A numerical algorithm based on Jacobi polynomials for FIDEs with error estimation

Document Type : Research Article

Authors

• K. Sadri ^{1, 2}
• D. Amilo ^{1, 2}
• E. Hincal ^{1, 3, 4}

¹ Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey.,

² Faculty of Art and Science, University of Kyrenia, Kyrenia, TRNC, Mersin 10, Turkey.

³ Faculty of Art and Science, University of Kyrenia, Kyrenia, TRNC, Mersin 10, Turkey.,

⁴ Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan

10.22067/ijnao.2025.92092.1596

Abstract

This study aims to address a specific class of mathematical problems known as fractional integro-differential equations. These equations are used to model various phenomena„ including heat conduction in materials with memory, damping laws, diffusion processes, earthquake models, fluid dy-namics, traffic flow, and continuum mechanics. This research focuses on problems where the fractional derivative operator is defined in the Caputo sense. Our proposed methodology employs an operational approach based on the use of shifted Jacobi polynomials. We derive operational matrices for fractional integration and product, which are then applied to approx-imate solutions for both linear and nonlinear problems. By using these matrices in conjunction with the collocation method, we transform the orig-inal problem into a system of algebraic equations. Notably, our approach is simpler and more cost-effective compared to established methods such as Adomian decomposition, Homotopy perturbation, Sinc-collocation, and Legendre wavelet techniques. We provide several illustrative examples to validate our method’s effectiveness and reliability. Additionally, we present theorems concerning the existence of a unique solution and the convergence of our proposed approach.

Keywords

• Fractional integro-differential equation
• Caputo derivative operator
• Jacobi polynomials
• Operational matrices
• Convergence

Main Subjects

• Numerical methods for integral equations and its applications

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References

[1] Abbasbandy, S. On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaes. Math. 36 (2013), 93–105.
[2] Abdelkawy, M.A., Lopes, A.M. and Babatin, M.M. Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order, Chaos. Soliton. Frac. 134 (2020), 109721.
[3] Abo-Gabal, H., Zaky, M.A. and Doha, E.H. Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions, Appl. Num. Math. 182 (2022), 214–234.
[4] Ahmad, W.M. and El–Khazali, R. Fractional-order dynamical models of love, Chaos. Solit. Frac. 33 (2007), 1367–1375.
[5] Babaei, A., Jafari, H. and Banihashemi, S. Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J. Comput. Appl. Math. 377 (2020), 112908.
[6] Bhrawy, A.H., Tharwat, M.M. and Yildirim, A. A new formula for frac-tional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations, App. Math. Modelling. 37 (2013), 4245–4252.

[7] Biazar, J. and Sadri, K. Solution of weakly singular fractional integro-differential equations by using a new operational approach, J. Comput. Appl. Math. 352 (2019), 453–477.
[8] Chen, Y. and Tang, T. Convergence analysis of the Jacobi spectral col-location methods for Volterra integral equations with a weakly singular kernel, Math. Comput. 79 (2010), 147–167.
[9] Darweesh, A., Al-Khaled, K. and Al-Yaqeen, O.A. Haar wavelets method for solving class of coupled systems of linear fractional Fredholm integro-differential equations, Heliyon. 9(9) (2023), e19717.
[10] Dehestani, H., Ordokhani, Y. and Razzaghi, M. Combination of Lu-cas wavelets with Legendre–Gauss quadrature for fractional Fredholm–Volterra integro-differential equations, J. Comput. Appl. Math. 382 (2021), 113070.
[11] Dehghan, M. and Fakhar-Izadi, F. The spectral collocation method with three different bases for solving a nonlinear partial differential equa-tion arising in modeling of nonlinear waves, Math. Comput. Model. 53 (2011), 1865–1877.
[12] Diethelm, A. and Free, D. On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: Scien-tific Computing in Chemical Engineering II: Computational Fluid Dy-namics, Reaction Engineering, and Molecular Properties (Eds. F. Keil, W. Mackens, H. Vob and J. Werther), Springer Verlag, Heidelberg, Pp.
217–224, 1999.
[13] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. A new Jacobi opera-tional matrix: An application for solving fractional differential equations, App. Math. Modelling. 36 (2012), 4931–4943.
[14] Ebrahimi, H. and Sadri, K. An operational approach for solving frac-tional pantograph differential equation, Iranian J. Numer. Anal. Opt. 9(1) (2019), 37–68.

[15] Elbeleze, A.A., Kilicman, A. and Taib, B.M. Approximate solution of integro-differential equation of fractional (arbitrary) order, J. King Saud. Univ. Sci. 28(1) (2016), 61–68.
[16] Ghazanfari, B., Ghazanfari, A.G. and Veisi, F. Homotopy Perturbation method for the nonlinear fractional integro-differential equations, Aust. J. Basic. App. Sci. 4(12) (2010), 5823–5829.
[17] Guo, B.Y. and Wang, L.L. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory. 128 (2004), 1–41.
[18] He, J.H. Nonlinear oscillation with fractional derivative and its appli-cations, In: International Conference on Vibrating Engineering, Dalian, China, pp. 288–291, 1998.
[19] He, J.H. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15 (1999), 86–90.
[20] Heydari, M.H., Zhagharian, Sh. and Razzaghi, M. Jacobi polynomi-als for the numerical solution of multi-dimensional stochastic multi-order time fractional diffusion-wave equations, Comput. Math. Appl. 152 (2023), 91–101.
[21] Hosseininia, M., Heydari, M.H. and Avazzadeh, Z. Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation, Chaos. Soliton. Frac. 155 (2022), 111729.
[22] Huang, L., Li, X.F., Zhaoa, Y. and Duan, X.Y. Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. App. 62 (2011), 1127–1134.
[23] Keskin, Y., Karaoglu, O., Servi, S. and Oturan, G. The approximate solution of high order linear fractional differential equations with variable coefficients in term of generalized Taylor series, Math. Comput. App. 16 (2011), 617–629.
[24] Khader, M.M. and Sweilam, N.H. On the approximate solutions for sys-tem of fractional integro-differential equations using Chebyshev pseudo-spectral method, Appl. Math. Modelling. 37(24) (2013), 9819–9828.

