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Shifted Legendre polynomials (SLPs) with the Riemann–Liouville fractional integral operator have been used to create a novel fractional integration tool. This tool will be called the fractional shifted Legendre integration matrix (FSL B-matrix). Two algorithms depending on this matrix are designed to solve two different types of integral equations. The first algorithm is to solve fractional Volterra integro-differential equations (VIDEs) with a non-singular kernel. The second algorithm is for Abel's integral equations. In addition, error analysis for the spectral expansion has been proven to ensure the expansion's convergence. Finally, several examples have been illustrated, including an application for the population model. [ABSTRACT FROM AUTHOR]

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]

We introduce new differentiation matrices based on the pseudospectral collocation method. Monic Chebyshev polynomials (MCPs) were used as trial functions in differentiation matrices (D-matrices). Those matrices have been used to approximate the solutions of higher-order ordinary differential equations (H-ODEs). Two techniques will be used in this work. The first technique is a direct approximation of the H-ODE. While the second technique depends on transforming the H-ODE into a system of lower order ODEs. We discuss the error analysis of these D-matrices in-depth. Also, the approximation and truncation error convergence have been presented to improve the error analysis. Some numerical test functions and examples are illustrated to show the constructed D-matrices' efficiency and accuracy. [ABSTRACT FROM AUTHOR]