Abstract: Wavelet transform or wavelet analysis is a recently
developed mathematical tool in applied mathematics. In numerical
analysis, wavelets also serve as a Galerkin basis to solve partial
differential equations. Haar transform or Haar wavelet transform has
been used as a simplest and earliest example for orthonormal wavelet
transform. Since its popularity in wavelet analysis, there are several
definitions and various generalizations or algorithms for calculating
Haar transform. Fast Haar transform, FHT, is one of the algorithms
which can reduce the tedious calculation works in Haar transform. In
this paper, we present a modified fast and exact algorithm for FHT,
namely Modified Fast Haar Transform, MFHT. The algorithm or
procedure proposed allows certain calculation in the process
decomposition be ignored without affecting the results.
Abstract: In this work, we derive two numerical schemes for
solving a class of nonlinear partial differential equations. The first
method is of second order accuracy in space and time directions, the
scheme is unconditionally stable using Von Neumann stability
analysis, the scheme produced a nonlinear block system where
Newton-s method is used to solve it. The second method is of fourth
order accuracy in space and second order in time. The method is
unconditionally stable and Newton's method is used to solve the
nonlinear block system obtained. The exact single soliton solution
and the conserved quantities are used to assess the accuracy and to
show the robustness of the schemes. The interaction of two solitary
waves for different parameters are also discussed.