Optimization-Homotopy-Taylor Series According to Laplace Technique for generalized Non-linear Pantograph Delay Differential Equations of Higher Order to Obtain Analytic and Closed Solutions

Abstract

In this study, optimization-homotopy technique which is a powerful tool to find the
analytic solution for non-linear generalized delay Pantograph differential equations of
higher order with the concept of homotopy approach is used. Taylor series with Laplace
technique is applied to approximate solution. The main advantage for this technique is to
get more an accurate solution with more extended of the convergence region for whole
time interval therefore, this technique is a capable of provide us a continuous
representation of approximate solution, which gives better information and behavior of
solution for over time interval for the problem under consideration. It is reducing more
complex calculations which need more times. Exact solution is conducted by expand the
iteration of classical homotopy results using Taylor series. Selected examples to
investigate the efficiency, capability, accurately and easily for the proposed technique.