Asnor, Asma Izzati
(2023)
Variable Step Variable Order Block Backward Differentiation Method For Solving Directly Higher Order Stiff Ordinary Differential Equations.
PhD thesis, Universiti Sains Malaysia.
Abstract
This thesis emphasises on developing Variable Step Variable Order Block Backward Differentiation Method (VSVO-BBDM) for solving directly higher-order stiff ordinary differential equations (ODEs). The scarcity of research on solving higher-order stiff ODEs directly, especially for order three and higher, is evident in the existing literature. As a result, it is crucial to take up the mantle of investigating and elucidating the direct solutions for these higher-order stiff ODEs, specifically for orders three and four. This method generates a set of new solutions in a block at each integration step along the interval. The first part of the thesis discusses the computational work mth-order Variable Step Block Backward Differentiation Formula (mVS-BBDF(3)) method for direct numerical solutions of third-order stiff ODEs. These problems are directly solved without going through the reduction process to the first-order system. The mVS-BBDF(3) method is implemented in the variable step size approach. Meanwhile, the second part of this thesis comprises the computational work of the VSVO-BBDM for solving the higher-order stiff ODEs directly. The computational work of the VSVO-BBDM is carried out using a strategy of varying the step size and varying the order. The advancement of this strategy is intended to enhance the efficiency of the proposed methods to approximate the solutions effectively. Besides, a detailed discussion of the convergence and stability properties of the proposed methods is also included.
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