| Shifted Legendre polynomials' derivatives for approximating some types of differential equations |
| Paper ID : 1031-ISCH |
| Authors |
|
Taha Abdelnaby Fahmy *1, Kamal Raslan Mohamed2, Mohamed Ahmed Abdelhakem3 1Teaching assistant at the Higher Institute of Engineering in 15 May City 2Mathematics Department, Faculty of Science, Al Azhar university 3Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt |
| Abstract |
| In the presented paper, the pseudo-spectral method is used to solve differential equations to employ shifted Legendre polynomials' derivatives (SLPs-FD) of the well-known Legendre polynomials as a novel basis function. These orthogonal polynomials are, namely, shifted Legendre polynomials' derivatives. Some essential relations are generated for these proposed polynomials. Shifted Legendre polynomials' derivatives Gauss-Lobatto quadrature weights and zeros have been calculated. The proposed approach differs from other numerical techniques as it is based on a differentiation matrix. Consequently, a matrix for differentiation (D- matrix) of shifted Legendre polynomials is constructed. The resulting matrix equation can be solved, and approximately the unknown shifted Legendre polynomials' derivatives coefficients can be found. Finally, the presented method has approximated some integer-order boundary value problems. 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. |
| Keywords |
| Differential Equations, Spectral methods, pseudo-Spectral methods, Legendre polynomials |
| Status: Abstract Accepted (Poster Presentation) |