An efficient new numerical algorithm for solving Emden–Fowler pantograph differential equation using Laguerre polynomials

• Published in: Journal of computational science Vol. 72; p. 102108
• Main Authors: Saha, Nikita, Singh, Randhir
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2023
• Subjects: Collocation method; Emden–Fowler equation; Laguerre polynomials; Operational matrix; Pantograph differential equation; Operational matrix; Laguerre polynomials; Collocation method; Emden–Fowler equation; Pantograph differential equation
• ISSN: 1877-7503; 1877-7511
• DOI: 10.1016/j.jocs.2023.102108

This paper proposes a new approach using the collocation technique for the numerical solution of a class of second-order Emden–Fowler pantograph differential equations. In the literature, all the numerical schemes are designed to deal with only the IVPs of the concerned model. This motivates us to extend the work by considering BVPs as well. First, the uniqueness of the solution to the problem is proved for all three sets of initial/boundary conditions, and then numerical algorithms are established. The algorithms are based on operational matrices of the derivatives, which helps in reducing the computational cost to a great extent. This is shown with the help of CPU time for all the illustrative examples. The formulated schemes handle singularity effectively by using suitable collocation points and simplify the mentioned problems into a nonlinear system of equations, which can be solved by any iterative method. Unlike other methods, the present approach requires no linearization, discretization, or perturbation. The computed numerical results confirm that the present method provides better accuracy than the Bessel collocation method (Izadi and Srivastava, 2021) and the Bernstein operational approach (Sriwastav and Barnwal, 2023). The error bound of the proposed method is also analyzed to support the numerical results. •A recently developed Emden–Fowler pantograph delay differential equation is considered.•The model is solved subject to boundary conditions for the first time.•A new approach polynomial-based operational methodology for faster computations is proposed.•Uniqueness of solution and Error analysis of the proposed method is provided.•Several problems are solved to demonstrate the efficiency and accuracy of the present method.