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Haddam, Qassim, Najafi, Esmaeil, Sohrabi, Saeed. (1404). Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases. سامانه مدیریت نشریات علمی, 13(3), 742-757. doi: 10.22034/cmde.2024.63023.2800
Qassim Hadi Haddam; Esmaeil Najafi; Saeed Sohrabi. "Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases". سامانه مدیریت نشریات علمی, 13, 3, 1404, 742-757. doi: 10.22034/cmde.2024.63023.2800
Haddam, Qassim, Najafi, Esmaeil, Sohrabi, Saeed. (1404). 'Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases', سامانه مدیریت نشریات علمی, 13(3), pp. 742-757. doi: 10.22034/cmde.2024.63023.2800
Haddam, Qassim, Najafi, Esmaeil, Sohrabi, Saeed. Jacobi collocation method for numerical solution of nonlinear weakly singular Volterra integro-differential equations: fractional and stochastic cases. سامانه مدیریت نشریات علمی, 1404; 13(3): 742-757. doi: 10.22034/cmde.2024.63023.2800

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

Computational Methods for Differential Equations
مقاله 2، دوره 13، شماره 3، مهر 2025، صفحه 742-757    اصل مقاله (493.6 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.63023.2800
نویسندگان
Qassim Hadi Haddam؛ Esmaeil Najafi* ؛ Saeed Sohrabi
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
چکیده
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. 
کلیدواژه‌ها
Fractional calculus؛ Multi-term Volterra integro-differential equations؛ Jacobi collocation method؛ Smoothing transformation؛ Sum of exponentials approximation؛ Stochastic
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