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<p>In order to model anomalous diffusion in various physical systems, this article presents a collocation technique for solving intricate, multidimensional stochastic integral equation that include fractional Brownian motion. The method utilizes two categories of orthogonal polynomials: second-kind shifted Chebyshev polynomials and shifted Legendre polynomials. It uses Newton-Cotes nodes as collocation points, and the Itô approximation to handle the stochastic integral of the equation. This approach transforms the equation into solvable algebraic systems, which may be either linear or nonlinear. This process effectively decreases computing complexity while preserving accuracy. Furthermore, a thorough convergence study, with meticulous proof, verifies the accuracy of the process. Additionally, four examples demonstrate the effectiveness of the method. A comparison test reveals that results from shifted Chebyshev polynomials are closer to the exact solutions than results from shifted Legendre polynomials. Moreover, a 99% confidence interval for the error mean is constructed, revealing that the approximations maintain high accuracy.</p>

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Spectral collocation method for multidimensional stochastic integral equation involving fractional Brownian motion

  • 期刊论文

  • 发表日期：

    2025年8月13日

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  • Reema Gupta,
  • Snehashish Chakraverty

摘要

In order to model anomalous diffusion in various physical systems, this article presents a collocation technique for solving intricate, multidimensional stochastic integral equation that include fractional Brownian motion. The method utilizes two categories of orthogonal polynomials: second-kind shifted Chebyshev polynomials and shifted Legendre polynomials. It uses Newton-Cotes nodes as collocation points, and the Itô approximation to handle the stochastic integral of the equation. This approach transforms the equation into solvable algebraic systems, which may be either linear or nonlinear. This process effectively decreases computing complexity while preserving accuracy. Furthermore, a thorough convergence study, with meticulous proof, verifies the accuracy of the process. Additionally, four examples demonstrate the effectiveness of the method. A comparison test reveals that results from shifted Chebyshev polynomials are closer to the exact solutions than results from shifted Legendre polynomials. Moreover, a 99% confidence interval for the error mean is constructed, revealing that the approximations maintain high accuracy.