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Numerical solution of stochastic integral equations by using Bernoulli operational matrix

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  • Zeghdane, Rebiha

Abstract

In this paper, a new computational method based on stochastic operational matrix for integration of Bernoulli polynomials is proposed for solving nonlinear Volterra–Fredholm–Hammerstein stochastic integral equations. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra–Fredholm–Hammerstein stochastic integral equation is reduced to nonlinear system of algebraic equations with unknown Bernoulli coefficients. This work is inspired by Bazm (2015), where the authors study the deterministic integral equations. In order to show the rate of convergence of the suggested approach, we present theorems on convergence analysis and error estimation. Some illustrative error estimations and examples are provided and included to demonstrate applicability and accuracy of the technique.

Suggested Citation

  • Zeghdane, Rebiha, 2019. "Numerical solution of stochastic integral equations by using Bernoulli operational matrix," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 238-254.
    • Handle: RePEc:eee:matcom:v:165:y:2019:i:c:p:238-254
    DOI: 10.1016/j.matcom.2019.03.005
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    References listed on IDEAS

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    1. Pierpaolo Natalini & Angela Bernardini, 2003. "A generalization of the Bernoulli polynomials," Journal of Applied Mathematics, Hindawi, vol. 2003, pages 1-9, January.
    2. K. Balachandran & J.-H. Kim, 2010. "Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2010, pages 1-16, March.
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    Cited by:

    1. Hossein Hassani & Zakieh Avazzadeh & Praveen Agarwal & Mohammad Javad Ebadi & Ali Bayati Eshkaftaki, 2024. "Generalized Bernoulli–Laguerre Polynomials: Applications in Coupled Nonlinear System of Variable-Order Fractional PDEs," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 371-393, January.
    2. Saha Ray, S. & Singh, P., 2021. "Numerical solution of stochastic Itô-Volterra integral equation by using Shifted Jacobi operational matrix method," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    3. Ahmadinia, M. & Afshariarjmand, H. & Salehi, M., 2023. "Numerical solution of multi-dimensional Itô Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process," Applied Mathematics and Computation, Elsevier, vol. 450(C).

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