Application of discrete wavelet transform to the solution of boundary value problems for quasi-linear parabolic equations

The wavelet method for solving the linear and quasi-linear parabolic equations under initial and boundary conditions is set out. By applying regular multi-resolution analysis and received formula for differentiating wavelet decompositions of functions of many variables the problem is reduced to a finite set of linear and accordingly nonlinear algebraic equations for the wavelet coefficients of the problem solution. The general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. For quasi-linear parabolic equations the convergence rate of an approximate weak solution to a classical one is estimated. The proposed method is used for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data.