Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation - Essam Roshdy El-Zahar; Abdelhalim Ebaid (Axioms, 2022)
In this paper, the pantograph delay differential equation <semantics>y′(t)=ay(t)+by<mfenced separators="" open="(" close=")">ct</mfenced></semantics> subject to the condition <semantics>y(0)=λ</semantics> is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for <semantics>λ=1</semantics>, is well-posed if <semantics>c<1</semantics>, but not if <semantics>c>1</semantics>. In addition, the solution is available in the form of a standard power series when <semantics>λ=1</semantics>. 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 <semantics>c∈(−1,1)</semantics> such that <semantics><mfenced separators="" open="|" close="|">ba</mfenced><1</semantics> and it also converges for <semantics>c>1</semantics> when <semantics>a<0</semantics>. For <semantics>c=−1</semantics>, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity <semantics>2π<msqrt>b2−a2</msqrt></semantics> when <semantics>b>a</semantics>. The current results are introduced for the first time and have not been reported in the relevant literature.