https://orcid.org/0009-0007-9726-5994
Abstract
In this article, we discuss the fractional temporal-spatial reaction-diffusion model with Neumann boundary conditions in one- and two-dimensional cases. The problem is solved by using a novel approach that depends on the approximation of a variable-order (VO) Caputo fractional derivative in the form of an operational matrix based on the shifted Vieta-Fibonacci (SVF) and collocation procedures. In this proposed scheme, first, the shifted Vieta-Fibonacci and operational matrix are used to approximate the dependent variable and Caputo derivatives of variable order, respectively, to construct the residual connected with the proposed problem. After that, the residual is collocated at some points of the domain, which produces a system of algebraic equations, and this system is solved by an appropriate numerical technique. The convergence and error analysis of the scheme are also analyzed. In this article, we also analyze the order of convergence for the solution of the considered problem. For validation purposes, the proposed scheme is applied to some particular cases of the proposed model, and the comparisons are made with the exact solution. It is found that the scheme is sufficiently accurate, and the accuracy enhances as the degree of approximating polynomials improves.
References
1.
Karpman
, V.
, 1996
, “Stabilization of Soliton Instabilities by Higher-Order Dispersion: Fourth-Order Nonlinear Schrödinger-Type Equations
,” Phys. Rev. E
, 53
(2
), pp. R1336
–R1339
.10.1103/PhysRevE.53.R13362.
Myers
, T. G.
, Charpin
, J. P.
, and Chapman
, S. J.
, 2002
, “The Flow and Solidification of a Thin Fluid Film on an Arbitrary Three-Dimensional Surface
,” Phys. Fluids
, 14
(8
), pp. 2788
–2803
.10.1063/1.14885993.
Myers
, T. G.
, and Charpin
, J. P.
, 2004
, “A Mathematical Model for Atmospheric Ice Accretion and Water Flow on a Cold Surface
,” Int. J. Heat Mass Transfer
, 47
(25
), pp. 5483
–5500
.10.1016/j.ijheatmasstransfer.2004.06.0374.
Halpern
, D.
, Jensen
, O.
, and Grotberg
, J.
, 1998
, “A Theoretical Study of Surfactant and Liquid Delivery Into the Lung
,” J. Appl. Physiol.
, 85
(1
), pp. 333
–352
.10.1152/jappl.1998.85.1.3335.
Toga
, A. W.
, 1998
, Brain Warping
, Elsevier
, Academic Press, New York.6.
Mémoli
, F.
, Sapiro
, G.
, and Thompson
, P.
, 2004
, “Implicit Brain Imaging
,” NeuroImage
, 23
, pp. S179
–S188
.10.1016/j.neuroimage.2004.07.0727.
Hornreich
, R.
, Luban
, M.
, and Shtrikman
, S.
, 1975
, “Critical Behavior at the Onset of k→-Space Instability on the Line
,” Phys. Rev. Lett.
, 35
(25
), pp. 1678
–1681
.10.1103/PhysRevLett.35.16788.
Aronson
, D. G.
, and Weinberger
, H. F.
, 1978
, “Multidimensional Nonlinear Diffusion Arising in Population Genetics
,” Adv. Math.
, 30
(1
), pp. 33
–76
.10.1016/0001-8708(78)90130-59.
Guozhen
, Z.
, 1982
, “Experiments on Director Waves in Nematic Liquid Crystals
,” Phys. Rev. Lett.
, 49
(18
), pp. 1332
–1335
.10.1103/PhysRevLett.49.133210.
Coullet
, P.
, Elphick
, C.
, and Repaux
, D.
, 1987
, “Nature of Spatial Chaos
,” Phys. Rev. Lett.
, 58
(5
), pp. 431
–434
.10.1103/PhysRevLett.58.43111.
Dee
, G.
, and van Saarloos
, W.
, 1988
, “Bistable Systems With Propagating Fronts Leading to Pattern Formation
,” Phys. Rev. Lett.
, 60
(25
), pp. 2641
–2644
.10.1103/PhysRevLett.60.264112.
Galaktionov
, V. A.
, 2010
, “Incomplete Self-Similar Blow-Up in a Semilinear Fourth-Order Reaction-Diffusion Equation
,” Stud. Appl. Math.
, 124
(4
), pp. 347
–381
.10.1111/j.1467-9590.2009.00474.x13.
