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Legendre polynomials' first derivatives have been used as the basis function via the pseudo-Galerkin spectral method. Operational matrices for derivatives have been used and extended to deal with the system of ordinary differential-algebraic equations. An algorithm via those matrices has been designed. The accuracy and efficiency of the proposed algorithm had been shown by two techniques, theoretically, via the boundedness of the approximated expansion and numerically through numerical examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This research aims to assemble two methodical spectral Legendre's derivative algorithms to numerically attack the Lane-Emden, Bratu's, and singularly perturbed type equations. We discretize the exact unknown solution as a truncated series of Legendre's derivative polynomials. Then, via tau and collocation methods for linear and nonlinear problems, respectively, we obtain linear/nonlinear systems of algebraic equations in the unknown expansion coefficients. Finally, with the aid of the Gaussian elimination technique in the linear case and Newton's iterative method for the non-linear case - with vanishing initial guess- we solve these systems to obtain the desired solutions. The stability and convergence analyses of the numerical schemes were studied in-depth. The schemes are convergent and accurate. Some numerical test problems are performed to verify the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]