International Journal of Nonlinear Analysis and Applications
A numerical scheme for solving variable order Caputo-Prabhakar fractional integro-differential equation
Document Type : Research Paper
Authors
1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Abstract
In this paper we apply the Chebyshev polynomials method for the numerical solution of a class of variable-order fractional integro-differential equations with initial conditions. Moreover, a class of variable-order fractional integro-differential equations with fractional derivative of Caputo-Prabhakar sense is considered. The main aim of the Chebyshev polynomials method is to derive four kinds of operational matrices of Chebyshev polynomials. With such operational matrices, an equation is transformed into the products of several dependent matrices, which can also be viewed as the system of linear equations after dispersing the variables. Finally, numerical examples have been presented to demonstrate the accuracy of the proposed method, and the results have been compared with the exact solution.
Keywords
[1] R. Agarwal, S. Jain, R.P. Agarwal, Analytic solution of generalized space time fractional reaction–diffusion equation, Fract Differ Calc, 7(2017), 169-84.
[2] H. Aminikhah, A.R. Sheikhani, H. Rezazadeh, Exact solutions for the fractional differential equations by using
the first integral method, Nonlinear Engineering, 4(1)(2015), 15-22.
[3] Y. Chen, M. Yi, C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets
method, Journal of Computational Science, 3(5)(2012), 367-373.
[4] E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin, A. M. Lopes, On spectral methods for solving variable-order fractional
integro–differential equations, Computational and Applied Mathematics, 37(3)(2018), 3937-3950.
[5] I.L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Applied
Mathematics Letters, 21(4)(2008), 372–376.[6] R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, ˇ Hilfer-Prabhakar derivatives and some applications, Applied
mathematics and computation, 242(2014), 576-589.
[7] R. Garra, R. Garrappa, The Prabhakar or three parameters Mittag-Leffler function: Theory and application,
Communications in Nonlinear Science and Numerical Simulation, 56(2018), 314-329.
[8] R. Garrappa, Gr¨unwald–Letnikov operators for fractional relaxation in Havriliak-Negamimodels, Commun Nonlinear Sci Numer Simul 38(2016), 178–191.
[9] R. Garra, R. Garrappa, The Prabhakar or three parameters Mittag–Leffler function: Theory and application,
Communications in Nonlinear Science and Numerical Simulation, 56(2018), 314-329.
[10] A. Giusti, I. Colombaro, Prabhakar-like fractional viscoelasticity, Communications in Nonlinear Science and Numerical Simulation, 56(2018), 138-143.
[11] A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio, F. Mainardi, A practical guide to
Prabhakar fractional calculus, Fractional Calculus and Applied Analysis, 23(1)(2020), 9-54.
[12] B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific Publishing, Singapore, 2015.
[13] R.K. Gupta, B.S. Shaktawat, D. Kumar, Certain relation of generalized fractional calculus associated with the
generalized Mittag-Leffler function. J. Raj. Acad. Phy. Sci, 15(3)(2016), 117-126.
[14] M.M. Hosseini, Adomian decomposition method for solution of nonlinear differential algebraic equations, Applied
mathematics and computation, 181(2)(2006), 1737-1744.
[15] M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of non-integer order transfer functions for analysis of
electrode processes, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 33(2)(1971), 253-265.
[16] S. Khubalkar, A. Junghare, M. Aware, S. Das, Unique fractional calculus engineering laboratory for learning and
research, International Journal of Electrical Engineering Education, 57(1)(2020), 3-33.
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations , Elsevier
Science Limited, 2006.
[18] J. Lai, S. Mao, J. Qiu, H. Fan, Q. Zhang, Z. Hu, J. Chen, Investigation progress and applications of fractional
derivative model in geotechnical engineering, Mathematical Problems in Engineering, 2016.
[19] F. Mainardi, R. Garrappa, On complete monotonicity of the Prabhakar function and non-Debye relaxation in
dielectrics, Journal of Computational Physics, 293(2015), 70-80.
[20] M. Mashoof, A.H.R. Sheikhani, Numerical Solution of Fractional Control System by Haar-wavelet Operational
Matrix Method, Int. J. Industrial Mathematics, 8(2016), 289-298.
[21] M. Mashoof, A.H.R. Sheikhani, H.S. Najafi, Stability Analysis of Distributed-Order Hilfer-Prabhakar Systems
Based on Inertia Theory, Mathematical Notes, 104(1-2)(2018), 74-85.
[22] M. Mashoof, A.H.R. Sheikhani, H.S. Naja, Stability analysis of distributed order Hilfer-Prabhakar differential
equations, Hacettepe Journal of Mathematics and Statistics, 47(2)(2018), 299-315.
[23] A. Mohebbi, M. Saffarian, Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order
Fractional Cable Equation, Journal of Applied and Computational Mechanics, 6(2)(2020), 235-247.
