A numerical technique for solving nonlinear fractional stochastic integro-differential equations with n-dimensional Wiener process

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, University of Mazandaran, P.O. Box: 47416-95447, Babolsar, Iran.

2 Department of Computer Sciences, University of Mazandaran, P.O. Box: 47416-95447, Babolsar, Iran.

Abstract

This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative, and a suitable numerical integration rule. Error estimate of the method is investigated in detail. In the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach. 

Keywords


  • [1]          A. Babaei, H. Jafari, and S. Banihashemi, 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 (2020), 112908.
  • [2]          A. Cardone, R. D’Ambrosio, and B. Patersone, A spectral method for stochastic fractional differential equa- tions, Applied Numerical Mathematics, 139(1) (2019), 115-119.
  • [3]          F. R. Chang, Stochastic Optimization in Continuous Time, Cambridge University Press, Cambridge, 2004.
  • [4]          G. H. Choe, Stochastic analysis for finance with simulations, Universitext, Springer, 2016.
  • [5]          S. I. Denisov, P. Hanggi, and H. Kantz, Parameters of the fractional Fokker-Planck equation, Europhys. Lett, 85(4) (2009), 40007.
  • [6]          M. Fallahpour, K. Maleknejad, and M. Khodabin, Approximation solution of two-dimensional linear stochastic fredholm integral equation by applying the haar wavelet, International Journal of Mathematical Modelling & Computations, 5(4) (2015), 361-372.
  • [7]          M. Girgoriu, Stochastic calculus: Applications in Science and Engineering, Springer, LLC, 2000.
  • [8]          D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, Society for Industrial and Applied Mathematics, 43(3) (2001), 525-546.
  • [9]          L. Huang, X. F. Li, Y. Zhao, and X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Computers and Mathematics with Applications, 62(3) (2011), 1127-1134.
  • [10]        D. Jabari Sabegh, R. Ezzati, and K. Maleknejad, Approximate solution of fractional integro-differentail equa- tions by least squares method, International Journal of Analysis and Applications, 17(2) (2019), 303-310.
  • [11]        M. Kamrani, Numerical solution of stochastic fractional differential equations, Numerical Algorithms, 68(1) (2014), 81-93.
  • [12]        M. Khodabin, K. Maleknejad, and T. Damercheli, Approximate solution of stochastic Volterra integral equa- tions via expansion method, Int. J. Industeria Mathematics, 6(1) (2014), 41-48.
  • [13]        A. A Kilbas, H. M. Sirvastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204 of North-Holland Mathematics Studies. Elesiver, Amesterdam, 2006.
  • [14]        P.E. Kloden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Velarge Berlin Heidelbreg, New York, 1995.
  • [15]        P. K. Kythe and M. R. Schaferkotter, Handbook of computational methods for integration, Chapman and Hall/CRC Press, New York, (2005).
  • [16]        M. Li, C. Huang, P. Hu, and J. Wen, Mean-square stability and convergence of a split-step theta method for stochastic Volterra integral equations, Journal of Computational and Applied Mathematics, 382(1) (2021), 113077.
  • [17]        L. Li and J. G. Liu, A discretization of Caputo derivatives with application to time fractional SDEs and  gradient flows, SIAM Journal on Numerical Analysis, 57(5) (2019), 2095-2120.
  • [18]        H. Liang, Z. Yang, and J. Gao, Strong superconvergence of the EulerMaruyama method for linear stochastic Volterra integral equations, Journal of Computational and Applied Mathematics, 317(1) (2017), 447-457.
  • [19]        X. Ma and Ch. Huang, Numerical solution of fractional integro differential equations by hybrid collocation method, Applied Mathematics and Computation, 219(12) (2013), 6750-6760.
  • [20]        K. Maleknejad, M. Khodabin, and M. Rostami, Numerical solution of stochastic Volterra integral  equations by  a stochastic operational matrix based on block pulse functions, Math. Comput. Model, 55(3-4) (2015), 791-800.
  • [21]        X. Mao, Stochastic Differential Equations and Applications, Second edition, Elsevier, New York, 2007.
  • [22]        F. Mirzaee and E. Hadadiyan, Numerical solution of Volterra-Fredholm integral equations via modification of hat functions, Applied Mathematics and Computations, 280(1) (2016), 110-123.
  • [23]        F. Mirzaee and N. Samadyar, Convergence of Legendre wavelet collocation method for solving nonlinear Stratonovich Volterra integral equations, Computational Methods for Differential Equations, 6(1) (2018), 80-  97.
  • [24]        B. P. Moghadam, A. Mendes Lopes, J. A. Tenreiro Machado, and Z. S. Mostaghim, Computational scheme for solving nonlinear fractional stochastic differential equations with delay, Stochastic Analysis and Applications, 37(6) (2019), 893-908.
  • [25]        F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations, International Journal of Applied Mathematics Research, 4(2) (2015), 217-227.
  • [26]        S. Nemati and P. M. Lima, Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Applied Mathematics and Computation, 327(1) (2018), 79-92 .
  • [27]        M. Nouri, Solving Ito integral equations with time delay via basis functions, Computational Methods for Differential Equations, 8(2) (2020), 268-281.
  • [28]        B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 5th edition, Springer Velarge, Berlin, 1998.
  • [29]        R. Panda and M. Dash, Fractional generalized splines and signal processing, Signal Process, 86(9) (2006), 2340-2350.
  • [30]        I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • [31]        M. Saffarzadeh, G. B. Loghmani, and M. Heydari, An iterative technique for the numerical solution of non- linear Ito-stochastic integral equations, Journal of Computational and Applied Mathematics, 333(1) (2018), 74-86.
  • [32]        M. Senol and H. D. Kasmaei, On the numerical solution of nonlinear fractional integro-differential equations, New Trends in Mathematical Sciences, 5(3) (2017), 118-127.
  • [33]        Z. Taheri, S. Javadi, and E. Babolian, Numerical solution  of stochastic fractional integro-differential equation  by the spectral collocation method, Journal of Computational and Applied Mathematics, 321(1) (2017), 336-  347.
  • [34]        Z. Yang, H. Yang, and Z. Yao, Strong convergence analysis for Volterra integro-differential equations with fractional Brownian motions, Journal of Computational and Applied Mathematics, 383(1) (2021), 113156.
Volume 10, Issue 1
January 2022
Pages 61-76
  • Receive Date: 04 August 2020
  • Revise Date: 11 December 2020
  • Accept Date: 21 December 2020
  • First Publish Date: 05 January 2021