Collection-based numerical method for multi-order fractional integro-differential equations

Document Type : Research Article


1 Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.

2 Federal College of Dental Technology and Therapy, Enugu, Nigeria.

3 Department of Mathematics, Adamawa State University, Mubi, Nigeria.


In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equations using Caputo sense. We obtain the integral form of the problem and transform it into a system of linear alge-braic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method’s uniqueness as well as the convergence of the method. Numerical examples show that the developed method is efficient in problem-solving and competes favorably with the existing method.


Main Subjects

[1] Agbolade, A.O. and Anake, T.A. Solution of first order Volterra lin-ear integro-differential equations by collocation method, J. Appl. Math. (2017), Article ID, 1510267.
[2] Ajileye, G. and Aminu, F.A. Approximate solution to first-order integro-differential equations Using polynomial collocation approach, J. Appl. Computat. Math. 11 (2022), 486.
[3] Ajileye, G., James, A., Abdullahi, A. and Oyedepo, T. Collocation ap-proach for the computational solution of Fredholm-Volterra fractional order of integro-differential equations, J. Niger. Soc. Phys. Sci. (2022), 834–834.
[4] Ghafoor, A., Haq, S., Rasool, A. and Baleanu, D. An efficient numerical algorithm for the study of time fractional Tricomi and Keldysh type equations, Engineering with Computers 38(4) (2022), 3185–3195.
[5] Gülsu, M., Öztürk, Y. and Anapalı, A. Numerical approach for solving fractional Fredholm integro-differential equation, Int. J. Comput. Math. 90(7) (2013), 1413–1434.
[6] Guo, N. and Ma, Y. Numerical algorithm to solve fractional integro-differential equations based on Legendre wavelets method, IAENG Int. J. Appl. Math. 48(2) (2018), 140–145.
[7] Huang, L., Li, X.,-F., Zhao, Y. and Duan, X.-Y. Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl. 62(3) (2011), 1127–1134.
[8] Irandoust-pakchin, S., Kheiri, H. and Abdi-mazraeh, S. Chebyshev car-dinal functions: an effective tool for solving nonlinear Volterra and Fred-holm integro-differential equations of fractional order, Iran. J. Sci. Tech-nol. Trans. A Sci. 37 (2013), 53–62.
[9] Khan, R.H. and Bakodah, H.O. Adomian decomposition method and its modification for nonlinear Abel’s integral equations, Int. J. Math. Anal. (Ruse) 7 (45-48) (2013), 2349–2358.
[10] Li, C. and Wang, Y. Numerical algorithm based on Adomian decompo-sition for fractional differential equations, Comput. Math. Appl. 57(10) (2009), 1672–1681.
[11] Lotfi, A. Dehghan, M. and Yousefi, S.A. A numerical technique for solving fractional optimal control problems, Comput. Math. Appl. 62(3) (2011), 1055–1067.
[12] Ma, Y., Wang, L. and Meng, Z. Numerical algorithm to solve fractional integro-differential equations based on operational matrix of generalized block pulse functions, CMES - Comput. Model. Eng. Sci. 96(1) (2013), 31–47.
[13] Mohammed, D.Sh. Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial, Math. Probl. Eng. (2014), Art. ID 431965, 5 pp.
[14] Nawaz, Y. Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Com-put. Math. Appl. 61(8) (2011), 2330–2341.
[15] Rani, D. and Mishra, V. Solutions of Volterra integral and integro-differential equations using modified Laplace Adomian decomposition method, J. Appl. Math. Stat. Inform. 15(1) (2019), 5–18.
[16] Rostamy, D., Alipour, M., Jafari, H. and Baleanu, D. Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis, Rom. Rep. Phys. 65(2) (2013), 334–349.
[17] Thabet, H., Kendre, S. and Unhale, S. Numerical analysis of iterative fractional partial integro-differential equations, J. Math. (2022), Art. ID 8781186, 14 pp.
[18] Yang, C. and Hou, J. Numerical solution of Volterra integro-differential equations of fractional order by Laplace decomposition method, Interna-tional Journal of Mathematical and Computational Sciences 7(5) (2013), 863–867.
[19] Zhou, Y. Basic theory of fractional differential equations, World Scien-tific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.