Wavelet and its variants based numerical methods for partial differential equations

Abstract

Since the 1990s, wavelet theory has been adopted for numerically solving the partial differential equations (PDEs). There is an immense research available on wavelet based techniques for numerically solving PDEs. But the theory of wavelet to numerically solve PDEs on arbitrary manifolds is yet in its emerging phase. Moreover, handling general boundary conditions using wavelets is also a tedious task. In this thesis, fast adaptive methods based on wavelets and their variants are developed which are easily extendable to general manifolds and can handle general boundaries. Second generation wavelet, spectral graph wavelet (SPGW) and curvelet are used for this purpose. Curvelet is already used in various areas of engineering, but to best of our knowledge it has very thin appearance in the field of PDEs. \par We started with Daubechies and second-generation wavelet to get a better understanding of wavelet-based methods for solving PDEs. Daubechies wavelet have been widely used for numerically solving PDEs, but we have used Daubechies wavelet for solving real life problems, $i.e.$, traffic flow problems. Furthermore, we developed a Matlab toolbox which contains the routines for the second generation wavelet transformation and inverse wavelet transformation on the space $\mathcal{L}_2([a,b])$. These wavelet transforms are further used for computing the wavelet and scaling function values ($\psi(x)$ and $\phi(x)$ respectively). We have also included the Matlab code for generating second generation wavelet based adaptive grid in our suite. \par As our aim was to tackle PDEs with different boundaries and Daubechies wavelet based methods are limited to periodic boundary conditions, we used second generation wavelet and third generation wavelet.