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Whether we realise it or not, differential equations are widely used in everyday life. Many physical phenomena problems involving rates of change can be expressed as initial value problems (IVPs) of higher order ordinary differential equations (ODEs) that may not have analytical solutions. Thus, num...
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| Format: | Thesis |
| Language: | eng eng |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://etd.uum.edu.my/9772/1/permission%20to%20deposit-grant%20the%20permission-901732.pdf https://etd.uum.edu.my/9772/2/s901732_01.pdf |
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| Summary: | Whether we realise it or not, differential equations are widely used in everyday life. Many physical phenomena problems involving rates of change can be expressed as initial value problems (IVPs) of higher order ordinary differential equations (ODEs) that may not have analytical solutions. Thus, numerical methods for solving IVPs of higher order ODEs are needed. Conventionally, reduction methods were employed to convert higher order ODEs into equivalent systems of first order ODEs and then solved using the available methods. However, this approach increases the number of equations and requires more computation which may jeopardise the accuracy of the solution. Direct methods, on the other hand, are able to solve IVPs of higher order ODEs without having to reduce them to first-order ODE systems. One of the direct methods is the block method which has the capability of approximating exact solutions at multiple points simultaneously. Nevertheless, the block method has zero stability barriers and these constraints lead to the development of hybrid block methods. Most of the existing hybrid block methods were found to only consider specific off-step points for deriving the methods. Although several methods have been derived based on generalised off-step point(s), they are limited to one-step hybrid block methods. Thus, this study proposed new generalised multi-step hybrid block methods for solving IVPs of higher ODEs directly, taking into account all possible arrangements of different points in the interval. The arrangement of off-step point(s) with interior step point(s) in two-step and three-step intervals was obtained through permutation. In deriving these methods, a power series that was adopted as an approximate solution to the problems of ODEs order m was interpolated at m points while its highest derivative was collocated at all points in the interval. Investigations on the properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also carried out. The eighty-four (84) newly developed methods were then used to solve a number of IVPs of higher order ODEs that had previously been tested using existing methods. The comparison in terms of errors revealed that all new methods produced more accurate solutions than existing methods when solving the same problems. Hence, the proposed methods, especially 2L4T hybrid block methods, are superior in solving IVPs of higher order ODEs. |
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