Authors : John Chuseh Ahmadu; Ayinde Muhammed Abdullahi; Onyeozili Ijeoma Abigail

Volume/Issue : Volume 11 - 2026, Issue 9 - September

Google Scholar : https://tinyurl.com/5h93abvd

DOI : https://doi.org/10.38124/ijisrt/26sep430

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Abstract : This article develops a direct power-series collocation method for smooth multi-term Caputo initial-value problems with variable coefficients. The contribution is a basis-level fractionalintegral construction that converts the differential model into an augmented algebraic system without first deriving a problem-specific operational matrix. Application of the Riemann–Liouville integral exposes the classical initial data, while gamma- and beta-function identities provide closed expressions for the action of the transformed operator on monomials. Sufficient conditions are established for equivalence of the differential and integral formulations, uniqueness through a contraction argument, conditional convergence of the collocation approximation, and residualbased a posteriori control. The resulting algorithm is implemented symbolically in MAPLE.

Keywords : Caputo Fractional Derivative; Multi-Term Fractional Equation; Power-Series Collocation; Variable Coefficients; Convergence Analysis; MAPLE.

References :

1. Diethelm, K. (2010). The analysis of fractional differential equations. Springer. https://doi.org/10.1007/978-3-642-14574-2
2. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier.
3. Huang, L., Li, X.-F., Zhao, Y., & Duan, X.-Y. (2011). Approximate solution of fractional integro-differential equations by Taylor expansion method. Computers & Mathematics with Applications, 62(3), 1127–1134. https://doi.org/10.1016/j.camwa.2011.03.037
4. Wu, C., & Wang, Z. (2022). The spectral collocation method for solving a fractional integrodifferential equation. AIMS Mathematics, 7(6), 9577–9587. https://doi.org/10.3934/math.2022532
5. Wang, C., & Chen, B. (2023). An hp-version spectral collocation method for fractional Volterra integro-differential equations with weakly singular kernels. AIMS Mathematics, 8(8), 19816– 19841. https://doi.org/10.3934/math.20231010
6. Kang, Y.-S., & Jo, S.-H. (2024). Spectral collocation method for solving multi-term fractional integro-differential equations with nonlinear integral. Mathematical Sciences, 18, 91–106. https://doi.org/10.1007/s40096-022-00487-9
7. Tedjani, A. H., Amin, A. Z., Abdel-Aty, A.-H., Abdelkawy, M. A., & Mahmoud, M. (2024). Legendre spectral collocation method for solving nonlinear fractional Fredholm integrodifferential equations with convergence analysis. AIMS Mathematics, 9(4), 7973–8000. https://doi.org/10.3934/math.2024388
8. Hamood, M. M., Sharif, A. A., & Ghadle, K. P. (2025). A numerical approach to fractional Volterra–Fredholm integro-differential problems using shifted Chebyshev spectral collocation. Scientific Reports, 15, 29678. https://doi.org/10.1038/s41598-025-13732-7
9. Saldır, O. (2025). Numerical solution of fractional integro-differential equations with non-local conditions using reproducing kernel method based on Chebyshev polynomials. Journal of Applied Mathematics and Computing, 71(Suppl 1), 1403–1431. https://doi.org/10.1007/s12190-02502518-9
10. Ahmed, A. I. (2026). A numerical method based on shifted fractional-order Chelyshkov functions for fractional delay integro-differential equations. Numerical Algorithms. https://doi.org/10.1007/s11075-026-02397-6
11. Nguyen, H. T., Nguyen, H. C., Wang, R., & Zhou, Y. (2021). Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 26(12), 6483–6510. https://doi.org/10.3934/dcdsb.2021030
12. Zhou, Y. (2014). Basic theory of fractional differential equations. World Scientific. https://doi.org/10.1142/9069
13. Rostamy, D., Alipour, M., Jafari, H., & Baleanu, D. (2013). Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis. Romanian Reports in Physics, 65(2), 334–349.
14. Uwaheren, O. A., Adebisi, A. F., & Taiwo, O. A. (2020). Perturbed collocation method for solving singular multi-order fractional differential equations of Lane–Emden type. Journal of theNigerian Society of Physical Sciences, 2(3), 141–148. https://doi.org/10.46481/jnsps.2020.69