Biorthogonal multiwavelets on the interval for numerical solutions of Burgers’ equation

Abstract

To numerically solve the Burgers’ equation, in this paper we propose a general method for constructing wavelet bases on the interval [0,1] derived from symmetric biorthogonal multiwavelets on the real line. In particular, we obtain wavelet bases with simple structures on the interval [0,1] from the Hermite cubic splines. In comparison with all other known constructed wavelets on the interval [0,1], our constructed wavelet bases on the interval [0,1] from the Hermite cubic splines not only have good approximation and symmetry properties with extremely short supports, but also employ a minimum number of boundary wavelets with a very simple structure. These desirable properties make them to be of particular interest in numerical algorithms. Our constructed wavelet bases on the interval [0,1] are then used to solve the nonlinear Burgers’ equation. Our method is based on the finite difference formula combined with the collocation method. Therefore, our proposed numerical scheme in this paper is abbreviated as MFDCM (Mixed Finite Difference and Collocation Method). Some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy.