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

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.