Spectral collocation algorithm for the fractional Bratu equation via Hexic shifted Chebyshev polynomials

Atta, Ahmed Gamal; Soliman, Jomana Farag; Elsaeed, Elaf Wael; Elsaeed, Mostafa Wael; Youssri, Youssri Hassan

doi:10.22034/cmde.2024.61045.2621

Spectral collocation algorithm for the fractional Bratu equation via Hexic shifted Chebyshev polynomials

Document Type : Research Paper

Authors

¹ Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt.

² Department of Mathematics, Faculty of Science, Galala University, Egypt.

³ Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.

10.22034/cmde.2024.61045.2621

Abstract

This paper offers a numerical collocation scheme for solving the fractional nonlinear Bratu differential equation. We obtain a system of nonlinear equations using our spectral collocation method, which we then solve iteratively using Newton’s method to obtain an approximate solution. Additionally, numerical comparisons are made between the proposed strategy and several numerical strategies documented in various literature. The numerical findings verify the accuracy, computational efficiency, and ease of use of the recommended approach

Keywords

Main Subjects

Fractional ordinary differential equations

References

[1] W. M. Abd-Elhameed, Y. H. Youssri, and A. G. Atta, Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials, J. Math. Model, (2024), 277–299.
[2] W. M. Abd-Elhameed and Y. H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul., 20(2) (2019), 191–203.
[3] W. M. Abd-Elhameed, Y. H. Youssri, and A. G. Atta, Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials, Boundary Value Probl., 2024(1) (2024), 1–24.
[4] A. M. Al-Bugami, M. A. Abdou, and A. Mahdy, Sixth-kind Chebyshev and Bernoulli polynomial numerical methods for solving nonlinear mixed partial integrodifferential equations with continuous kernels, J. Funct. Spaces, (2023).
[5] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, and Y. H. Youssri, A fast Galerkin approach for solving the fractional Rayleigh–Stokes problem via sixth-kind Chebyshev polynomials, Math., 10(11) (2022), 1843.
[6] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, and Y. H. Youssri, Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem, Math. Sci., 17(4) (2023), 415–429.
[7] A. G. Atta, Spectral collocation approach with shifted Chebyshev third-kind series approximation for nonlinear generalized fractional Riccati equation, Int. J. Appl. Comput. Math., 10(2) (2024), 59.
[8] S. Aydınlık and A. Kırı¸s, An efficient method for solving fractional integral and differential equations of Bratu type, TWMS J. Appl. Eng. Math., (2024).
[9] M. Bisheh-Niasar, A. Saadatmandi, and M. Akrami-Arani, A new family of high-order difference schemes for the solution of second order boundary value problems, Iran. J. Math. Chem., 9(3) (2018), 187–199.
[10] V. P. Dubey, R. Kumar, and D. Kumar, Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration, Physica A: Stat. Mech. Appl., 521 (2019), 762–772.
[11] A. Ebrahimzadeh, R. Khanduzi, and Z. Ebrahimzadeh, Coupling Chebyshev Collocation with TLBO to Optimal Control Problem of Reservoir Sedimentation: A Case Study on Golestan Dam, Gonbad Kavous City, Iran, Math. Interdisc. Res., 8(2) (2023), 141–159.
[12] H. Jafari and H. Tajadodi, Electro-spun organic nanofibers elaboration process investigations using BPs operational matrices, Math. Interdisc. Res., 7(1) (2016), 19–27.
[13] A. Khalouta and A. Kadem, Solution of the fractional Bratu-type equation via fractional residual power series method, Tatra Mt. Math. Publ., 76(1) (2020), 127–142.
[14] A. Khalouta, A novel representation of numerical solution for fractional Bratu-type equation, Adv. Stud. EuroTbilisi Math. J., 15(1) (2022), 93–109.
[15] M. Masjed-Jamei, Some new classes of orthogonal polynomials and special functions, University of Kassel, 2006.
[16] M. Moustafa, Y. H. Youssri, and A. G. Atta, Explicit Chebyshev Galerkin scheme for the time-fractional diffusion equation, Int. J. Mod. Phys. C. (IJMPC), 35(01) (2024), 1–15.
[17] O. Odetunde, A. Ajanı, A. Taıwo, and S. Onıtılo, Numerical solution of Bratu-type initial value problems by Aboodh Adomian decomposition method, Cankaya Univ. J. Sci. Eng., 20(2) (2023), 64–75.
[18] P. Pirmohabbati, A. H. Refahi Sheikhani, and A. Abdolahzadeh Ziabari, Numerical solution of nonlinear fractional Bratu equation with hybrid method, Int. J. Appl. Comput. Math., 6 (2020), 1–22.
[19] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[20] M. A. Z. Raja and S. U. I. Ahmad, Numerical treatment for solving one-dimensional Bratu problem using neural networks, Neural Comput. Appl., 24(3) (2014), 549–561.
[21] K. Sadri, D. Amilo, K. Hosseini, E. Hin¸cal, and A. R. Seadawy, A Tau-Gegenbauer spectral approach for systems of fractional integro-differential equations with the error analysis, AIMS Math., 9(2) (2024), 3850–3880.
[22] S. M. Sayed, A. S. Mohamed, E. M. Abo-Eldahab, and Y. H. Youssri, A compact combination of second-kind Chebyshev polynomials for Robin boundary value problems and Bratu-type equations, J.Umm Al-Qura Univ. Appl. Sci., (2024).
[23] J. Shen, T. Tang, and L. Wang, Spectral methods: algorithms, analysis and applications, Springer Science & Business Media, 2011.
[24] H. Singh, A. K. Singh, R. K. Pandey, D. Kumar, and J. Singh, An efficient computational approach for fractional Bratu’s equation arising in electrospinning process, Math. Methods Appl. Sci., 44(13) (2021), 10225–10238.
[25] G. O. Theophilus, A. G. Olusegun, and M. Umaru, K-step block hybrid Nystr¨om-type method for the solution of Bratu problem with impedance boundary condition, Cent. Asian J. Math. Theory Comput. Sci., 5(1) (2024), 34–46.
[26] N. S. Yakusak and E. O. Taiwo, Approximate solution of Bratu differential equation using Falker-type method, Scholar J., 2(1) (2024).
[27] Y. H. Youssri and A. G. Atta, Fej´er-quadrature collocation algorithm for solving fractional integro-differential equations via Fibonacci polynomials, Contemp. Math., (2024), 296–308.
[28] F. Yin and Y. Fu, Explicit high accuracy energy-preserving Lie-group sine pseudo-spectral methods for the coupled nonlinear Schrodinger equation, Appl. Numer. Math., 195 (2024), 1–16.
[29] X. Zhao, L. L. Wang, and Z. Xie, Sharp error bounds for Jacobi expansions and Gegenbauer–Gauss quadrature of analytic functions, SIAM J. Numer. Anal., 51(3) (2013), 1443–1469.
[30] W. Zheng, Y. Chen, and J. Zhou, A Legendre spectral method for multidimensional partial Volterra integrodifferential equations, J. Comput. Appl. Math., 436 (2024), 115302.