| ime-dependent Schrodinger and vibration problems are of much physical and engineering significance. In this dissertation, we study the fully discrete high-order approximate solution of initial-boundary value problems for linear and nonlinear Schrodinger equations and vibration equation in two space variables on a unit square. We employ orthogonal spline collocation (OSC) with ;For linear Schrodinger problems, we formulate Crank-Nicolson and ADI OSC schemes, and verify the existence and uniqueness of the approximations. We prove that the schemes are stable and are of second-order accuracy in time and ;We consider vibrating plate problems with three kinds of boundary conditions, in which the plate is (a) clamped on all sides; (b) hinged on all sides; and (c) hinged on two vertical sides and clamped on two horizontal sides. We reformulate the vibration equation as a system of Schrodinger equations in two new variables, and consider Crank-Nicolson schemes for this system. Stability and optimal order ;For nonlinear Schrodinger problems, we present extrapolated Crank-Nicolson and ADI OSC schemes, and examine existence and uniqueness of the approximations. The schemes involve the solution of linear algebraic systems at each time step. We derive the stability and an optimal order ;Matrix forms of the ADI schemes are given, and for their implementation, algorithms are described which cost... |
