Solving pantograph type through shifted Vieta-Pell polynomials

One of the basic mathematical models in the study of differential equations is the Pantograph delay differential equation. This article effectively solves the nonlinear pantograph delay equations through Vieta-Pell polynomials using the sub-interval collocation method. When the solution score is less than or equal to n, the approach discussed in this article ensures an exact solution. The method that can be directed reduces the problem that is related to solving a system of algebraic equations. A number of completed test instances are provided to demonstrate the accuracy and potency of the novel approach. Additionally, many calculations of the residual errors are used to determine the accuracy of the numerical results. Such residuals are demonstrated to tend to zero.