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A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations

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  • Torkaman, Soraya
  • Heydari, Mohammad
  • Loghmani, Ghasem Barid

Abstract

In this paper, an iterative scheme including a combination of the quasilinearization technique and multi-dimensional linear barycentric rational interpolation is applied to solve nonlinear multi-dimensional Volterra integral equations. First, employing the quasilinearization method, the nonlinear multi-dimensional Volterra integral equation is reduced to a sequence of linear Volterra integral equations. Under appropriate assumptions, the constructed iterative sequence is uniformly convergent to the unique solution of the problem. In general, finding an analytical solution for linear Volterra integral equations is impossible. Hence, in each iteration, using a collocation method based on multi-dimensional barycentric rational basis functions, the solution to the linear integral equation is approximated. The quadratic convergence of the quasilinearization approach and the error estimation of the combined method are investigated theoretically. In the end, the efficiency and the validity of this method are illustrated with some numerical examples and compared with those of the existing numerical methods.

Suggested Citation

  • Torkaman, Soraya & Heydari, Mohammad & Loghmani, Ghasem Barid, 2023. "A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 366-397.
    • Handle: RePEc:eee:matcom:v:208:y:2023:i:c:p:366-397
    DOI: 10.1016/j.matcom.2023.01.039
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    References listed on IDEAS

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    1. Liu, Hongyan & Huang, Jin & Zhang, Wei & Ma, Yanying, 2019. "Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 295-304.
    2. Assari, Pouria & Dehghan, Mehdi, 2019. "A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 249-265.
    3. Sudhakar G. Pandit, 1997. "Quadratically converging iterative schemes for nonlinear Volterra integral equations and an application," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-10, January.
    4. Laeli Dastjerdi, H. & Nili Ahmadabadi, M., 2017. "The numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 545-554.
    5. Pan, Yubin & Huang, Jin & Ma, Yanying, 2019. "Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 149-161.
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