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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
مراجع
  • [1] A. Aghajani, Y. Jalilian, and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fractional Calculus and Applied Analysis, 15, 44–69.
  • [2] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, Solitons & Fractals, 40(2) (2009), 521–529.
  • [3] R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210.
  • [4] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, World Scientific Publishing Co. Pte. Ltd., Singapore, (2012).
  • [5] G. Ben-Yu and L. L. Wang, Jacobi interpolation approximations and their applications to singular differential equations, Advances in Computational Mathematics, 14(3) (2001), 227–276.
  • [6] J. Biazar and K. Sadri, Solution of weakly singular fractional integro-differential equations by using a new operational approach, Journal of Computational and Applied Mathematics, 352 (2019), 453-477.
  • [7] H. Brunner, Volterra Integral Equations; An Introduction to Theory and Applications, Cambridge University Press, (2017).
  • [8] A. Cardone M. Donatelli, F. Durastante, R. Garrappa, M. Mazza, and M. Popolizio, Fractional differential equations: modeling, discretization, and numerical solvers, Springer INdAM Series 50, Springer Nature Singapore Pte Ltd., (2023).
  • [9] K. Diethelm, The Analysis of Fractional Differential Equations, an Application Oriented, Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer, Berlin, (2010).
  • [10] S. Das, Kindergarten of Fractional Calculus, Cambridge Scholars Publishing, (2020).
  • [11] O. Farkhondeh Rouz, S. Shahmorad, and D. Ahmadian, Double weakly singular kernels in stochastic Volterra integral equations with application to the rough Heston model, Appl. Math. Comput., 475 (2024), 128720.
  • [12] Q. H. Haddam, E. Najafi, and S. Sohabi, Numerical solution of nonlinear multi-term weakly singular fractional Volterra integro-differential equations using Jacobi collocation method, (2024), PREPRINT (Version 1) available at Research Square.
  • [13] S. Jiang, J. Zhang, Q. Zhang, and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678.
  • [14] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, (2010).
  • [15] P. Mokhtary, Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integrodifferential equations, Applied Numerical Mathematics, 121 (2017), 52–67.
  • [16] E. Najafi, Nystr¨om-quasilinearization method and smoothing transformation for the numerical solution of nonlinear weakly singular Fredholm integral equations, Journal of Computational and Applied Mathematics, 368 (2020), 112538.
  • [17] E. Najafi, Smoothing transformation for numerical solution of nonlinear weakly singular Volterra integral equations using quasilinearization and product integration methods, Applied Numerical Mathematics, 153 (2020), 540–557.
  • [18] A. Pedas, E. Tamme, and M. Vikerpuur, Spline collocation for fractional weakly singular integro-differential equations, Applied Numerical Mathematics, 110 (2016), 204–214.
  • [19] A. Pedas, E. Tamme, and M. Vikerpuur, Numerical solution of linear fractional weakly singular integro-differential equations with integral boundary conditions, Applied Numerical Mathematics, 149 (2020), 124-140.
  • [20] A. Pedas and M. Vikerpuur, Spline collocation for multi-term fractional integro-differential equations with weakly singular kernels, Fractal and Fractional, 5(3) (2021), 90.
  • [21] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Ukraine, Elsevier Science, (1998).
  • [22] T. Rezazadeh and E. Najafi, Jacobi collocation method and smoothing transformation for numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations, Applied Numerical Mathematics, 181 (2022), 135–150.
  • [23] E. Shishkina and S. Sitnik, Transmutations Singular and Fractional Differential Equations with Applications to Mathematical Physics, London: Academic Press, (2020).
  • [24] M. Vikerpuur, Two Collocation Type Methods for Fractional Differential Equations with Non-Local Boundary Conditions, Mathematical Modelling and Analysis, 22(5) (2017), 654-670.
  • [25] S. Xiang, On interpolation approximation: convergence rates for polynomial interpolation for functions of limited regularity, SIAM Journal on Numerical Analysis, 54(4) (2016), 2081–2113.
  • [26] M. Yi, L. Wang, and J. Huang, Legendre wavelets method for the numerical solution of fractional integrodifferential equations with weakly singular kernel, Applied Mathematical Modelling, 40(4) (2016), 3422–3437.
  • [27] J. Zhao, J. Xiao, and N. J Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algor., 65 (2014), 723–743.
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