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An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation

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  • Alipour, Sahar
  • Mirzaee, Farshid

Abstract

The authors propose a numerical iterative algorithm based on a combination of the successive approximations method and the bilinear spline interpolation. This algorithm is used to obtain an approximate solution of two-dimensional nonlinear stochastic Ito^-Volterra integral equation. In fact, this algorithm is an attractive extension of the numerical iterative approach for a class of two-dimensional nonlinear stochastic Itô-Volterra integral equations. To reach this aim, the bilinear spline interpolation, Gauss-Legendre quadrature formulas for double integrals and two dimensional Ito^ approximation are presented. The effectiveness of the method is shown for three examples. The obtained results and the convergence analysis theorems reveal that the suggested algorithm is very efficient and the convergence rate is O(h2).

Suggested Citation

  • Alipour, Sahar & Mirzaee, Farshid, 2020. "An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    • Handle: RePEc:eee:apmaco:v:371:y:2020:i:c:s0096300319309397
    DOI: 10.1016/j.amc.2019.124947
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    References listed on IDEAS

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    1. Mei, Hongwei & Yin, George & Wu, Fuke, 2016. "Properties of stochastic integro-differential equations with infinite delay: Regularity, ergodicity, weak sense Fokker–Planck equations," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3102-3123.
    2. Dareiotis, Konstantinos & Leahy, James-Michael, 2016. "Finite difference schemes for linear stochastic integro-differential equations," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3202-3234.
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    Cited by:

    1. Solhi, Erfan & Mirzaee, Farshid & Naserifar, Shiva, 2023. "Approximate solution of two dimensional linear and nonlinear stochastic Itô–Volterra integral equations via meshless scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 369-387.
    2. María Isabel Berenguer & Manuel Ruiz Galán, 2022. "An Iterative Algorithm for Approximating the Fixed Point of a Contractive Affine Operator," Mathematics, MDPI, vol. 10(7), pages 1-10, March.
    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).
    4. Zhang, Zhiguo & Kon, Mark A., 2022. "Wavelet matrix operations and quantum transforms," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    5. Wen, Xiaoxia & Huang, Jin, 2021. "A combination method for numerical solution of the nonlinear stochastic Itô-Volterra integral equation," Applied Mathematics and Computation, Elsevier, vol. 407(C).

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