An Analysis of the Theta-Method for Pantograph-Type Delay Differential Equations

Abstract

The pantograph equation arises in electrodynamics as a delay differential equation (DDE). In this article, we provide the $\vartheta$ -method for numerical solutions of pantograph equations. We investigate the stability conditions for the numerical schemes. The theoretical results are verified by numerical simulations. The theoretical results and numerical simulations show that implicit or partially implicit $\vartheta$ -methods, with $\vartheta >\left(1/2\right)$ , are effective in resolving stiff pantograph problems.