","volume":"21","author":"Padhi","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"496","DOI":"10.3906\/mat-2010-70","article-title":"On a coupled Caputo conformable system of pantograph problems","volume":"45","author":"Thabet","year":"2021","journal-title":"Turk. J. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.jmaa.2004.07.039","article-title":"Adomian decomposition: A tool for solving a system of fractional differential equations","volume":"301","author":"Jafari","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jmaa.2005.05.009","article-title":"An iterative method for solving nonlinear functional equations","volume":"316","author":"Jafari","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s13226-014-0102-7","article-title":"An efficient method for nonlinear fractional differential equations: Combination of the Adomian decomposition method and spectral method","volume":"45","author":"Babolian","year":"2014","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"382","DOI":"10.2478\/s13540-014-0176-2","article-title":"Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method","volume":"17","author":"Khodabakhshi","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_51","first-page":"315","article-title":"Numerical methods for sequential fractional differential equations for Caputo operator","volume":"35","author":"Loghmani","year":"2012","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Granas, A., and Dugundji, J. (2003). Elementary Fixed Point Theorems. Fixed Point Theory, Springer.","DOI":"10.1007\/978-0-387-21593-8"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1341\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:34:39Z","timestamp":1760164479000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1341"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,25]]},"references-count":52,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081341"],"URL":"https:\/\/doi.org\/10.3390\/sym13081341","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,25]]}}}