{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T14:20:41Z","timestamp":1769005241833,"version":"3.49.0"},"reference-count":50,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T00:00:00Z","timestamp":1676419200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["2764"],"award-info":[{"award-number":["2764"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems.<\/jats:p>","DOI":"10.3390\/sym15020519","type":"journal-article","created":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T03:09:21Z","timestamp":1676430561000},"page":"519","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Method for Solving Time-Fractional Initial Boundary Value Problems of Variable Order"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2744-6320","authenticated-orcid":false,"given":"Kinda","family":"Abuasbeh","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asia","family":"Kanwal","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3610-339X","authenticated-orcid":false,"given":"Ramsha","family":"Shafqat","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bilal","family":"Taufeeq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Lahore Pakistan, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3102-166X","authenticated-orcid":false,"given":"Muna A.","family":"Almulla","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6447-6361","authenticated-orcid":false,"given":"Muath","family":"Awadalla","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,15]]},"reference":[{"key":"ref_1","unstructured":"Lazarevi\u0107, M.P., Rapai\u0107, M.R., \u0160ekara, T.B., Mladenov, V., and Mastorakis, N. (2014). Advanced Topics on Applications of Fractional Calculus on Control Problems, System Stability and Modeling, WSEAS Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"549","DOI":"10.4310\/ATMP.2012.v16.n2.a5","article-title":"Geometry of fractional spaces","volume":"16","author":"Calcagni","year":"2012","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_3","unstructured":"Sar, E.Y., and Giresunlu, I.B. (2016). Fractional differential equations. Pramana J. Phys., 87."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Klafter, J., Lim, S.C., and Metzler, R. (2012). Fractional Dynamics: Recent Advances, World Scientific.","DOI":"10.1142\/8087"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2019). On history of mathematical economics: Application of fractional calculus. Mathematics, 7.","DOI":"10.3390\/math7060509"},{"key":"ref_6","first-page":"1021","article-title":"Applications of fractional calculus","volume":"4","author":"Dalir","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1122\/1.4819083","article-title":"Generalization of a theoretical basis for the application of fractional calculus to viscoelasticity","volume":"57","author":"Wharmby","year":"2013","journal-title":"J. Rheol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"168026","DOI":"10.1016\/j.ijleo.2021.168026","article-title":"A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus","volume":"247","author":"Yilmaz","year":"2021","journal-title":"Optik"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gonzalez-Lee, M., Vazquez-Leal, H., Morales-Mendoza, L.J., Nakano-Miyatake, M., Perez-Meana, H., and Laguna-Camacho, J.R. (2021). Statistical assessment of discrimination capabilities of a fractional calculus based image watermarking system for Gaussian watermarks. Entropy, 23.","DOI":"10.3390\/e23020255"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"552","DOI":"10.2478\/s13540-014-0185-1","article-title":"Some pioneers of the applications of fractional calculus","volume":"17","author":"Machado","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_12","unstructured":"Tarasov, V.E. (2019). Handbook of Fractional Calculus with Applications, De Gruyter."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10846-022-01597-1","article-title":"Applications of fractional operators in robotics: A review","volume":"104","year":"2022","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_14","first-page":"228","article-title":"Some applications of fractional calculus in technological development","volume":"10","author":"Mishra","year":"2019","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.cnsns.2017.04.001","article-title":"The role of fractional calculus in modeling biological phenomena: A review","volume":"51","author":"Ionescu","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"860","DOI":"10.1016\/j.amc.2011.03.062","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/0315-0860(77)90039-8","article-title":"The development of fractional calculus 1695\u20131900","volume":"4","author":"Ross","year":"1977","journal-title":"Hist. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific.","DOI":"10.1142\/8180"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1016\/j.amc.2007.09.020","article-title":"Finite difference methods and a Fourier analysis for the fractional reaction\u2013subdiffusion equation","volume":"198","author":"Chen","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_20","unstructured":"Birajdar, G.A., and Dhaigude, D.B. (2014, January 17\u201319). An implicit numerical method for semi-linear fractional diffusion equation. Proceedings of the International Conference on Mathematical Sciences, Chennai, India."},{"key":"ref_21","first-page":"1","article-title":"Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term","volume":"26","author":"Zhang","year":"2008","journal-title":"J. Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"C488","DOI":"10.21914\/anziamj.v46i0.973","article-title":"Analysis of a discrete non-Markovian random walk approximation for the time-fractional diffusion equation","volume":"46","author":"Liu","year":"2004","journal-title":"Anziam J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","article-title":"Finite difference\/spectral approximations for the time-fractional diffusion equation","volume":"225","author":"Lin","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF02832352","article-title":"Implicit difference approximation for the two-dimensional space-time-fractional diffusion equation","volume":"25","author":"Zhuang","year":"2007","journal-title":"J. Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1137\/060673114","article-title":"New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation","volume":"46","author":"Zhuang","year":"2008","journal-title":"Siam J. Numer. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1016\/j.camwa.2008.02.015","article-title":"Implicit finite difference approximation for time-fractional diffusion equations","volume":"56","author":"Murio","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_27","first-page":"1","article-title":"Crank-Nicolson finite difference method for solving time-fractional diffusion equation","volume":"2","author":"Sweilam","year":"2012","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_28","first-page":"1","article-title":"Stability of nonlinear fractional diffusion equation","volume":"36","author":"Birajdar","year":"2016","journal-title":"Lib. Math."},{"key":"ref_29","first-page":"42","article-title":"Adomain decomposition method for fractional Benjamin-Bona-Mahony-Burger\u2019s equations","volume":"8","author":"Dhaigude","year":"2012","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_30","first-page":"1","article-title":"Numerical solution of system of fractional partial differential equations by discrete Adomian decomposition method","volume":"3","author":"Dhaigude","year":"2012","journal-title":"J. Frac. Cal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"107","DOI":"10.4208\/aamm.12-m12105","article-title":"Numerical solution of fractional partial differential equations by discrete Adomian decomposition method","volume":"6","author":"Dhaigude","year":"2014","journal-title":"Adv. Appl. Math. Mech."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Mehmood, Y., Shafqat, R., Sarris, I.E., Bilal, M., Sajid, T., and Akhtar, T. (2022). Numerical Investigation of MWCNT and SWCNT Fluid Flow along with the Activation Energy Effects over Quartic Auto Catalytic Endothermic and Exothermic Chemical Reactions. Mathematics, 10.","DOI":"10.3390\/math10244636"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Boulares, H., Benchaabane, A., Pakkaranang, N., Shafqat, R., and Panyanak, B. (2022). Qualitative properties of positive solutions of a kind for fractional pantograph problems using technique fixed point theory. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6100593"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3559035","DOI":"10.1155\/2022\/3559035","article-title":"Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation","volume":"2022","author":"Abuasbeh","year":"2022","journal-title":"J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Abuasbeh, K., Shafqat, R., Alsinai, A., and Awadalla, M. (2023). Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay. Symmetry, 15.","DOI":"10.3390\/sym15020290"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Abuasbeh, K., Shafqat, R., Alsinai, A., and Awadalla, M. (2023). Analysis of the Mathematical Modelling of COVID-19 by Using Mild Solution with Delay Caputo Operator. Symmetry, 15.","DOI":"10.3390\/sym15020286"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Alnahdi, A.S., Shafqat, R., Niazi, A.U.K., and Jeelani, M.B. (2022). Pattern Formation Induced by Fuzzy Fractional-Order Model of COVID-19. Axioms, 11.","DOI":"10.3390\/axioms11070313"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1007\/s11071-016-2716-2","article-title":"Two analytical methods for time-fractional nonlinear coupled Boussinesq\u2013Burger\u2019s equations arise in propagation of shallow water waves","volume":"85","author":"Kumar","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.amc.2009.02.047","article-title":"Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation","volume":"212","author":"Lin","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1137\/080730597","article-title":"Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term","volume":"47","author":"Zhuang","year":"2009","journal-title":"Siam J. Numer. Anal."},{"key":"ref_41","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. Stat. Mech. Its Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s11075-012-9622-6","article-title":"Numerical approximation for a variable-order nonlinear reaction\u2013subdiffusion equation","volume":"63","author":"Chen","year":"2013","journal-title":"Numer. Algorithms"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1090\/S0025-5718-2011-02447-6","article-title":"Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation","volume":"81","author":"Chen","year":"2012","journal-title":"Math. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1250085","DOI":"10.1142\/S021812741250085X","article-title":"Finite difference schemes for variable-order time-fractional diffusion equation","volume":"22","author":"Sun","year":"2012","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s11071-008-9385-8","article-title":"Nonlinear dynamics and control of a variable-order oscillator with application to the van der Pol equation","volume":"56","author":"Diaz","year":"2009","journal-title":"Nonlinear Dyn."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1002\/andp.20055170602","article-title":"The variable viscoelasticity oscillator","volume":"14","author":"Soon","year":"2005","journal-title":"Ann. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"60","DOI":"10.22436\/jmcs.029.01.06","article-title":"Approximate solutions of linear time-fractional differential equations","volume":"29","author":"Oderinu","year":"2023","journal-title":"J. Math. Comput. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"119","DOI":"10.22436\/jmcs.022.02.03","article-title":"New group iterative schemes for solving the two-dimensional anomalous fractional sub-diffusion equation","volume":"22","author":"Alia","year":"2021","journal-title":"J. Math. Comput. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s40314-022-01934-y","article-title":"Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems","volume":"41","author":"Guo","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"85","DOI":"10.22436\/jmcs.022.01.08","article-title":"A numerical study on time-fractional Fisher equation using an extended cubic B-spline approximation","volume":"22","author":"Akram","year":"2021","journal-title":"J. Math. Comput. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/519\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:36:06Z","timestamp":1760121366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/519"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,15]]},"references-count":50,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020519"],"URL":"https:\/\/doi.org\/10.3390\/sym15020519","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,15]]}}}