Exact and Numerical Analysis of the Pantograph Delay Differential Equation via the Homotopy Perturbation Method

The delay differential equations are of great importance in real-life phenomena. A special type of these equations is the Pantograph delay differential equation. Generally, solving a delay differential equation is a challenge, especially when the complexity of the delay terms increases. In this paper, the homotopy perturbation method is proposed to solve the Pantograph delay differential equation via two different canonical forms; thus, two types of closed-form solutions are determined. The first gives the standard power series solution while the second introduces the exponential function solution. It is declared that the current solution agrees with the corresponding ones in the literature in special cases. In addition, the properties of the solution are provided. Furthermore, the results are numerically validated through performing several comparisons with the available exact solutions. Moreover, the calculated residuals tend to zero, even in a huge domain, which reflects the high accuracy of the current analysis. The obtained results reveal the effectiveness and efficiency of the current analysis which can be further extended to other types of delay equations.