Block backward differentiation formulas for solving first and second order fuzzy differential equations
In this thesis, the concerns are mainly in modifying existence method of Block Backward Differentiation Formula (BBDFs) for solving first order fuzzy differential equation, second order non-stiff and stiff fuzzy differential equations (FDEs). This method will solve the Initial Value Problems (IVP...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2016
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| Online Access: | http://psasir.upm.edu.my/id/eprint/69281/1/FS%202016%2071%20IR.pdf http://psasir.upm.edu.my/id/eprint/69281/ |
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| Summary: | In this thesis, the concerns are mainly in modifying existence method of Block
Backward Differentiation Formula (BBDFs) for solving first order fuzzy differential
equation, second order non-stiff and stiff fuzzy differential equations (FDEs). This
method will solve the Initial Value Problems (IVPs) of FDEs using constant step size.
The first part of the thesis discussed the combination of BBDF and Block Simpson into
Hybrid method for solving first order FDEs. The subsequent part of the thesis focuses
on the modification of BBDF into fuzzy version of BBDF for solving second order
non-stiff FDEs and second orders stiff FDEs.
Algorithm was developed to run the FDEs problems in Microsoft Visual C++
environment to obtain exact and approximate solutions. The algorithm of existing
BBDF was modified into fuzzy version. The BBDFs method approximates the solution
at two points concurrently. Therefore, numerical results show that the proposed
methods reduce the execution time when compared to the Backward Differentiation
Formula (BDF). In order to compute the error norm, the difference between the
approximate solutions and the exact solutions was calculated. The numerical results
also show the proposed method produces smaller errors when compared to modified
Euler method. The accuracy of the solutions obtained by BBDF and BDF are
comparable particularly when the finer step sizes are used. However, in term of
execution time, the proposed method BBDF outperformed BDF method. The solutions
obtained were illustrated by graphs.
In conclusion, the numerical results clearly demonstrate the efficiency of using BBDF
methods proposed in this study for solving fuzzy differential equations. From the
results of tests problems, the modified BBDF method reveals that the execution time
has been reduced and the numerical result is accurate, which proves its superiority on
the existing methods. |
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