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Pseudo-Spectral Galerkin Method Using Shifted Vieta-Fibonacci Polynomials for Stochastic Models: Existence, Stability, and Numerical Validation

Author

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  • Reema Gupta

    (Department of Mathematics, National Institute of Technology)

  • Snehashish Chakraverty

    (Department of Mathematics, National Institute of Technology)

Abstract

This article introduces a novel methodology for tackling stochastic Itô Volterra-Fredholm integral equations through a pseudo-Spectral Galerkin method utilizing shifted Vieta-Fibonacci polynomials. The method is very efficient in solving integral and integro-differential equations, which often arise in many scientific and engineering fields. This approach is widely applicable in many domains, including signal processing, electromagnetic theory, population dynamics, finance, and related areas. By employing this method, the intricate task of solving such equations is simplified into a set of linear algebraic equations that are efficiently solvable via the Gauss elimination method for numerical solutions. The article also presents the existence, uniqueness, and stability of the solution. The convergence of the proposed method is rigorously established, ensuring its reliability. To validate its effectiveness, two numerical examples are provided. In addition to the examples, a comparison study is conducted between the orthonormal Bernoulli polynomials-based, second-kind shifted Chebyshev polyunomials-based pseudo-Spectral Galerkin approach and existing results based on improved hat functions. Numerical results demonstrate the superiority of the suggested approach in terms of accuracy and computational efficiency, showcasing its applicability across various scientific and engineering domains.

Suggested Citation

  • Reema Gupta & Snehashish Chakraverty, 2024. "Pseudo-Spectral Galerkin Method Using Shifted Vieta-Fibonacci Polynomials for Stochastic Models: Existence, Stability, and Numerical Validation," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-20, December.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10114-w
    DOI: 10.1007/s11009-024-10114-w
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    References listed on IDEAS

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    1. Yousefi, S. & Razzaghi, M., 2005. "Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 1-8.
    2. Deng, Guoting & Yang, Yin & Tohidi, Emran, 2021. "High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels," Applied Mathematics and Computation, Elsevier, vol. 396(C).
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