Exponentially Fitted Element-Free Galerkin Approach for Nonlinear Singularly Perturbed Problems

Jagbir Kaur,Vivek Sangwan,M M Bhatti

doi:10.1155/2021/4165954

Jagbir Kaur, Vivek Sangwan + Show 1 more

Open Access

https://doi.org/10.1155/2021/4165954

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Abstract

As it is well recognized that conventional numerical schemes are inefficient in approximating the solutions of the singularly perturbed problems (SPP) in the boundary layer region, in the present work, an effort has been made to propose a robust and efficient numerical approach known as element-free Galerkin (EFG) technique to capture these solutions with a high precision of accuracy. Since a lot of weight functions exist in the literature which plays a crucial role in the moving least square (MLS) approximations for generating the shape functions and hence affect the accuracy of the numerical solution, in the present work, due emphasis has been given to propose a robust weight function for the element-free Galerkin scheme for SPP. The key feature of nonrequirement of elements or node connectivity of the EFG method has also been utilized by proposing a way to generate nonuniformly distributed nodes. In order to verify the computational consistency and robustness of the proposed scheme, a variety of linear and nonlinear numerical examples have been considered and L ∞ errors have been presented. Comparison of the EFG solutions with those available in the literature depicts the superiority of the proposed scheme.

Highlights

Perturbed problems (SPP) and their numerical solutions have attained great attention from researchers from the past few decades
We have proposed the elementfree Galerkin (EFG) method, one of the most popular and versatile mesh-free methods, for approximating nonlinear Singularly perturbed problems (SPP) solutions. e element-free Galerkin (EFG) technique is based on the theory of moving least squares (MLS) approximation and standard weight functions to construct the shape functions
The element-free Galerkin method has not been utilized for solving SPP to the best of the authors’ knowledge

Summary

Introduction

Perturbed problems (SPP) and their numerical solutions have attained great attention from researchers from the past few decades. Geng and Qian [10] employed the reproducing kernel method for obtaining numerical solutions of singularly perturbed turning point problems having dual boundary layers. Nadjafi and Ghassabzade [12] proposed another MLS-based meshless technique to deal with singularly perturbed delay differential equations (DDEs) Another meshless approach, namely, the radial basis collocation method with the coordinate stretching procedure was presented by Akhavan Ghassabzade et al [13] for the mathematical treatment of singularly perturbed DDEs. In this study, we have investigated the following nonlinear singularly perturbed problem: εv′′ f v′, v, x􏼁, x ∈ Ω (a, b),. E EFG technique is based on the theory of moving least squares (MLS) approximation and standard weight functions to construct the shape functions. We can observe that the shape functions are bell-shaped. e derivatives of the MLS shape functions can be obtained by differentiating (16) with respect to the spatial coordinates

Choice of Weight Functions

Quasi-Linearization Technique

Element-Free Galerkin Formulation

Numerical Results and Discussion

Conclusion