Block hybrid methods for numerical treatment of differential equations with applications
This thesis focuses mainly on deriving block hybrid methods for solving Ordinary Differential Equations (ODEs). Block hybrid methods are the methods that generate a block of new solutions at the main and off-step points concurrently. The first part of the thesis is about the derivation of the exp...
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| Format: | Thesis |
| Language: | English |
| Published: |
2016
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/69781/1/IPM%202016%207%20-%20IR.pdf |
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| Summary: | This thesis focuses mainly on deriving block hybrid methods for solving Ordinary Differential
Equations (ODEs). Block hybrid methods are the methods that generate a block
of new solutions at the main and off-step points concurrently. The first part of the thesis
is about the derivation of the explicit block hybrid methods based on Newton-Gregory
backward difference interpolation formula for solving first order ODEs. The regions
of stability are presented. The numerical results are shown in terms of total steps and
accuracy.
The second part of the thesis describes the mathematical formulation of explicit and
implicit one-point block hybrid methods for first order ODEs whereby the derivation
involves the divided differences relative to main and off-step points. The stability properties
are discussed. The explicit and implicit block hybrid methods are implemented
in predictor-corrector mode of constant step size to obtain the numerical approximation
for first order ODEs. The implementation of block hybrid methods in variable step size
is also presented. Some numerical examples are given to illustrate the efficiency of the
methods.
The one-point block hybrid methods are then implemented for numerical solution of first
order delay differential equations (DDEs). The Q-stability of the methods is investigated.
Since the block hybrid methods include the approximate solution at both the main and
additional off-steps points, more computed values that surrounding the delay term can be
used to provide a better estimation in interpolating the delay term.
The third part of the thesis is mainly focused on block hybrid collocation methods for obtaining
direct solution of second-, third- and fourth-order ODEs. The derivation involves
interpolation and collocation of the basic polynomial. The stability properties are investigated.
Illustrative examples are presented to demonstrate the efficiency of the methods.
The block hybrid collocation methods are also applied to solve the physical problems
such as Lane-Emden equation, Van Der Pol oscillator, Fermi-Pasta-Ulam problem, the
nonlinear Genesio equation, the problem in thin film flow and the fourth order problem from ship dynamics.
As a whole, the block hybrid methods for solving different orders of ordinary differential
equations have been presented. The illustrative examples demonstrate the accuracy
advantage of the block hybrid methods. |
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