Cutting-Edge Spectral Solutions for Differential and Integral Equations Utilizing Legendre’s Derivatives

This research introduces a spectral numerical method for solving some types of integral equations, which is the pseudo-Galerkin spectral method. The presented method depends on Legendre’s first derivative polynomials as basis functions. Subsequently, an operational integration matrix has been constructed to express integrals as a linear combination of these basis functions. This process transforms the given integral equation into a system of algebraic equations. The unknowns of the obtained system are the spectral expansion constants. Then, we solve the obtained algebraic system using the Gauss elimination method for linear systems or Newton’s iteration method for nonlinear systems. This approach yields the desired semi-analytic approximate solution. Additionally, our method extends to the solution of ordinary differential equations, as every initial value problem can be equivalently represented as a corresponding Volterra integral equation. On the other hand, every boundary value problem can be transformed into a corresponding Fredholm integral equation. This transformation is achieved by incorporating the given conditions. Moreover, convergence and error analyses are thoroughly examined. Finally, to validate the efficiency and accuracy of the proposed method, we conduct numerical test problems.

Keywords:

Legendre polynomials , pseudo-Galerkin spectral method , Integral equations , Lane–Emden equation , stable population model