‎A spectral collocation method‎ for solving stochastic fractional integro-differential equation

Document Type : Original Article

Authors

Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.

Abstract
In this paper‎, ‎a numerical scheme based on shifted Vieta-Lucas polynomials is utilised to solve mentioned equation‎. ‎The main characteristic of the presented method is to approximate Brownian motion with help of the Gauss-Legendre quadrature‎, ‎which makes calculations easier‎. ‎Another characteristic of this method are employed suitable collocation points to convert the stochastic equation under the study into a system of algebraic equations by using the operational matrices‎. ‎So that‎, ‎Newton's method is applied to solve them‎. The convergence analysis and error bound of the suggested method are well established‎. ‎Additionally‎, ‎the proofs related to the existence and uniqueness of the solutions for the equations under investigation have been provided‎. ‎In order to illustrate the effectiveness‎, ‎compatibility and plausibility of the proposed technique‎, ‎four numerical examples are presented.

Keywords

Subjects


[1] Azimi, R. & Mohagheghy Nezhad, M. & Foadian,S. (2022) Shifted Legendre Tau Method for Solving the Fractional Stochastic Integro-Differential Equations, 6 (2) 221-241. 1
[2] Babaei, A., Jafari, H. & Banihashemi, S. (2020) Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, Journal of Computational and Applied Mathematics, 112908. 1
[3] Gupta, R. & Saha Ray, S. (2023) A new effective coherent numerical technique based on shifted Vieta-Fibonacci polynomials for solving stochastic fractional integro-differential equation, Computational and Applied Mathematics, 1-25. 1
[4] Horadam, A. F. (2000) Vieta Polynomials, The University of New England, Armidaie, Australia. 2.2
[5] Jabari Sabegh, D., Ezzati, R. & Maleknejad, K. (2019) Approximate solution of fractional integro-differentail equations by least squares method, International Journal of Analysis and Applications, 17, 303-310. 1
[6] Jafari, H., Ganji, R.M., Salati, S. &, Johnston, S.J. (2024) A mixed-method to numerical simulation of variable order
stochastic advection diffusion equations, Alexandria Engineering Journal, Volume 89, 60-70. 1
[7] Lord, G. J. & Powell, C.E. & Shardlow, T. (2014) An Introduction to Computational Stochastic PDEs, Vol. 50: Cambridge University Press. 2.3, 2.4
[8] Masti, I. & Sayevand, Kh. (2020) On collocation-Galerkin method and fractional B-spline functions for a class of
stochastic fractional integro-differential equations, Mathematics and Computers in Simulation, 216, 263-287. 1
[9] Mirzaee, F. & Alipour, S. (2020) Cubic B-spline approximation for linear stochastic integro-differential equation
of fractional order. J Comput Appl Math (366) 112440. 7, 3
[10] Mirzaee, F., Alipour, S. & Samadyar, N. (2019) Numerical solution based on hybrid of block-pulse and parabolic
functions for solving a system of nonlinear stochastic Itô -Volterra integral equations of fractional order. J Comput
Appl Math 349:157-171. 1
[11] Mirzaee, F., & Samadyar, N. (2017) Application of orthonormal Bernstein polynomials to construct a efficient
scheme for solving fractional stochastic integro-differential equation. Optik 132:262-273. 1
[12] Morters, P. & Peres, Y. (2010) Brownian Motion, Cambridge University Press, Cambridge 30. 1
[13] Mousavi, B. KH., Askari Hemmat, A. & M. H. Heydari (2017). An application of Wilson system in numerical
solution of Fredholm integral equations, Poincare Journal of Analysis and Applications, 2, 61-72. 1
[14] Podlubny, I. (1999) Fractional Diferential Equations, Academic Press,New York, NY, USA. 2.1, 2.2
[15] Robbins, N. (1991) Vietas triangular array and a related family of polynomials, Int. J. Math. Math. Sci. 14 (2),
239-244. 2.2
[16] Sayevand, Kh., Tenreiro Machado, J. & Masti, I. (2020) On dual Bernstein polynomials and stochastic fractional
integro-differential equations, Math Meth Appl Sci., 1-20. 1
[17] Sayevand, Kh. (2015) Analytical treatment of Volterra integro-differential equations of fractional order, Applied
Mathematical Modelling, 39 (15) 4330-4336. 1
[18] Sayevand, Kh. & Tenreiro Machado, J. A.(2019) A Survey on Fractional Asymptotic Expansion Method: A Forgot-
ten Theory, 22, 1165-1176. 1
[19] Sayevand, Kh. & Rostami, M. (2022) Numerical solution of multi order fractional differential equations using
Lucas polynomials, Mathematical Researches, 8 (1), 184-204. 1
[20] Senol, M., & Kasmaei, H. D. (2017) On the numerical solution of nonlinear fractional integro-differential equations,
New Trends in Mathematical Sciences, 5(3), 118-127. 1
[21] Singh, P.K. & Saha Ray, S. (2023) Numerical treatment for the solution of stochastic fractional differential equation
using Lerch operational matrix method, Journal of Computational and Nonlinear Dynamics, 1-27. 1, 7, 3
[22] Singh, P.K. & Saha Ray, S. (2021) Numerical solution of stochastic Itô-Volterra integral equation by using Shifted Jacobi operational matrix method. Appl Math Comput 410:126440. 3.1, 3.1
[23] Stoer, J. & Bulirsch, R. (2002) Introduction to numerical analysis. 2nd ed. Berlin: Springer. 2.5
[24] Wazwaz, A. M. (2011) Linear and Nonlinear Integral Equations Methods and Applications, Springer, New York.1
[25] Yousefi, A., Javadi, S. & Babolian, E. (2019). A computational approach for solving fractional integral equations based on Legendre collocation method, Mathematical Sciences, 13, 231240. 1
Volume 6, Issue 2
Spring 2025
Pages 1-31

  • Receive Date 08 August 2024
  • Accept Date 19 February 2025