A construction of multiple knot B-spline wavelets has been given in [C. K. Chui and E. Quak,
Wavelet on a bounded interval, In: D. Braess and L. L. Schumaker, editors. Numerical methods of
approximation theory. Basel: Birkhauser Verlag; (1992), pp. 57-76].
In this work, we first modify these wavelets to solve the elliptic (partially)
Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.
We generalize this construction to two dimensional case by Tensor product space.
In addition, the solution of the system discretized by Galerkin method
with modified multiple knot B-spline wavelets is discussed.
We also consider a nonlinear partial differential equation for unsteady
flows in an open channel called Saint-Venant. Since the solving of this
problem by some methods such as finite difference and finite element
produce unsuitable approximations specially in the ends of channel,
it is solved by multiple knot B-spline wavelet method that yields a very well approximation.
Finally, some numerical examples are given to support our theoretical results.