Numerical solution of arbitrary-order linear partial differential equations using an optimal control technique

Abstract

This study aims to present a new approach for finding the numerical solution of linear partial differential equations using an optimal control technique. First, a partial differential equation is transformed into an equivalent optimal control problem and then, control and state variables are approximated by Chebychev series. Therefore, the obtained problem is converted to an optimization problem subject to algebraic equality constraints. Finally, a suitable iterative optimization technique is implemented to approximate the unknown coefficients of Chebychev polynomials to find the numerical solution of the original problem. In this method, a new idea is used, enabling us to deal with almost all kinds of linear partial differential equations with different types of initial and boundary conditions. Several kinds of numerical examples are solved to illustrate the accuracy and efficiency of the proposed method.