Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method
Gamal, M., El-Kady, M., Abdelhakem, M. (2024). Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method. EKB Journal Management System, 2(1), -. doi: 10.21608/abas.2023.238304.1031
Marwa Gamal; Mamdouh El-Kady; Mohamed A Abdelhakem. "Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method". EKB Journal Management System, 2, 1, 2024, -. doi: 10.21608/abas.2023.238304.1031
Gamal, M., El-Kady, M., Abdelhakem, M. (2024). 'Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method', EKB Journal Management System, 2(1), pp. -. doi: 10.21608/abas.2023.238304.1031
Gamal, M., El-Kady, M., Abdelhakem, M. Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method. EKB Journal Management System, 2024; 2(1): -. doi: 10.21608/abas.2023.238304.1031

Solving some types of ordinary differential equations by using Chebyshev derivatives direct residual spectral method

Advances in Basic and Applied Sciences
Article 1, Volume 2, Issue 1, January 2024  XML PDF (520.85 K)
Document Type: Original Article
DOI: 10.21608/abas.2023.238304.1031
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Authors
Marwa Gamalorcid 1; Mamdouh El-Kadyorcid 2, 3, 4; Mohamed A Abdelhakem email orcid 2, 3, 4
1Basic Science Department, School of Engineering, May University in Cairo (MUC), Cairo, Egypt
2Mathematics Department, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt,
3Basic Science Department, School of Engineering, Canadian Intentional College, New Cairo, Egypt
4Helwan School of Numerical Analysis in Egypt (HSNAE), Egypt
Abstract
Herein, novel basis orthogonal polynomials have developed. These developed polynomials have been used to find the approximation solutions for some types of linear and nonlinear ordinary differential equations by direct numerical method. This numerical method depends on the Chebyshev polynomials' derivatives. We shall present these solutions in the form of a finite sum of the Chebyshev polynomials' derivatives and unknown coefficients involving these polynomials. By substituting into the differential equation, the given differential equation will be converted into a system of algebraic equations. The obtained algebraic system can be solved easily to get the values of the spectral expansion constant. In addition, an algorithm for the approximated process has been designed to be easily used in the coding process. Consequently, some ordinary differential equations have been solved via the introduced Chebyshev polynomials' derivatives. Finally, the approximated solutions have been compared with exact and other methods solutions to illustrate the efficiency and accuracy of the used method.
Keywords
Chebyshev polynomials' derivatives; Spectral method; Bratu equation; Lane-Emden equation
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