Hybrid methods for solving higher order ordinary differential equations

In this thesis, a class of numerical integrators for solving special higher order ordinary differential equations (ODEs) is proposed. The methods are multistage and multistep in nature. This class of integrators is called ”hybrid methods”, specifically, hybrid methods for directly solving special...

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Main Author:	Jikantoro, Yusuf Dauda
Format:	Thesis
Language:	English
Published:	2018
Subjects:	Differential equations - Numerical solutions Numerical analysis
Online Access:	http://psasir.upm.edu.my/id/eprint/76922/1/FS%202018%2092%20-%20IR.pdf
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spelling	my-upm-ir.769222020-02-11T01:36:41Z Hybrid methods for solving higher order ordinary differential equations 2018-04 Jikantoro, Yusuf Dauda In this thesis, a class of numerical integrators for solving special higher order ordinary differential equations (ODEs) is proposed. The methods are multistage and multistep in nature. This class of integrators is called ”hybrid methods”, specifically, hybrid methods for directly solving special third order ODEs denoted by HMTD and for directly solving special fourth order ODEs denoted by HMFD are proposed. B-series approach is developed and used in deriving their algebraic order conditions and analyzing the order of convergence of the methods. Using the algebraic order conditions, a class of explicit HMTD and HMFD are derived. The methods are applied to some test problems alongside some existing integrators in the literature for the purpose of validation. Results obtained show that the proposed methods in this thesis are a better alternatives. To analyze the methods further, convergence analysis is conducted via consistency and zero stability, where the methods are found to be consistent and zero stable, hence, they are convergent. Absolute stability of the methods is also investigated, where stability polynomials of the methods are presented for obtaining intervals and regions of absolute stability. Finally, a set of embedded pairs of two-step hybrid methods for solving special second order ODEs are proposed and investigated. The methods are tested on some model problems using different error tolerances. Results obtained are compared with those of existing embedded methods possessing similar properties. From the comparison, it is found that the new embedded methods possess better accuracy and efficiency. Differential equations - Numerical solutions Numerical analysis 2018-04 Thesis http://psasir.upm.edu.my/id/eprint/76922/ http://psasir.upm.edu.my/id/eprint/76922/1/FS%202018%2092%20-%20IR.pdf text en public doctoral Universiti Putra Malaysia Differential equations - Numerical solutions Numerical analysis
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Differential equations - Numerical solutions Numerical analysis
spellingShingle	Differential equations - Numerical solutions Numerical analysis Jikantoro, Yusuf Dauda Hybrid methods for solving higher order ordinary differential equations
description	In this thesis, a class of numerical integrators for solving special higher order ordinary differential equations (ODEs) is proposed. The methods are multistage and multistep in nature. This class of integrators is called ”hybrid methods”, specifically, hybrid methods for directly solving special third order ODEs denoted by HMTD and for directly solving special fourth order ODEs denoted by HMFD are proposed. B-series approach is developed and used in deriving their algebraic order conditions and analyzing the order of convergence of the methods. Using the algebraic order conditions, a class of explicit HMTD and HMFD are derived. The methods are applied to some test problems alongside some existing integrators in the literature for the purpose of validation. Results obtained show that the proposed methods in this thesis are a better alternatives. To analyze the methods further, convergence analysis is conducted via consistency and zero stability, where the methods are found to be consistent and zero stable, hence, they are convergent. Absolute stability of the methods is also investigated, where stability polynomials of the methods are presented for obtaining intervals and regions of absolute stability. Finally, a set of embedded pairs of two-step hybrid methods for solving special second order ODEs are proposed and investigated. The methods are tested on some model problems using different error tolerances. Results obtained are compared with those of existing embedded methods possessing similar properties. From the comparison, it is found that the new embedded methods possess better accuracy and efficiency.
format	Thesis
qualification_level	Doctorate
author	Jikantoro, Yusuf Dauda
author_facet	Jikantoro, Yusuf Dauda
author_sort	Jikantoro, Yusuf Dauda
title	Hybrid methods for solving higher order ordinary differential equations
title_short	Hybrid methods for solving higher order ordinary differential equations
title_full	Hybrid methods for solving higher order ordinary differential equations
title_fullStr	Hybrid methods for solving higher order ordinary differential equations
title_full_unstemmed	Hybrid methods for solving higher order ordinary differential equations
title_sort	hybrid methods for solving higher order ordinary differential equations
granting_institution	Universiti Putra Malaysia
publishDate	2018
url	http://psasir.upm.edu.my/id/eprint/76922/1/FS%202018%2092%20-%20IR.pdf
_version_	1747813192447295488

Hybrid methods for solving higher order ordinary differential equations

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