Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem (Advances in Applied Mathematics and Mechanics, 2021)
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 ofapproximation 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 unsteadyflows in an open channel called Saint-Venant. Since the solving of thisproblem by some methods such as finite difference and finite elementproduce 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.