Abstract
In contrast to the traditional trigonometric basis functions which have infinite support, wavelets may have compact support, thus being able to approximate a functions not by cancellation, but through placement of the right wavelets at appropriate locations. The multi-resolution analysis (MRA) properties of wavelets render them attractive candidates for functions in terms of which numerical solutions of differential equations can be represented. In particular, Mallat’s MRA algorithm applied on orthonormal or semi-orthogonal wavelets can be used to generate adaptive grid methods. In this work, two classes of compactly supported wavelets, namely, Daubechies’ orthonormal wavelet bases and Chui-Wang’s semi-orthogonal B-spline wavelets, are used with the method of weighted residuals for numerical solution of differential equations. A variation relying on Laplace transform is also examined. Based on the MRA properties of wavelet bases, two-scale multi-level methods are proposed, which generate higher level approximate wavelet solutions from the results of lower level calculations. Adaptive grid refinement can be applied only to regions where the residual at lower level is relatively large. The proposed methods may be advantageous when solutions exhibit abrupt changes. Several numerical examples based on these methods are given. Comparisons with traditional methods for numerical solution of differential equations are made and directions for further development are given.
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Nikolaou, M., You, Y. (1994). Use of Wavelets for Numerical Solution of Differential Equations. In: Motard, R.L., Joseph, B. (eds) Wavelet Applications in Chemical Engineering. The Kluwer International Series in Engineering and Computer Science, vol 272. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2708-4_7
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