Direct solution of higher order ordinary differential equations using one-step hybrid block methods with generalised off-step points In the presence of higher derivative
A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of t...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://etd.uum.edu.my/7523/1/Depositpermission_s900374.pdf https://etd.uum.edu.my/7523/2/s900374_01.pdf https://etd.uum.edu.my/7523/ http://sierra.uum.edu.my/record=b1697805~S1 |
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| Summary: | A great number of physical phenomena can be expressed as initial or boundary value problems of higher order ordinary differential equations (ODEs) which may not have analytical solutions. Thus, there is a need to develop numerical methods for approximating the solution of higher order ODEs. One of the well-known direct methods which frequently employed is block method. Even though this method is capable of finding the approximate solutions at several points simultaneously, it fails to overcome the zero-stability barrier. Thus, a hybrid block method was introduced to tackle this drawback. The main benefit of this method is its ability of using data at off-step points which contribute to better accuracy. Most of the existing hybrid block methods, however, only focus on specific off-step point(s) in deriving the methods with the exception of the method proposed by Abdelrahim in 2016. Although he has successfully developed one-step hybrid block methods with generalised off-step point(s) for solving high order ODEs directly, nevertheless, the methods are only confined to initial value problems. Moreover, he did not consider higher derivative in developing those methods. Thus, this study introduced new one-step hybrid block methods with generalised off-step point(s) in the presence of higher derivative for directly solving higher order ODEs. In developing these methods, a power series was used as an approximate solution to the problems of ODEs of order m. The power series was interpolated at m points, while its mth and (m+1)th derivatives were collocated at all points in the given 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. Several initial and boundary value problems of higher order ODEs considered in literature were then solved by using the newly developed methods in order to investigate the accuracy of the solution in terms of error. The numerical results revealed that, in general, the new methods were able to produce smaller errors compared to the existing methods in solving the same problems. In conclusion, this study has successfully developed viable methods for directly solving both initial and boundary value problems of higher order ODEs. |
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