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

Articles in Press

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 Jacobi collocation method based
on the orthogonal polynomials. Since the solution of the proposed equation is not smooth
enough in the origin, thus the idea of the smoothing transformation is used on the equation
to incearse the smoothness of the solution. We represent an operator-based discussion of
smoothing transformation and Gauss-Jacobi quadrature for Riemann-Liouville integral op-
erators 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 ex-
amples are solved by the proposed method and the obtained error results are in accordance
with the convergence analysis of the method. Finally we brought an example regarding the
stochastic Volterra integro-differential equations of one singular kernel function.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 08 October 2024

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History

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

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