Adaptive mesh based Haar wavelet approximation for a singularly perturbed integral boundary problem

Document Type : Research Article

Authors

• P. Shukla ¹
• S. Saini ²
• V. Devi ³

¹ Department of Engineering Sciences, Indian Institute of Information Technology and Management Gwalior, Gwalior, Madhya Pradesh, 474015, India.

² Department of Mathematics, Vellore Institute of Technology, Vellore, Tamilnadu, 632014, India.

³ Department of Mathematics, Bhakt Darshan Govt. P.G. College, Pauri Garhwal, Uttrakhand, 246193, India.

10.22067/ijnao.2024.87550.1421

Abstract

This research presents a nonuniform Haar wavelet approximation of a singularly perturbed convection-diffusion problem with an integral boundary. The problem is discretized by approximating the second derivative of the solution with the help of a nonuniform Haar wavelets basis on an arbitrary nonuniform mesh. To resolve the multiscale nature of the problem, adaptive mesh is generated using the equidistribution principle. This approach allows for the dynamical adjustment of the mesh based on the solution’s behavior without requiring any information about the solution. The combination of nonuniform wavelet approximation and the use of adaptive mesh leads to improved accuracy, efficiency, and the ability to handle the multiscale behavior of the solution. On the adaptive mesh rigorous error analysis is performed showing that the proposed method is a second-order parameter uniformly convergent. Numerical stability and computational efficiency are validated in various tables and plots for numerical results obtained by the implementation of two test examples. 

Keywords

• Singular perturbation problems
• Adaptive mesh
• Nonuniform Haar wavelet
• Parameter uniform convergence

Main Subjects

• Numerical methods for differential equations and its applications

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