Extended two-point and three-point block backward differentiation formulas for solving first order stiff ordinary differential equations
This thesis focuses on solving first order stiff Ordinary Differential Equations (ODEs) using 2-point and 3-point block methods. The 2-point block method will compute the solutions yn+1 and yn+2 at points xn+1 and xn+2 simultaneously in a block at each step. Thus, the derivations of 2-point block me...
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| Format: | Thesis |
| Language: | English |
| Published: |
2018
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/83301/1/FS%202019%2066%20ir.pdf |
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| Summary: | This thesis focuses on solving first order stiff Ordinary Differential Equations (ODEs) using 2-point and 3-point block methods. The 2-point block method will compute the solutions yn+1 and yn+2 at points xn+1 and xn+2 simultaneously in a block at each step. Thus, the derivations of 2-point block methods of third and fifth order are presented. Order and error constant of the methods are determined. Newton’s Method is used to implement in the 2-point block methods. The numerical results for each method are presented and compared with the existing methods. Furthermore, the stability properties of all 2-point block methods are analysed to ensure that the methods are A(a)-stable. Hence, its suitable for solving stiff problems. Convergence characteristics of the methods are also investigated. The 2-point block method with fifth order is then extended to 3-point block method with same order. Advantage of the 3-point block method is the solutions will be approximated at three points concurrently which are xn+1, xn+2 and xn+3. Thus, the derivation of 3-point block method using Taylor’s series expansion is presented. Order and error constant of the method are verified. Stability and convergence properties of the method are investigated by determining the zero-stable, stability region, A(a)-stable and consistency. The 3-point block method is implemented by using Newton’s iteration to measure its efficiency. Numerical results of the method are presented and performance of the method are compared with the existing methods. An application problem of SIR model is solved by using the proposed methods. The numerical results are presented in tables for s, i and r groups for each method. Based on the analysis, the proposed methods can be an alternative solver for solving the application problem. |
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