×

Solving the sideways heat equation by a wavelet-Galerkin method. (English) Zbl 0883.35123

The authors discuss a heat problem in a quarter of the plane. The initial problem is ill-posed, in the sense that the existing solution does not depend continuously on the data. In order to solve this problem one uses wavelet techniques, so that the high-frequency components are filtered away. However, when the obtained problem is solved numerically, the high-frequency perturbations induced by round off errors of the computer can completely destroy the solution.
It is shown in the paper, that when using a wavelet-Galerkin approach then the problem with data being filtered away is numerically stabilized. This is done by projecting the data onto a wavelet subspace and, as the result, in the obtained space expanding the solution in terms of wavelets. The test function for a weakly formulated problem is taken from this space, too. It is shown that the obtained problem with initial conditions for ordinary differential equations is well-posed. The error estimates of the solution are given as well as a rule for choosing the corresponding wavelet space. Some numerical examples are given.

MSC:

35R30 Inverse problems for PDEs
35K05 Heat equation
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI