Applications of differential transform to boundary value problems for delayed differential equations

Abstract

An application of the differential transformation is proposed in this paper which is convenient for finding approximate solutions to boundary value problems for functional differential equations. We focus on two-point boundary value problem for equations with constant delays. Delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into recurrence relation in one variable. Based on the structure of the studied boundary value problem, the solution to the recurrence relation depends on one real parameter. Using the boundary conditions leads to an equation with the unknown parameter as the single variable which occurs generally in infinitely many terms. Approximate solution has the form of a Taylor polynomial. Coefficients of the polynomial are determined by solving the recurrence relation and a truncated equation with respect to the unknown parameter. Particular steps of the algorithm are demonstrated in an example of two-point boundary value problem for a differential equation with one constant delay.