A Master of Science thesis in Mathematics by Heba Qasim Alkafri entitled, "Solving The 2D Biharmonic Equation: A Numerical Approach Based On Spline Collocation," submitted in July 2016. Thesis advisor is Dr. Suheil Khoury and thesis co-advisor is Dr. Ali Sayfy. Soft and hard copy available.
In this thesis, several numerical methods, that are based on spline basis functions, are suggested for the solution of a general class of fourth order boundary value problems. In particular, The two-dimensional biharmonic equation complemented with Dirichlet boundary conditions is considered. The methods include the bivariate spline collocation using B-splines of degree 5 and 7. Moreover, a combination of finite difference and spline collocation is suggested. Numerical experiments are included to demonstrate the applicability and accuracy of the proposed schemes and to compare them with other techniques that are available in the literature. The numerical results include a special case of the problem which models the two-dimensional steady state incompressible Navier-Stokes equations in streamfunction formulation.