Wavelet Multiresolution Analyses Adapted For The Fast Solution Of Boundary Value Ordinary Differential Equations
Bjoern Jawerth, Wim Sweldens · NASA Technical Reports Server (NASA) · 1993
We present ideas on how to use wavelets in the solution of boundary value ordinary differential equations. Rather than using classical wavelets, we adapt their construction so that they become (bi)orthogonal with respect to the inner product defined by the operator. The stiffness matrix in a Galerkin method then becomes diagonal and can thus be trivially inverted. We show how one can construct an O(N) algorithm for various constant and variable coefficient operators. 1 Introduction The purpose of this paper is to use wavelets in the solution of certain linear ordinary differential equations of the form Lu(x) = f(x) for x 2 [0; 1]; where L = m X j=0 a j (x) D j ; and with appropriate boundary conditions on u(x) for x = 0; 1. Currently there exist two major solution techniques. First, if the coefficients a j (x) of the operator are constants, then the Fourier transform is well suited for solving these equations. The underlying reason is that the complex exponentials are eigenfunct...