Danumjaya
, P.
, and Pani
, A. K.
, 2012
, “Mixed Finite Element Methods for a Fourth Order Reaction Diffusion Equation
,” Numer. Methods Partial Differ. Equations
, 28
(4
), pp. 1227
–1251
.10.1002/num.2067914.
Hilfer
, R.
, 2000
, Applications of Fractional Calculus in Physics
, World Scientific
, Germany.15.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, Theory and Applications of Fractional Differential Equations
, Vol. 204
, Elsevier
, New York.16.
Mittal
, R.
, and Jiwari
, R.
, 2012
, “A Differential Quadrature Method for Numerical Solutions of Burgers'-Type Equations
,” Int. J. Numer. Methods Heat Fluid Flow
, 22
(7
), pp. 880
–895
.10.1108/0961553121125576117.
Yadav
, S.
, Kumar
, D.
, and Nisar
, K. S.
, 2021
, “A Reliable Numerical Method for Solving Fractional Reaction-Diffusion Equations
,” J. King Saud Univ.-Sci.
, 33
(2
), p. 101320
.10.1016/j.jksus.2020.10132018.
Mubaraki
, A. M.
, Nuruddeen
, R. I.
, and Gómez-Aguilar
, J.
, 2023
, “Modeling the Dispersion of Waves on a Loaded Bi-Elastic Cylindrical Tube With Variable Material Constituents
,” Results Phys.
, 53
, p. 106927
.10.1016/j.rinp.2023.10692719.
Islam
, M. T.
, Sarkar
, T. R.
, Abdullah
, F. A.
, and Gómez-Aguilar
, J.
, 2023
, “Characteristics of Dynamic Waves in Incompressible Fluid Regarding Nonlinear Boiti-Leon-Manna-Pempinelli Model
,” Phys. Scr.
, 98
(8
), p. 085230
.10.1088/1402-4896/ace74320.
Antil
, H.
, and Bartels
, S.
, 2017
, “Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling
,” Comput. Methods Appl. Math.
, 17
(4
), pp. 661
–678
.10.1515/cmam-2017-003921.
Antil
, H.
, and Rautenberg
, C. N.
, 2019
, “Sobolev Spaces With non-Muckenhoupt Weights, Fractional Elliptic Operators, and Applications
,” SIAM J. Math. Anal.
, 51
(3
), pp. 2479
–2503
.10.1137/18M122497022.
Li
, H.
, Ur Rahman
, G.
, Naz
, H.
, and Gómez-Aguilar
, J.
, 2024
, “Modeling of Implicit Multi Term Fractional Delay Differential Equation: Application in Pollutant Dispersion Problem
,” Alexandria Eng. J.
, 94
, pp. 1
–22
.10.1016/j.aej.2024.03.02123.
Lorenzo
, C. F.
, and Hartley
, T. T.
, 2002
, “Variable Order and Distributed Order Fractional Operators
,” Nonlinear Dyn.
, 29
(1/4
), pp. 57
–98
.10.1023/A:101658690565424.
Garrappa
, R.
, Giusti
, A.
, and Mainardi
, F.
, 2021
, “Variable-Order Fractional Calculus: A Change of Perspective
,” Commun. Nonlinear Sci. Numer. Simul.
, 102
, p. 105904
.10.1016/j.cnsns.2021.10590425.
Inc
, M.
, 2008
, “The Approximate and Exact Solutions of the Space-and Time-Fractional Burgers Equations With Initial Conditions by Variational Iteration Method
,” J. Math. Anal. Appl.
, 345
(1
), pp. 476
–484
.10.1016/j.jmaa.2008.04.00726.
Sakar
, M. G.
, and Ergören
, H.
, 2015
, “Alternative Variational Iteration Method for Solving the Time-Fractional Fornberg–Whitham Equation
,” Appl. Math. Modell.
, 39
(14
), pp. 3972
–3979
.10.1016/j.apm.2014.11.04827.
Jafari
, H.
, and Daftardar-Gejji
, V.
, 2006
, “Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by Adomian Decomposition
,” Appl. Math. Comput.
, 180
(2
), pp. 488
–497
.10.1016/j.amc.2005.12.03128.
Zeng
, F.
, Liu
, F.
, Li
, C.
, Burrage
, K.
, Turner
, I.
, and Anh
, V.
