NUMERICAL METHODS FOR EIGENVALUE COMPUTATION OF FOURTH-ORDER SELF-ADJOINT ORDINARY DIFFERENTIAL OPERATORS
Alalyani, Ahmad Ali
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This dissertation is mainly concerned with three new and very different numerical methods for computing eigenvalues of fourth-order Sturm-Liouville differential operators on finite intervals with regular boundary conditions. The first method is a fourth-order shooting method based on Magnus expansions (MG4). This method is similar to the SLEUTH method of Greenberg and Marletta  which uses MG2 shooting for the integrator (also known as second-order Pruess method ). The second method utilizes two first-order systems of nonlinear Riccati equations, which are integrated alternately to avoid integrating into singularities (conjugate points) when shooting across the Sturm-Liouville interval, and this provides good stabilization for the numerical integration, a fact also noticed by Scott, Shampine and Wing  for second-order equations. The third method, presented here for the first time, converts to a linear 7×7 first-order system which computes the quantities whose zeros determine the eigenvalues for a certain special class of boundary conditions; it also exhibits relatively good numerical stability. All three methods can very often achieve high accuracy on the order of machine precision, and some comparisons of their performance against the well known SLEUTH software package  are presented.