Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Abstract

In this study, we present a modified configuration, including an exact formulation, for the operational matrix form of the integration, differentiation, and product operators applied in the Galerkin method. Previously, many studies have investigated the methods for obtaining operational matrices (derivative, integral, and product) for Fourier, Chebyshev, Legendre, and Jacobi polynomials, and some have considered the non-orthogonal bases that almost all of them operate on approximately. However, in this study, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to previous approaches, this method transforms the original problem into a system of nonlinear algebraic equations. To retain the simplicity of the procedure, the samples are considered in one-dimensional contexts, although the proposed technique can also be employed for two- and three-dimensional problems. Two examples are presented to verify the accuracy of the proposed new approach and to demonstrate the superior performance of EOMs compared with ordinary operational matrices. The corresponding results demonstrate the increased accuracy of the new method. In addition, the convergence of the EOM method is studied numerically and analytically to prove the efficiency of the method.