A generalized novel approach based on orthonormal polynomial wavelets with an application to Lane-Emden equation

TL;DR

This paper introduces a versatile wavelet-based method using orthogonal polynomials to efficiently solve singular boundary value problems like Lane-Emden equations, demonstrating high accuracy and convergence.

Contribution

It develops a general wavelet framework based on orthogonal polynomials for solving nonlinear singular boundary value problems, with comprehensive comparisons of different wavelets.

Findings

Wavelet methods effectively handle singularities in SBVPs.

Solutions converge to known solutions with increased resolution.

The approach performs well on Lane-Emden type problems.

Abstract

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton-Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane-Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. We observe that the proposed technique performs well on a class of Lane-Emden type BVPs. As the paper deals with singularity, non-linearity significantly and different wavelets are used to compare the results.