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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my-upm-ir.927732022-05-10T04:17:03Z Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications 2021-02 M., Allogmany Reem Ayed 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. Differential equations - Numerical solutions Differential equations 2021-02 Thesis http://psasir.upm.edu.my/id/eprint/92773/ http://psasir.upm.edu.my/id/eprint/92773/1/FS%202021%2033%20-%20IR.pdf text en public doctoral Universiti Putra Malaysia Differential equations - Numerical solutions Differential equations Ismail, Fudziah |
| institution |
Universiti Putra Malaysia |
| collection |
PSAS Institutional Repository |
| language |
English |
| advisor |
Ismail, Fudziah |
| topic |
Differential equations - Numerical solutions Differential equations |
| spellingShingle |
Differential equations - Numerical solutions Differential equations M., Allogmany Reem Ayed Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| description |
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. |
| format |
Thesis |
| qualification_level |
Doctorate |
| author |
M., Allogmany Reem Ayed |
| author_facet |
M., Allogmany Reem Ayed |
| author_sort |
M., Allogmany Reem Ayed |
| title |
Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| title_short |
Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| title_full |
Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| title_fullStr |
Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| title_full_unstemmed |
Implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| title_sort |
implicit block methods with extra derivatives for solving general higher-order ordinary differential equations with applications |
| granting_institution |
Universiti Putra Malaysia |
| publishDate |
2021 |
| url |
http://psasir.upm.edu.my/id/eprint/92773/1/FS%202021%2033%20-%20IR.pdf |
| _version_ |
1747813763352887296 |