[24] S. Momani, Z. Odibat, V.S. Erturk, Generalized differential transform method for solving a space and time
fractional diffusion wave equation, Physics Letters A, 370(5-6)(2007), 379-387.
[25] H.S. Najafi, S.A. Edalatpanah, A.R.H. Sheikhani, Convergence analysis of modified iterative methods to solve
linear systems, Mediterranean journal of mathematics, 11(3)(2014), 1019-1032.
[26] Z.M. Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling,
51(9-10)(2010), 1181-1192.
[27] M.D. Ortigueira, Fractional calculus for scientists and engineers, Springer Science & Business Media, 2011.
[28] S.C. Pandey, The Lorenzo–Hartley’s function for fractional calculus and its applications pertaining to fractional
order modelling of anomalous relaxation in dielectrics, Computational and Applied Mathematics, 37(3)(2018),
2648-2666.
[29] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential
equations, to methods of their solution and some of their applications, Elsevier, 1998.
[30] T.R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, 1971.
[31] F. Shariffar, A.H.R. Sheikhani, A New Two-stage Iterative Method for Linear Systems and Its Application in
Solving Poissons Equation, International Journal of Industrial Mathematics, 11(4)(2019), 283–291.
[32] F. Shariffar, A.H.R. Sheikhani, H.S. Najafi, An efficient chebyshev semi-iterative method for the solution of
large systems, University Politehnica of Bucharest Scientific Bulletin-Series A Applied Mathematics and Physics.
80(4)(2018), 239-252.
[33] A.H.R. Sheikhani, M. Mashoof, A Collocation Method for Solving Fractional Order Linear System, Journal of the
Indonesian Mathematical Society, 23(1)(2017), 27-42.[34] M.A. Snyder, Chebyshev methods in numerical approximation , Prentice-Hall, 1966.
[35] H.M. Srivastava, R.K. Saxena, T.K. Pogany, R. Saxena, Integral transforms and special functions, Applied Mathematics and Computation, 22(7)(2011), 487-506.
[36] K. Sun, M. Zhu, Numerical algorithm to solve a class of variable order fractional integral-differential equation
based on Chebyshev polynomials, Mathematical Problems in Engineering, 2015.
[37] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional
calculus in science and engineering, Communications in Nonlinear Science and Numerical Simulation, 64(2018),
213-231.
[38] Y. Xu, V. Ert¨urk, A finite difference technique for solving variable order fractional integro-differential equations,
Bulletin of the Iranian Mathematical Society, 40(3)(2014), 699-712
[39] M. Zayernouri, G.E. Karniadakis, Fractional spectral collocation methods for linear and nonlinear variable order
FPDEs, Journal of Computational Physics, 293(2015), 312-338.
[2] H. Aminikhah, A.R. Sheikhani, H. Rezazadeh, Exact solutions for the fractional differential equations by using
the first integral method, Nonlinear Engineering, 4(1)(2015), 15-22.
[3] Y. Chen, M. Yi, C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets
method, Journal of Computational Science, 3(5)(2012), 367-373.
[4] E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin, A. M. Lopes, On spectral methods for solving variable-order fractional
integro–differential equations, Computational and Applied Mathematics, 37(3)(2018), 3937-3950.
[5] I.L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Applied
Mathematics Letters, 21(4)(2008), 372–376.[6] R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, ˇ Hilfer-Prabhakar derivatives and some applications, Applied
mathematics and computation, 242(2014), 576-589.
[7] R. Garra, R. Garrappa, The Prabhakar or three parameters Mittag-Leffler function: Theory and application,
Communications in Nonlinear Science and Numerical Simulation, 56(2018), 314-329.
[8] R. Garrappa, Gr¨unwald–Letnikov operators for fractional relaxation in Havriliak-Negamimodels, Commun Nonlinear Sci Numer Simul 38(2016), 178–191.
[9] R. Garra, R. Garrappa, The Prabhakar or three parameters Mittag–Leffler function: Theory and application,
Communications in Nonlinear Science and Numerical Simulation, 56(2018), 314-329.
[10] A. Giusti, I. Colombaro, Prabhakar-like fractional viscoelasticity, Communications in Nonlinear Science and Numerical Simulation, 56(2018), 138-143.
[11] A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio, F. Mainardi, A practical guide to
Prabhakar fractional calculus, Fractional Calculus and Applied Analysis, 23(1)(2020), 9-54.
[12] B. Guo, X. Pu, F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific Publishing, Singapore, 2015.
[13] R.K. Gupta, B.S. Shaktawat, D. Kumar, Certain relation of generalized fractional calculus associated with the
generalized Mittag-Leffler function. J. Raj. Acad. Phy. Sci, 15(3)(2016), 117-126.
