Pseudo-spectral matrices as a numerical tool for dealing BVPs, based on Legendre polynomials’ derivatives
M. Abdelhakem, Hanaa Moussa
Abstract
The pseudo-spectral method is used as a technique to employ the first derivative of the well-known Legendre polynomials (FDLs) as novel basis functions. Then, the FDLs Gauss- Lobatto quadrature weights (FDLs-GLQWs) and zeros (FDLs-GLQZs) have been calculated. Consequently, a matrix for differentiation (D-matrix) and another for integration (B-matrix) have been created. To solve several types of ordinary differential problems (ODPs), BVPs, integro–differential equations (IDEs), optimal control problems (OCPs), we designed three algorithms that relied on those matrices. Each algorithm for each type. The convergence of the designed algorithms has been verified theoretically by the error analysis. Finally, the proposed algorithms are applied to several numerical examples.