Block hybrid methods for numerical treatment of differential equations with applications

This thesis focuses mainly on deriving block hybrid methods for solving Ordinary Differential Equations (ODEs). Block hybrid methods are the methods that generate a block of new solutions at the main and off-step points concurrently. The first part of the thesis is about the derivation of the explicit block hybrid methods based on Newton-Gregory backward difference interpolation formula for solving first order ODEs. The regions of stability are presented. The numerical results are shown in terms of total steps and accuracy. The second part of the thesis describes the mathematical formulation of explicit and implicit one-point block hybrid methods for first order ODEs whereby the derivation involves the divided differences relative to main and off-step points. The stability properties are discussed. The explicit and implicit block hybrid methods are implemented in predictor-corrector mode of constant step size to obtain the numerical approximation for first order ODEs. The implementation of block hybrid methods in variable step size is also presented. Some numerical examples are given to illustrate the efficiency of the methods. The one-point block hybrid methods are then implemented for numerical solution of first order delay differential equations (DDEs). The Q-stability of the methods is investigated. Since the block hybrid methods include the approximate solution at both the main and additional off-steps points, more computed values that surrounding the delay term can be used to provide a better estimation in interpolating the delay term. The third part of the thesis is mainly focused on block hybrid collocation methods for obtaining direct solution of second-, third- and fourth-order ODEs. The derivation involves interpolation and collocation of the basic polynomial. The stability properties are investigated. Illustrative examples are presented to demonstrate the efficiency of the methods. The block hybrid collocation methods are also applied to solve the physical problems such as Lane-Emden equation, Van Der Pol oscillator, Fermi-Pasta-Ulam problem, the nonlinear Genesio equation, the problem in thin film flow and the fourth order problem from ship dynamics. As a whole, the block hybrid methods for solving different orders of ordinary differential equations have been presented. The illustrative examples demonstrate the accuracy advantage of the block hybrid methods.

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