[14] M.M. Hosseini, Adomian decomposition method for solution of nonlinear differential algebraic equations, Applied
mathematics and computation, 181(2)(2006), 1737-1744.
[15] M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of non-integer order transfer functions for analysis of
electrode processes, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 33(2)(1971), 253-265.
[16] S. Khubalkar, A. Junghare, M. Aware, S. Das, Unique fractional calculus engineering laboratory for learning and
research, International Journal of Electrical Engineering Education, 57(1)(2020), 3-33.
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations , Elsevier
Science Limited, 2006.
[18] J. Lai, S. Mao, J. Qiu, H. Fan, Q. Zhang, Z. Hu, J. Chen, Investigation progress and applications of fractional
derivative model in geotechnical engineering, Mathematical Problems in Engineering, 2016.
[19] F. Mainardi, R. Garrappa, On complete monotonicity of the Prabhakar function and non-Debye relaxation in
dielectrics, Journal of Computational Physics, 293(2015), 70-80.
[20] M. Mashoof, A.H.R. Sheikhani, Numerical Solution of Fractional Control System by Haar-wavelet Operational
Matrix Method, Int. J. Industrial Mathematics, 8(2016), 289-298.
[21] M. Mashoof, A.H.R. Sheikhani, H.S. Najafi, Stability Analysis of Distributed-Order Hilfer-Prabhakar Systems
Based on Inertia Theory, Mathematical Notes, 104(1-2)(2018), 74-85.
[22] M. Mashoof, A.H.R. Sheikhani, H.S. Naja, Stability analysis of distributed order Hilfer-Prabhakar differential
equations, Hacettepe Journal of Mathematics and Statistics, 47(2)(2018), 299-315.
[23] A. Mohebbi, M. Saffarian, Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order
Fractional Cable Equation, Journal of Applied and Computational Mechanics, 6(2)(2020), 235-247.
[24] S. Momani, Z. Odibat, V.S. Erturk, Generalized differential transform method for solving a space and time
fractional diffusion wave equation, Physics Letters A, 370(5-6)(2007), 379-387.
[25] H.S. Najafi, S.A. Edalatpanah, A.R.H. Sheikhani, Convergence analysis of modified iterative methods to solve
linear systems, Mediterranean journal of mathematics, 11(3)(2014), 1019-1032.
[26] Z.M. Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling,
51(9-10)(2010), 1181-1192.
[27] M.D. Ortigueira, Fractional calculus for scientists and engineers, Springer Science & Business Media, 2011.
[28] S.C. Pandey, The Lorenzo–Hartley’s function for fractional calculus and its applications pertaining to fractional
order modelling of anomalous relaxation in dielectrics, Computational and Applied Mathematics, 37(3)(2018),
2648-2666.
[29] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential
equations, to methods of their solution and some of their applications, Elsevier, 1998.
[30] T.R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, 1971.
[31] F. Shariffar, A.H.R. Sheikhani, A New Two-stage Iterative Method for Linear Systems and Its Application in
Solving Poissons Equation, International Journal of Industrial Mathematics, 11(4)(2019), 283–291.
[32] F. Shariffar, A.H.R. Sheikhani, H.S. Najafi, An efficient chebyshev semi-iterative method for the solution of
large systems, University Politehnica of Bucharest Scientific Bulletin-Series A Applied Mathematics and Physics.
80(4)(2018), 239-252.
[33] A.H.R. Sheikhani, M. Mashoof, A Collocation Method for Solving Fractional Order Linear System, Journal of the
Indonesian Mathematical Society, 23(1)(2017), 27-42.[34] M.A. Snyder, Chebyshev methods in numerical approximation , Prentice-Hall, 1966.
[35] H.M. Srivastava, R.K. Saxena, T.K. Pogany, R. Saxena, Integral transforms and special functions, Applied Mathematics and Computation, 22(7)(2011), 487-506.
[36] K. Sun, M. Zhu, Numerical algorithm to solve a class of variable order fractional integral-differential equation
based on Chebyshev polynomials, Mathematical Problems in Engineering, 2015.
[37] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional
calculus in science and engineering, Communications in Nonlinear Science and Numerical Simulation, 64(2018),
213-231.
[38] Y. Xu, V. Ert¨urk, A finite difference technique for solving variable order fractional integro-differential equations,
Bulletin of the Iranian Mathematical Society, 40(3)(2014), 699-712
[39] M. Zayernouri, G.E. Karniadakis, Fractional spectral collocation methods for linear and nonlinear variable order
FPDEs, Journal of Computational Physics, 293(2015), 312-338.
)
Volume 13, Issue 1
March 2022Pages 467-484
- Receive Date: 23 August 2020
- Revise Date: 01 October 2020
- Accept Date: 20 October 2020
- First Publish Date: 17 September 2021