, 2014
, “A Crank–Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation
,” SIAM J. Numer. Anal.
, 52
(6
), pp. 2599
–2622
.10.1137/13093419229.
Zheng
, R.
, Liu
, F.
, and Jiang
, X.
, 2020
, “A Legendre Spectral Method on Graded Meshes for the Two-Dimensional Multi-Term Time-Fractional Diffusion Equation With Non-Smooth Solutions
,” Appl. Math. Lett.
, 104
, p. 106247
.10.1016/j.aml.2020.10624730.
Ali
, I.
, Haq
, S.
, Hussain
, M.
, Nisar
, K. S.
, and Arifeen
, S. U.
, 2024
, “On the Analysis and Application of a Spectral Collocation Scheme for the Nonlinear Two-Dimensional Fractional Diffusion Equation
,” Results Phys.
, 56
, p. 107222
.10.1016/j.rinp.2023.10722231.
Shi
, Z.
, Zhao
, Y.
, Liu
, F.
, Tang
, Y.
, Wang
, F.
, and Shi
, Y.
, 2017
, “High Accuracy Analysis of an H1-Galerkin Mixed Finite Element Method for Two-Dimensional Time Fractional Diffusion Equations
,” Comput. Math. Appl.
, 74
(8
), pp. 1903
–1914
.10.1016/j.camwa.2017.06.05732.
Abbaszadeh
, M.
, and Dehghan
, M.
, 2019
, “Analysis of Mixed Finite Element Method (MFEM) for Solving the Generalized Fractional Reaction–Diffusion Equation on Nonrectangular Domains
,” Comput. Math. Appl.
, 78
(5
), pp. 1531
–1547
.10.1016/j.camwa.2019.03.04033.
Saadatmandi
, A.
, Dehghan
, M.
, and Azizi
, M.-R.
, 2012
, “The Sinc–Legendre Collocation Method for a Class of Fractional Convection–Diffusion Equations With Variable Coefficients
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(11
), pp. 4125
–4136
.10.1016/j.cnsns.2012.03.00334.
Shirzadi
, A.
, Ling
, L.
, and Abbasbandy
, S.
, 2012
, “Meshless Simulations of the Two-Dimensional Fractional-Time Convection–Diffusion–Reaction Equations
,” Eng. Anal. Boundary Elem.
, 36
(11
), pp. 1522
–1527
.10.1016/j.enganabound.2012.05.00535.
Yang
, J.
, Zhao
, Y.
, Liu
, N.
, Bu
, W.
, Xu
, T.
, and Tang
, Y.
, 2015
, “An Implicit MLS Meshless Method for 2-D Time Dependent Fractional Diffusion–Wave Equation
,” Appl. Math. Modell.
, 39
(3–4
), pp. 1229
–1240
.10.1016/j.apm.2014.08.00536.
Amin
, R.
, Hadi
, F.
, Altanji
, M.
, Nisar
, K. S.
, Sumelka
., and W.
Hafsa
, 2023
, “Solution of Variable-Order Nonlinear Fractional Differential Equations Using Haar Wavelet Collocation Technique
,” Fractals
, 31
(2
), p. 2340022
.10.1142/S0218348X2340022437.
Yadav
, P.
, Jahan
, S.
, and Nisar
, K. S.
, 2023
, “Fibonacci Wavelet Method for Time Fractional Convection–Diffusion Equations
,” Math. Methods Appl. Sci.
, 47
(4
), pp. 2639
–2655
.10.1002/mma.977038.
Ahmed
, S.
, Jahan
, S.
, and Nisar
, K. S.
, 2024
, “Haar Wavelet Based Numerical Technique for the Solutions of Fractional Advection Diffusion Equations
,” J. Math. Comput. Sci.
, 34
(3
), pp. 217
–233
.10.22436/jmcs.034.03.0239.
Liu
, Y.
, Du
, Y.
, Li
, H.
, He
, S.
, and Gao
, W.
, 2015
, “Finite Difference/Finite Element Method for a Nonlinear Time-Fractional Fourth-Order Reaction–Diffusion Problem
,” Comput. Math. Appl.
, 70
(4
), pp. 573
–591
.10.1016/j.camwa.2015.05.01540.
Liu
, Y.
, Du
, Y.
, Li
, H.
, Li
, J.
, and He
, S.
