Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.

Abstract

This paper deals with the numerical solution of a class of nonlinear multi-term weakly singular fractional Volterra integro- differential equations by the Jacobi collocation method based on Jacobi orthogonal polynomials. Since the solution of the proposed equation is not smooth enough at the origin, the idea of a smoothing transformation is used to increase the smoothness of the solution. We represent an operator-based discussion of the smoothing transformation and Gauss Jacobi quadrature for Riemann-Liouville integral operators and weakly singular integral operators using their similar constructions and extend it to the error analysis of the proposed method and obtain an error bound for the discrete  collocation solution. In addition, we propose an improved stochastic method, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the proposed method. To test the efficiency and accuracy, various numerical examples are solved by the proposed method and the obtained error results are in accordance with the convergence analysis of the method. Finally, we present an example regarding the stochastic Volterra integro-differential equations with one singular kernel function. 

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References

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Computational Methods for Differential Equations
Volume 13, Issue 3
July 2025
Pages 742-757

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History

  • Receive Date: 19 August 2024
  • Revise Date: 19 September 2024
  • Accept Date: 07 October 2024

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