An efficient technique for approximated BVPs via the second derivative Legendre polynomials pseudo-Galerkin method: Certain types of applications

This paper will present a highly efficient technique for solving linear and nonlinear differential equations. We will use the second derivative of Legendre polynomials as new base functions via a pseudo-Galerkin method. These base functions produce a new operational matrix for derivatives. The main idea is to convert the differential equations into linear or nonlinear algebraic equations with unknown coefficients. Consequently, these coefficients can be determined and used to get the approximate solution. Then, we studied the proposed strategy’s convergence and error analysis. Additionally, accuracy, efficiency, and stability were verified by applying the presented method to some types of ordinary differential equations, Mainly Land–Emden for astrophysics, Bratu for solid fuel ignition mode, Riccati equations, and real-life applications for fluid flow and population model