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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 3-4


Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions

S. Saha Ray / S. Singh
Published Online: 2018-04-24 | DOI: https://doi.org/10.1515/ijnsns-2017-0038


In this article, the numerical solutions of stochastic Volterra–Fredholm integral equations have been obtained by hybrid Legendre block-pulse functions (BPFs) and stochastic operational matrix. The hybrid Legendre BPFs are orthonormal and have compact support on [0,1). The numerical results obtained by the above functions have been compared with those obtained by second kind Chebyshev wavelets. Furthermore, the results of the proposed computational method establish its accuracy and efficiency.

Keywords: stochastic Volterra–Fredholm integral equation; hybrid Legendre block-pulse functions; stochastic operational matrix; second kind Chebyshev wavelets

MSC 2010: 60H05; 60H20; 60H30; 60H35


  • [1]

    K. Maleknejad, B. Basirat and E. Hashemizadeh, Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra–Fredholm integro-differential equations, Comput. Math. Appl. 61 (2011), 2821–2828.Web of ScienceCrossrefGoogle Scholar

  • [2]

    A.-M. Wazwaz, A first course in integral equations, World Scientific Publishing, London, 2015.Google Scholar

  • [3]

    C.P. Tsokos and W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974.Google Scholar

  • [4]

    G.J. Lord, C.E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, New York, USA, 2014.Google Scholar

  • [5]

    B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed, Springer-Verlag, New York, 1998.Google Scholar

  • [6]

    J.C. Cortés, L. Jordan and L. Villafuerte, Numerical solution of random differential equations: A mean square approach, Math. Comput. Model. 45 (2007), 757–765.CrossrefWeb of ScienceGoogle Scholar

  • [7]

    D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001), 525–546.CrossrefGoogle Scholar

  • [8]

    P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Springer-Verlag, New York, 1992.Web of ScienceGoogle Scholar

  • [9]

    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 (2012), 791–800.Web of ScienceCrossrefGoogle Scholar

  • [10]

    M. Khodabin, K. Maleknejad, M. Rostami and M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl. 64 (2012), 1903–1913.CrossrefWeb of ScienceGoogle Scholar

  • [11]

    K. Maleknejad, E. Hashemizadeh and B. Basirat, Numerical solvability of Hammerstein integral equations based on hybrid Legendre and Block-Pulse functions, Int. Conf. Parallel Distrib. Process. Tech. Appl. 2010 (2010), 172–175.Google Scholar

  • [12]

    K. Maleknejad and M.T. Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput. 145 (2003), 623–629.Google Scholar

  • [13]

    S. Saha Ray and A.K. Gupta, Numerical solution of fractional partial differential equation of parabolic type with Dirichlet boundary conditions using two-dimensional Legendre wavelets method, J. Comput. Nonlinear Dynam. 11 (1) (2016), 9.Google Scholar

  • [14]

    F. Mohammadi and P. Adhami, Numerical study of stochastic Volterra-Fredholm integral equations by using second kind Chebyshev wavelets, Random Operators Stochastic Equ. 24(2) (2016), 129–141.Web of ScienceGoogle Scholar

About the article

Published Online: 2018-04-24

Published in Print: 2018-06-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 289–297, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0038.

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