Numerical solution of multi-Pantograph delay boundary value problems via an efficient approach with the convergence analysis

Abstract

This present investigation is contemplated to provide Legendre spectral collocation method for solving multi-Pantograph delay boundary value problems (BVPs). In this regard, an equivalent integral form of such BVPs has been considered. The proposed method is based on Legendre-Gauss collocation nodes and Legendre-Gauss quadrature rule. Convergence analysis associated to the presented scheme has been provided to show its applicability theoretically. Some numerical examples are given to demonstrate the efficiency, accuracy, and versatility of our method. Numerical results confirm the theoretical predictions and are superior with respect to several recent numerical methods including Hermite collocation approach, Laguerre collocation technique and the reproducing kernel method.

Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations

“…The said equations have applicability in diverse range of subject areas, for instance, coherent states in quantum theory [2], control system [3] and cell-growth modeling in biology [4]. According to recent literature available, a number of numerical solvers have shown a great potential to solve [20], Laplace transform method [21], multistep block method [22], computational Legendre Tau method [23], fully-geometric mesh one-leg methods [24], Euler-Maruyama method [25]and so on. In all of these methods, the deterministic solution is generally given in different forms with stability and convergence analysis, while the outcomes show that all of these methods have their own limitations and advantages in comparison to others in certain applications.…”

Optimal Type-3 Fuzzy System for Solving Singular Multi-Pantograph Equations

“…Bilal et al [4], present the Boubeker polynomial approach to construct a numerical solver for SMDEs and its convergence is studied. In [5], the Legendre-Gauss collocation approach is presented in contrast to Hermite collocation method and its versatility and convergence are examined. Doha et al [6], suggest the Jacobi rational-Gauss function and a semi-analytical method.…”

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

“…The most common methodologies found in the literature are spectral method. Yang and Tohidi 5 adopted the Legendre collocation approximation for the differential equations with multipantograph delay. Meanwhile, Saadatmandi and Dehghan 6 and Wang et al 7 extended the Legendre collocation method to the nonlinear fractional differential equation.…”