An improved Haar wavelet quasilinearization technique for a class of generalized Burger's equation

TL;DR

This paper introduces an enhanced Haar wavelet quasilinearization method for numerically solving a generalized form of Burgers' equation, demonstrating improved accuracy and efficiency over existing techniques, especially for low viscosity scenarios.

Contribution

The paper develops a novel Haar wavelet quasilinearization approach for generalized Burgers' equation, with proven convergence and superior accuracy for small viscosity values.

Findings

Method achieves high accuracy with fewer grid points.

Error norms are lower compared to existing methods.

Convergence of the proposed technique is established.

Abstract

Solving Burgers' equation always poses challenge to researchers as for small values of viscosity the analytical solution breaks down. Here we propose to compute numerical solution for a class of generalised Burgers' equation described as $$ \frac{\partial w}{\partial t}+ w^{\mu}\frac{\partial w}{\partial x_{*}}=\nu w^{\delta}\frac{\partial^2 w}{\partial x^{2}_{*}},\hspace{0.2in}a \leq x_{*}\leq b,~~t\geq 0,$$ based on the Haar wavelet (HW) coupled with quasilinearization approach. In the process of numerical solution, finite forward difference is applied to discretize the time derivative, Haar wavelet to spatial derivative and non-linear term is linearized by quasilinearization technique. To discuss the accuracy and efficiency of the method $L_{\infty}$ and $L_{2}$-error norm are computed and they are compared with some existing results. We have proved the convergence of the proposed method. Computer simulations show that the present method gives accurate and better result even for small number of grid points for small values of viscosity.