A new semi-analytical approach for numerical solving of Cauchy problem for functional differential equations

TL;DR

This paper introduces a semi-analytical method combining the method of steps and differential transformation to efficiently solve Cauchy problems for functional differential equations, avoiding complex symbolic calculations.

Contribution

It presents a novel semi-analytical approach that simplifies solving FDEs by reducing them to ODEs and eliminates the need for initial guesses or symbolic computations.

Findings

Effective for FDEs with multiple constant delays

Outperforms existing methods like homotopy and Adomian decomposition

Applicable to neutral type differential equations

Abstract

One of the major challenges of contemporary mathematics is numerical solving of various problems for functional differential equations (FDE), in particular Cauchy problem for delayed and neutral differential equations. Recently large variety of methods to handle this task appeared. In the paper, we present new semi-analytical approach for FDE's consisting in combination of the method of steps and a technique called differential transformation method (DTM). This approach reduces the original Cauchy problem for delayed or neutral differential equation to Cauchy problem for ordinary differential equation for which DTM is convenient and efficient method. Moreover, there is no need of any symbolic calculations or initial approximation guesstimates in contrast to methods like homotopy analysis method, homotopy perturbation method, variational iteration method or Adomian decomposition method. The efficiency of the proposed method is shown on certain classes of FDE's with multiple constant delays including FDE of neutral type. We also compare it to the current approach of using DTM and the Adomian decomposition method where Cauchy problem is not well posed.