A Galerkin-like method for solving linear functional differential equations underinitial conditions

Abstract

In this paper, we present a weighted residual Galerkin method to solve linear functional differential equations. We consider the problem with variable coefficients under initial conditions. Assuming the exact solution of the problem has a Taylor series expansion convergent in the relevant domain, we seek a solution of the given problem in the form of a polynomial having degree $N$ of our choice. Substituting this polynomial with unknown coefficients in the given equation yields an expression linear in these coefficients. We then proceed as in the weighted residual method and take inner product of this expression with monomials up to degree $N$, resulting in $N+1$ linear algebraic equations. Appropriately incorporating the initial conditions and solving the resulting linear system, we obtain the approximate solution to the given problem. Additionally, we present a way of estimating the absolute error of the obtained approximation, which is then used to improve the original approximation through a method called residual correction. We also show that the upper bound for the error of the proposed method depends on the Taylor truncation error of the exact solution. The proposed scheme and the residual correction technique are illustrated in several example problems.