The Wavelet Analysis Is Used To Solve Differential Equations Research

Wavelet analysis is a new realm in the applied mathematics with rapid development, because the wavelets have the smooth and local compact property, compared with traditional finite element method and finite difference method, it is a more useful method for the question with local singularity, so, wavelet analysis now more and more applied into the numerical solution of partial deference equations. In this paper, the derivative operator's expression by wavelets and numerical solution of partial difference equations that based on heat-conduction equation were investigated.Wavelet analysis on difference equations' substance is to put the equation into wavelet domain. Leland Jameson has given one-order derivative operator's expression on the wavelets at scale j=1, based on that, one-order and two-order derivative operator's expression at scale j = 1,2 was given and one-order and two-order derivative operator's expression on the wavelet domain was proved in this paper.By the theory of derivative operator's expression on wavelets domain, this paper discuss two kinds of computer formats to heat-conduction equation. The first, on the scaling function space V_j, using like-Shannon wavelet to construct basis functions , the numerical algorithm to solve heat-conduction equation was established; The second, on the multi-resolution space V_j+1 = V_j+ W_j(j∈z), using the theory of derivative operator's expression on wavelets domain deduced by this paper, the numerical algorithm to solve heat-conduction equation was established. Finally , compared with finite difference method, the computation results show that wavelet method's accuracy is more higher, and do not have the oscillation phenomenon.