Wavelets adapted for the solution of ordinary differential equations

Björn Jawerth and Wim Sweldens

Department of Mathematics, University of South Carolina, Columbia SC29208


Abstract

In this paper we study the solution of ordinary differential equations of the form

Lu(x) = f(x) x in [a,b],

with appropriate boundary conditions at x=a,b. A typical solution method is to use a Galerkin approach, i.e. to find a function u such that

< x,Lu > = < v,f >,

for u and v belonging to specific spaces. By just using a wavelet basis for u and v, one does not make explicit use of their (bi)orthogonality properties. The idea is to adapt the construction of the wavelets to the operator and find two families of dual functions, one biorthogonal with respect to the energy inner product and one biorthogonal with respect to the L^2 inner product, each with an associated multiresolution analysis (mra) and a fast wavelet transform involving finite impulse response filters. The energy biorthogonality property will yield a diagonal matrix after discretization which can trivially be inverted. The algorithm then corresponds to decomposing the right hand side f using the L^2 mra, applying the inverse of the diagonal matrix and reconstructing using the energy inner product mra. Using wavelets on closed sets, the boundary conditions can be incorporated on the coarsest level. We show here how this construction, in case of the Laplace operator, corresponds to using antiderivatives of wavelets in the Galerkin method. In some cases these functions coincide with the hierarchical basis functions. This can trivially be generalized to the polyharmonic operator. In case of the Helmholz operator, the construction starts from wavelets which are (bi)orthogonal to an exponential weight function.