[25] Kumar, K., Pandey, R.K. and Sharma, S. Comparative study of three numerical schemes for fractional integro-Differential equations, J. Com-put. App. Math. 315 (2017), 287–302.
[26] Li, X.J. and Xu, C.J. A space-time spectral method for the time fractional diffusion equation, SIAM. J. Numer. Anal. 47(3) (2009), 2108–2131.
[27] Mainardi, F. Fractals and Fractional Calculus Continuum Mechanics, Springer Verlag, Pp, 291–348, 1997.
[28] Meng, Z., Wang, L., Li, H. and Zhang, W. Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math. 92(6) (2015), 1275–1291.
[29] Mokhtary, P. Discrete Galerkin method for fractional integro-differential equations, Acta. Math. Sci. 36B(2) (2016), 560–578.
[30] Momani, S. and Noor, M.A. Numerical methods for fourth-order frac-tional integro-differential equations, App. Math. Comput. 182 (2006), 754–760.
[31] Nazari Susahab, D., Shahmorad, S. and Jahanshahi, M. Efficient quadrature rules for solving nonlinear fractional integro-differential equa-tions of the Hammerstein type, App. Math. Modelling. 39(18) (2015), 5452–5458.
[32] Rahimkhani, P. and Ordokhani, Y. A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional par-tial differential equations with Dirichlet boundary conditions. Numer. Methods Partial Differ. Equ. 35(1) (2019), 34–59.
[33] Rahimkhani, P., Ordokhani, Y. and Babolian, E. An efficient approxi-mate method for solving delay fractional optimal control problems, Non-linear. Dyn. 86 (2016), 1649—1661.
[34] Rahimkhani, P., Ordokhani, Y. and Babolian, E. A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer. Algor. 74 (2017), 223—245.

[35] Rahimkhani, P., Ordokhani, Y. and Babolian, E. Numerical solu-tion of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math. 309 (2017), 493–510.
[36] Rahimkhani, P., Ordokhani, Y. and Babolian, E. Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations. Appl. Numer. Math. 122 (2017), 66–81.
[37] Rahimkhani, P., Ordokhani, Y. and Babolian, E. Muntz–Legendre wavelet operational matrix of fractional-order integration and its ap-plications for solving the fractional pantograph differential equations, Numer. Algor. 77 (2018), 1283—1305.
[38] Sadek, L. A cotangent fractional derivative with the application, Fractal Fract. 7(6) (2023), 444.
[39] Sadek, L., Baleanu, D., Abdo, M.S. and Shatanawi, W. Introducing novel Θ-fractional operators: Advances in fractional calculus, J. King Saud University - Sci. 36(9) (2024), 103352.
[40] Sadek, L. and Bataineh, A.S. The general Bernstein function: Appli-cation to χ-fractional differential equations, Math. Method. Appl. Sci. 47(7), (2024) 6117–6142.
[41] Sadek, L., Bataineh, A.S., Isik, O.R., Alaoui, H.T. and Hashim, I. A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations, Math. Comput. Simul. 212 (2023), 475–488.
[42] Sadek, L. and Lazar, T.A. On Hilfer cotangent fractional derivative and a particular class of fractional problems, AIMS Mathematics. 8(12) (2023), 28334–28352.
[43] Sadek, L., Ounamane, S., Abouzaid, B. and Sadek, E.M. The Galerkin Bell method to solve the fractional optimal control problems with inequal-ity constraints, J. Comput. Sci. 77 (2024), 102244.

[44] Sadek, L. and Algefary, A. On quantum trigonometric fractional calculus, Alex. Eng. J. 120 (2025), 371–377.
[45] Sadri, K., Amini, A. and Cheng, C. Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials, J. Comput. App. Math. 319 (2017), 493–513.
[46] Sadri, K., Amini, A. and Cheng, C. A new numerical method for delay and advanced integro-differential equations, Numer. Algor. 77 (2018), 381–412.
[47] Szego, G. Orthogonal polynomials, American Mathematical Society, Providence, Rhode Island, 1939.
[48] Tavares, D., Almeida, R. and Torres, D.F.M. Caputo derivatives of frac-tional variable order: Numerical approximations, Commun. Nonlinear Sci. Numer. Simul. 35 (2016), 69–87.
[49] Varol Bayram, D. and Daşcıoğlu, A. A method for fractional Volterra integro-differential equations by Laguerre polynomials, Adv. Differ. Equ. 2018 (2018), 466.
[50] Wang, J., Xu, T.Z., Wei, Y.Q. and Xie, J.Q. Numerical simulation for coupled systems of nonlinear fractional order integro-differential equa-tions via wavelets method, Appl. Math. Comput. 324 (2018), 36–50.
[51] Yaghoubi, S., Aminikhah, H. and Sadri, K. A new efficient method for solving yystem of weakly singular fractional integro-differential equa-tions by shifted sixth-kind Chebyshev polynomials, J. Math. 2022 (2022), 9087359.
[52] Yaghoubi, S., Aminikhah, H. and Sadri, K. A spectral shifted Gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis, Sadhana. 48 (2023), 213.
[53] Yang, Y. Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations, Acta Math. Sci. 34(3) (2014), 673–690.