Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation

In this paper, the pantograph delay differential equation (Formula presented.) subject to the condition (Formula presented.) is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for (Formula presented.), is well-posed if (Formula presented.), but not if (Formula presented.). In addition, the solution is available in the form of a standard power series when (Formula presented.). In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for (Formula presented.) such that (Formula presented.) and it also converges for (Formula presented.) when (Formula presented.). For (Formula presented.), the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity (Formula presented.) when (Formula presented.). The current results are introduced for the first time and have not been reported in the relevant literature.