On wavelet-based algorithms for solving differential equations. (English) Zbl 0883.65067

Benedetto, John J. (ed.) et al., Wavelets: mathematics and applications. Boca Raton, FL: CRC Press. Studies in Advanced Mathematics. 449-466 (1994).
Summary: We describe an order \(N\) method for computing the Green’s function of the two-point boundary value problem for elliptic differential operators in the wavelet “system of coordinates”. For simplicity, we consider the ordinary \(O(h^2)\) finite difference scheme, and use wavelets only to perform the “linear algebra”. Our main tool is the diagonal preconditioning available for the periodized differential operators in the wavelet bases.
For the entire collection see [Zbl 0840.00013].


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B27 Green’s functions for ordinary differential equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65F35 Numerical computation of matrix norms, conditioning, scaling
34B05 Linear boundary value problems for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations