A comparison of numerical methods for solving the bratu and bratu-type problems
The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique solution is obtained when A = Ac where Ac = 3.513830719 is the critical value. The First Bratu-Type problem corresponds A = _7[2 while the...
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| Format: | Thesis |
| Language: | English |
| Published: |
2005
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| Subjects: | |
| Online Access: | http://eprints.uthm.edu.my/7437/1/24p%20RUHAILA%20MD.KASMANI.pdf |
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| Summary: | The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two
exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique
solution is obtained when A = Ac where Ac = 3.513830719 is the critical
value. The First Bratu-Type problem corresponds A = _7[2 while the Second
Bratu-Type problem is ul/(x) + 7[ 2e-u(x) = o. The exact solution of the First
Bratu-Type problem blows up at x = 0.5 while the Second Bratu-Type problem
is continuous. The present work seeks to compare various numerical methods
for solving the Bratu and Bratu-Type problems. The numerical methods are the
standard Adomian decomposition method, the modified Adomian decomposition
method, the shooting method and the finite difference method. These methods
are implemented using Maple. Convergence is achieved by applying the four
methods when 0 < A ::; 2, however the shooting method is the most effective
method as it gives the smallest maximum absolute error. ·When A = Ac, none of
these methods give the convergence solutions. Due to the nature of the solution
of the First Bratu-Type problem, only the shooting method and the modified
Adomian decomposition method can give the convergence values to the exact
solution. The finite difference method is proved to be the most effective method
for the Second Bratu-Type problem compared to other methods.
Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposi�tion method, modified Adomian decomposition method, shooting method, finite
difference method. |
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