A reliable scheme for nonlinear delay differential equations of pantograph-type

In this article, an iterative method to obtain approximate analytical solutions to delay differential equations of the pantograph type is presented. The primary goal of this approach to solving such problems is to transform them into integral problems. The proposed method involves two stages. First, create an integral operator for the problem. Applying the Normal-S iterative scheme to this integral operator yields the iterative scheme for such problems. The results show that the suggested method converges quickly, is computationally efficient, and produces accurate approximation solutions. The article also addresses the convergence of the iterative approach. To demonstrate the effectiveness of the method, we examine various numerical examples. The numerical simulations show that the method is an efficient tool for solving functional differential equations. The method is also applicable to delay boundary value problems and to illustrate the scheme's applicability, an example is provided. The comparisons demonstrate the relevance and efficacy of the proposed work.