Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications
Traditionally, higher order ordinary differential equations (ODEs) are solved by reducing them to an equivalent system of first order ODEs. However, it is more cost effective if they can be solved directly by numerical methods. Block methods approximate the solutions of the ODEs at more than one...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2021
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/92773/1/FS%202021%2033%20-%20IR.pdf |
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| Summary: | Traditionally, higher order ordinary differential equations (ODEs) are solved
by reducing them to an equivalent system of first order ODEs. However, it is
more cost effective if they can be solved directly by numerical methods. Block
methods approximate the solutions of the ODEs at more than one point at
one time step, hence faster solutions can be obtained. It is well-known too
that a more accurate numerical results can be obtained by incorporating the
higher derivatives of the solutions in the method. Based on these arguments,
we are focused on developing block methods with extra derivatives for directly
solving second, third and fourth order ODEs.
In this study, two-point and three-point implicit block methods with extra
derivatives are derived using Hermite Interpolating polynomial as the basis
function. The technique of integration is used in the derivation as it is more
straight forward and can easily be carried out compared to the existing technique
of collocation and interpolation in which the points need to be collocated
and interpolated resulting in a huge system of linear equations which need to
be solved simultaneously.
The thesis consists of three parts, the first part of the thesis described the
derivation of two-point and three-point implicit block methods which incorporated
the second, third and fourth derivatives of the solution for directly
solving general second order ODEs. Absolute stability for both block methods
is presented. The second part of the thesis is focused on the derivation of twoi point and three-point implicit block methods which include the third, fourth
and fifth derivatives of the solutions for directly solving general third order
ODEs. The last part of the thesis concerned with the construction of twopoint
and three-point implicit block method which involved the fourth and the
fifth derivatives of the solution for directly solving general fourth order ODEs.
The basic properties of all the methods, such as algebraic order, zero-stability,
and convergence are established. Numerical results clearly show that the new
proposed methods are more efficient in terms of accuracy and computational
time when compared with well-known existing methods. Applications in several
real fields also illustrate the efficiency of the proposed methods.
In conclusion, the new block methods with extra derivatives and codes developed
based on the methods are suitable for solving second, third and fourth
order ODEs respectively and can be applied to solve physical problems. |
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