, 2015
, “A Two-Grid Mixed Finite Element Method for a Nonlinear Fourth-Order Reaction–Diffusion Problem With Time-Fractional Derivative
,” Comput. Math. Appl.
, 70
(10
), pp. 2474
–2492
.10.1016/j.camwa.2015.09.01241.
Sun
, H.
, Sun
, Z-Z.
, and Du
, R.
, 2019
, “A Linearized Second-Order Difference Scheme for the Nonlinear Time-Fractional Fourth-Order Reaction-Diffusion Equation
,” Numer. Math.: Theory, Methods Appl.
, 12
(4
), pp. 1168
–1190
.10.4208/nmtma.OA-2017-014442.
Abbaszadeh
, M.
, and Dehghan
, M.
, 2020
, “Direct Meshless Local Petrov–Galerkin (DMLPG) Method for Time-Fractional Fourth-Order Reaction–Diffusion Problem on Complex Domains
,” Comput. Math. Appl.
, 79
(3
), pp. 876
–888
.10.1016/j.camwa.2019.08.00143.
Nikan
, O.
, Machado
, J. T.
, and Golbabai
, A.
, 2021
, “Numerical Solution of Time-Fractional Fourth-Order Reaction-Diffusion Model Arising in Composite Environments
,” Appl. Math. Modell.
, 89
, pp. 819
–836
.10.1016/j.apm.2020.07.02144.
Sun
, H.
, Cao
, W.
, and Zhang
, M.
, 2023
, “Energy Stability of a Temporal Variable-Step Difference Scheme for Time-Fractional Nonlinear Fourth-Order Reaction–Diffusion Equation
,” Int. J. Comput. Math.
, 100
(5
), pp. 991
–1008
.10.1080/00207160.2023.216751745.
Haghi
, M.
, Ilati
, M.
, and Dehghan
, M.
, 2023
, “A Fourth-Order Compact Difference Method for the Nonlinear Time-Fractional Fourth-Order Reaction–Diffusion Equation
,” Eng. Comput.
, 39
(2
), pp. 1329
–1340
.10.1007/s00366-021-01524-246.
Zeng
, Y.
, Li
, Y.
, Zeng
, Y.
, Cai
, Y.
, and Luo
, Z.
, 2024
, “The Dimension Reduction Method of Two-Grid Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for Nonlinear Fourth-Order Reaction Diffusion Equation With Temporal Fractional Derivative
,” Commun. Nonlinear Sci. Numer. Simul.
, 133
, p. 107962
.10.1016/j.cnsns.2024.10796247.
Sharma
, R.
, and Rajeev 2023
, “An Operational Matrix Approach to Solve a 2D Variable-Order Reaction Advection Diffusion Equation With Vieta–Fibonacci Polynomials
,” Spec. Top. Rev. Porous Media: Int. J.
, 14
(5
), pp. 79
–96
.10.1615/SpecialTopicsRevPorousMedia.202304803448.
Sharma
, R.
, and Rajeev
, 2024
, “A Numerical Approach Based on Vieta–Fibonacci Polynomials to Solve Fractional Order Advection–Reaction Diffusion Problem
,” J. Anal.
, pp. 1
–25
.10.1007/s41478?024?00804?649.
Singh
, M.
, Das, S., and Rajeev
, 2024
, “An Innovative Vieta–Fibonacci Wavelet Collocation Method for the Numerical Solution of Three-Component Brusselator Reaction Diffusion System of Fractional Order
,” J. Math. Chem.
, 62
(7
), pp. 1558
–1594
.10.1007/s10910-024-01621-950.
Zhang
, H.
, Yang
, X.
, and Xu
, D.
, 2019
, “A High-Order Numerical Method for Solving the 2D Fourth-Order Reaction-Diffusion Equation
,” Numer. Algorithms
, 80
(3
), pp. 849
–877
.10.1007/s11075-018-0509-z51.
Vong
, S.
, and Wang
, Z.
, 2014
, “Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System
,” Adv. Appl. Math. Mech.
, 6
(4
), pp. 419
–435
.10.4208/aamm.2014.4.s152.
Zhang
, P.
, and Pu
, H.
, 2017
, “A Second-Order Compact Difference Scheme for the Fourth-Order Fractional Sub-Diffusion Equation
,” Numer. Algorithms
, 76
(2
), pp. 573
–598
.10.1007/s11075-017-0271-7Copyright © 2025 by ASME
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