Spectral matrix computational Tau approach for fractional differential equations via fifth-kind chebyshev polynomials
Abstract
This paper presents Tau approach for solving fractional differential equations (FDEs) via shifted Chebyshev polynomials of the fifth kind. By leveraging the unique properties of these polynomials, we develop operational matrices that facilitate the approximation of solutions to both linear and nonlinear FDEs. The proposed method employs a tau technique in the matrix form to transform the problem into a solvable algebraic system, ensuring computational efficiency and accuracy. This work presents a rigorous convergence analysis and demonstrates the efficacy of the proposed approach through a series of illustrative examples, showcasing a marked improvement in solution precision relative to conventional methodologies. This research contributes to the growing of work in fractional calculus and offers a robust tool for researchers and practitioners in applied mathematics and engineering.
Downloads
References
Iqbal, M. A., Miah, M. M., Ali, H. S., Shahen, N. H. M., & Deifalla, A. (2024). New applications of the fractional derivative to extract abundant soliton solutions of the fractional order PDEs in mathematics physics. Partial Differential Equations in Applied Mathematics, 9, 100597.
Cui, P., & Jassim, H. K. (2024). Local fractional Sumudu decomposition method to solve fractal PDEs arising in mathematical physics. Fractals, 32(04), 2440029.
Adel, W., Elsonbaty, A., & Mahdy, A. M. S. (2024). On some recent advances in fractional order modeling in engineering and science. Computation and Modeling for Fractional Order Systems, 169-197.
Song, L., Yu, W., Tan, Y., & Duan, K. (2024). Calculations of fractional derivative option pricing models based on neural network. Journal of Computational and Applied Mathematics, 437, 115462.
Gao, X. L., Li, Z. Y., & Wang, Y. L. (2024). Chaotic dynamic behavior of a fractional-order financial system with constant inelastic demand. International Journal of Bifurcation and Chaos, 34(09), 2450111.
Podlubny, I. (1999). Fractional Differential Equations. Academic Press.
Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier.
Agarwal, R., & Purohit, S. D. (2024). Modeling Calcium Signaling: A Fractional Perspective. Springer Nature.
Boyd, J. P. (2001). Chebyshev and Fourier Spectral Methods (2nd ed.). Dover Publications.
Shen, J., Tang, T., & Wang, L. (2011). Spectral Methods: Algorithms, Analysis, and Applications. Springer.
Gotz, M., Hein, J., & Pannek, J. (2012). Chebyshev spectral methods for fractional differential equations. SIAM Journal on Numerical Analysis, 50(6), 2866-2888.
Sweilam, N. H., Nagy, A. M., & El-Sayed, A. A. (2015). Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation. Chaos, Solitons & Fractals, 73, 141-147.
Abd-Elhameed, W. M., & Alkenedri, A. M. (2021). Spectral solutions of linear and nonlinear BVPs using certain Jacobi polynomials generalizing third-and fourth-kinds of Chebyshev polynomials. Computer Modeling in Engineering & Sciences, 126(3), 955-989.
Abdelkawy, M. A., Lopes, A. M., & Babatin, M. M. (2020). Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order. Chaos, Solitons & Fractals, 134, 109721.
Heydari, M. H., Avazzadeh, Z., & Atangana, A. (2021). Shifted Jacobi polynomials for nonlinear singular variableorder time fractional Emden–Fowler equation generated by derivative with non-singular kernel. Advances in Difference Equations, 2021, 1-15.
Masjed-Jamei, M. (2006). Some new classes of orthogonal polynomials and special functions: A symmetric generalization of Sturm-Liouville problems and its consequences. Department of Mathematics, University of Kassel.
Obeid, M., Abd El Salam, M. A., & Mohamed, M. S. (2023). A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel. AIMS Mathematics, 8(7), 16724-16747.
Cao, Y., Zhong, W., & Li, X. (2017). Numerical solutions of nonlinear fractional differential equations using Chebyshev polynomials of the fifth kind. Computers & Mathematics with Applications, 73(7), 1465-1480.
Zheng, X., Liu, Z., & Zhang, Q. (2020). Numerical solution of fractional differential equations with generalized Chebyshev polynomials of the fifth kind. Mathematics and Computers in Simulation, 177, 194-208.
