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The Galerkin method for singular nonoscillatory two-point boundary value problems
January 1994, KFUPM, Saudi Arabia
Abstract: The Galerkin method with special patch basis is used for the approximation of the solution of a general class of second order singular two-point boundary value problems. Convergence is analysed in several norms. Higher order convergence in the uniform norm are obtained for special data. The results are new both for special and general data. The class of problems treated in this dissertation extends the class of problems treated in the literature in many directions. The existence, uniqueness and the regularity of the solutions are not assumed; these are studied in proper places as needed. Both linear and nonlinear problems are treated in this dissertation.
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Stiffly stable 3-step linear multistep methods
March 1986, KFUPM, Saudi Arabia
Abstract: The whole class of 3-step linear multistep methods for solving system of ordinary differential equations is shown in a parameter space. All A -stable methods are shown as a subclass. Almost all of the existing stiffly stable methods of this class are found. The angle for A (a)-stability of these methods are shown. Particular emphasis is given to develop methods with least error constant and having large region of instability in the right half plane which are suitable for stiff systems with the capability of detecting unstable behavior of a problem. Variable stepsize variable formula methods are developed from this class. Algorithms are developed for different cases:- fixed formula with equal stepsize; fixed formula with variable stepsize using Hermite interpolation; and variable stepsize variable formula method. The algorithms are embodied in computer codes and these are applied to some examples. The idea can be extended to higher step linear multistep methods.