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

In this paper, the pantograph delay differential equation y′(t)=ay(t)+byct subject to the condition y(0)=λ is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for λ=1, is well-posed if c<1, but not if c>1. In addition, the solution is available in the form of a standard power series when λ=1. 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 c∈(−1,1) such that ba<1 and it also converges for c>1 when a<0. For c=−1, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2πb2−a2 when b>a. The current results are introduced for the first time and have not been reported in the relevant literature.