Abd-Elhameed, W. M., Alkhamisi, S. O., Amin, A. K., & Youssri, Y. H. (2023). Numerical contrivance for Kawaharatype differential equations based on fifth-kind Chebyshev polynomials. Symmetry, 15(1), 138.
Ali, K. K., Abd El Salam, M. A., Mohamed, E. M., Samet, B., Kumar, S., & Osman, M. S. (2020). Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Advances in Difference Equations, 2020, 1-23.
Abd El Salam, M. A., Ramadan, M. A., Nassar, M. A., Agarwal, P., & Chu, Y. M. (2021). Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations. Advances in Difference Equations, 2021, 1-17.
Abd-Elhameed, W. M., & Alkhamisi, S. O. (2021). New results of the fifth-kind orthogonal Chebyshev polynomials. Symmetry, 13(12), 2407.
Ali, K. K., Abd El Salam, M. A., & Mohamed, M. S. (2022). Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations. AIMS Math, 7(5), 7759-7780.
Bagley, R. L., & Torvik, P. J. (1983). A model for viscoelastic behavior based on fractional calculus. Journal of Rheology, 27(3), 201-210.
Oliveira, D. S., & Capelas de Oliveira, E. (2019). On a Caputo-type fractional derivative. Advances in Pure and Applied Mathematics, 10(2), 81-91.
Teodoro, G. S., Machado, J. T., & De Oliveira, E. C. (2019). A review of definitions of fractional derivatives and other operators. Journal of Computational Physics, 388, 195-208.
Abd-Elhameed, W. M., & Youssri, Y. (2018). Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. Computational and Applied Mathematics, 37, 2897-2921.
Abd-Elhameed, W. M., & Youssri, Y. (2019). Sixth-kind Chebyshev spectral approach for solving fractional differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 20(2), 191-203.
Obeid, M., Abd El Salam, M. A., & Younis, J. A. (2023). Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials. Applied Mathematics in Science and Engineering, 31(1), 2187388.
Ganji, R. M., Jafari, H., & Baleanu, D. (2020). A new approach for solving multi variable orders differential equations with Mittag–Leffler kernel. Chaos, Solitons & Fractals, 130, 109405.
Osler, T. J. (1971). Taylor’s series generalized for fractional derivatives and applications. SIAM Journal on Mathematical Analysis, 2(1), 37-48.
Bhrawy, A. H., Ezz-Eldien, S. S., Doha, E. H., Abdelkawy, M. A., & Baleanu, D. (2017). Solving fractional optimal control problems within a Chebyshev–Legendre operational technique. International Journal of Control, 90(6), 1230–1244.
Abdelhakem, M., Baleanu, D., Agarwal, P., & Moussa, H. (2023). Approximating system of ordinary differentialalgebraic equations via derivative of Legendre polynomials operational matrices. International Journal of Modern Physics C, 34(03), 2350036.
Ali, K. K., Abd El salam, M. A., & Mohamed, E. M. (2021). A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument. International Journal of Nonlinear Sciences and Numerical Simulation, 22(1), 83-91.
Srivastava, H. M., Shah, F. A., & Abass, R. (2019). An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation. Russian Journal of Mathematical Physics, 26(1), 77-93.
Zakian, V. (1969). Numerical inversion of Laplace transform. Electronics Letters, 5(6), 120-121.
Weideman, J. A. C. (2019). Gauss–Hermite quadrature for the Bromwich integral. SIAM Journal on Numerical Analysis, 57(5), 2200-2216.
Dubner, H., & Abate, J. (1968). Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. Journal of the ACM (JACM), 15(1), 115-123.
Ahmed, H. M. (2024). Enhanced shifted Jacobi operational matrices of integrals: spectral algorithm for solving some types of ordinary and fractional differential equations. Boundary Value Problems, 2024(1), 75.
Youssri, Y. H. (2017). A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation. Advances in Difference Equations, 2017, 1-17.
Doha, E. H., Bhrawy, A. H., & Ezz-Eldien, S. S. (2011). Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Applied Mathematical Modelling, 35(12), 5662-5672.
Arqub, O. A., & Al-Smadi, M. (2018). Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlev´e equations in Hilbert space. Chaos, Solitons , Fractals, 117, 161-167.
Abu Arqub, O., & Maayah, B. (2018). Solutions of Bagley–Torvik and Painlev´e equations of fractional order using iterative reproducing kernel algorithm with error estimates. Neural Computing and Applications, 29, 1465-1479.
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática

This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).