Pseudo-Spectral and Kronecker Product Methods for Fourth Order Partial Differential Equations

A general technique for solving linear partial differential equations in two dimensions with constant and variable coefficients is developed. The technique is based on the pseudo-spectral method and it translates a PDE into a matrix equation which is then converted via the Kronecker product into a single linear system that can be solved readily in MATLAB. Three application areas are studied in detail. They are the lid-driven cavity, motion of a liquid slug in a channel and deflection of a rectangular plate. The lid-driven cavity problem uses a single nonlinear equation which has several variants. In the simplest case, it reduces to the standard biharmonic equation. In two other cases, it has a steady state solution that is obtained by iteration. In the most general cases, after substituting the time derivative with a discrete forward difference expression, two different inductive formulas are derived which lead to time-dependent solutions. The lid-driven cavity application illustrates the ability to handle nonlinear equations that have variable coefficients. The liquid slug and plate deflection problems are fundamentally two-dimensional on a rectangular domain and a variety of aspect ratios are handled. The numerical method can solve equations of considerably greater generality than those illustrated in these three applications. Any linear operator can be built up from partial derivatives, variable coefficients and compositions as well as linear combinations of basic components. Using the Kronecker product the equation is translated into a linear system and is solved in MATLAB. By inductively creating a sequence of solutions, nonlinear problems can be solved and time-dependent solutions can be obtained. We also introduce a method of replacing the main equations at certain grid points with boundary condition equations to insure that there are the same number of equations as unknowns and to avoid duplicating equations in the corners of the computational grid.