Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems

Dina Abdelhamid; Wedad Albalawi; Kottakkaran Sooppy Nisar; A. Abdel-Aty; Suliman Alsaeed; M. Abdelhakem; Dina Abdelhamid; Wedad Albalawi; Kottakkaran Sooppy Nisar; A. Abdel-Aty; Suliman Alsaeed; M. Abdelhakem

doi:10.3934/math.20231255

AIMS Mathematics

2023, Volume 8, Issue 10: 24609-24631. doi: 10.3934/math.20231255

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Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems

1.
Basic Science Department, Faculty of Engineering, May University in Cairo, Cairo, Egypt
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al Kharj 11942, Saudi Arabia
4.
Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
5.
Applied Sciences College, Department of Mathematical Sciences, Umm Al-Qura University P.O. Box 715, Makkah 21955, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
7.
Basic Science Department, School of Engineering, Canadian International College (CIC), New Cairo, Egypt
8.
Helwan School of Numerical Analysis in Egypt (HSNAE), Egypt

Received: 03 March 2023 Revised: 20 July 2023 Accepted: 06 August 2023 Published: 21 August 2023
MSC : 33C45, 34A12, 65L60

A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices.
- Chebyshev polynomials,
- Legendre polynomials,
- pseudospectral differential matrices,
- error analysis,
- IBVPs,
- Ricatti equation
Citation: Dina Abdelhamid, Wedad Albalawi, Kottakkaran Sooppy Nisar, A. Abdel-Aty, Suliman Alsaeed, M. Abdelhakem. Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems[J]. AIMS Mathematics, 2023, 8(10): 24609-24631. doi: 10.3934/math.20231255

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Abstract

A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices.

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