Direct methods via multiple shooting technique for solving boundary value problems

In this thesis, nonlinear two-point second order boundary value problems (BVPs) are solved using the one-point, two-point block and three-point block direct method. Subsequently, the two-point direct block method is extended to solve third order BVPs. It also elaborates on the computational complexity, stability analysis, consistency, convergent and the order of the methods. Multiple shooting technique via the three-step iterative method is implemented in order to solve the BVPs. This approach can avoid the sensitive BVPs when choosing the wrong initial guessing value and converge faster compared to the existing method. Variable step size strategy is adapted for solving second order and third order BVPs respectively. Furthermore the variable step size and order strategy is developed to solve second order BVPs directly. Besides that, a two-point direct block method is proposed to solve three applications of BVPs in fluid dynamics. These applications are modelled as third order BVPs and system with combination of third and second order BVPs. Numerical results showed that the performance of the developed methods can obtain better results in terms of maximum error, total step, total function calls and execution time when compared to existing method. In conclusion, the proposed direct block methods in this thesis are appropriate for solving second order and third order nonlinear boundary value problem.

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