Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations

In this research, methods that will be able to solve the second order initial value problem (IVP) directly are developed. These methods are in the scheme of a multi-step method which is known as the two-step method. The two-step method has an advantage as it can estimate the solution with less fu...

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Main Author:	Md Ariffin, Latifah
Format:	Thesis
Language:	English
Published:	2016
Subjects:	Differential equations
Online Access:	http://psasir.upm.edu.my/id/eprint/69295/1/FS%202016%2081%20UPM%20IR.pdf
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spelling	my-upm-ir.692952022-03-11T02:21:10Z Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations 2016-12 Md Ariffin, Latifah In this research, methods that will be able to solve the second order initial value problem (IVP) directly are developed. These methods are in the scheme of a multi-step method which is known as the two-step method. The two-step method has an advantage as it can estimate the solution with less function evaluations compared to the one-step method. The selection of step size is also important in obtaining more accurate and efficient results. Smaller step sizes will produce a more accurate result, but it lengthens the execution time. Two-Step Runge-Kutta (TSRK) method were derived to solve first-order Ordinary Differential Equations (ODE). The order conditions of TSRK method were obtained by using Taylor series expansion. The explicit TSRK method was derived and its stability were investigated. It was then analyzed experimentally. The numerical results obtained were analyzed by making comparisons with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The explicit Two-Step Runge-Kutta-Nyström (TSRKN) method was derived with reference to the technique of deriving the TSRK method. The order conditions of TSRKN method were also obtained by using Taylor series expansion. The strategies in choosing the free parameters were also discussed. The stability of the methods derived were also investigated. The explicit TSRKN method was then analyzed experimentally and comparisons of the numerical results obtained were made with the existing methods in terms of maximum global error, number of steps taken and function evaluations. Next, we discussed the derivation of an embedded pair of the TSRKN (ETSRKN) methods for solving second order ODE. Variable step size codes were developed and numerical results were compared with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The ETSRKN were then used to solve second-order Fuzzy Differential Equation (FDE). We observe that ETSRKN gives better accuracy at the end point of fuzzy interval compared to other existing methods. In conclusion, the methods developed in this thesis are able to solve the system of second-order differential equation (DE) which consists of ODE and FDE directly. Differential equations 2016-12 Thesis http://psasir.upm.edu.my/id/eprint/69295/ http://psasir.upm.edu.my/id/eprint/69295/1/FS%202016%2081%20UPM%20IR.pdf text en public doctoral Universiti Putra Malaysia Differential equations Suleiman, Mohamed
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
advisor	Suleiman, Mohamed
topic	Differential equations
spellingShingle	Differential equations Md Ariffin, Latifah Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
description	In this research, methods that will be able to solve the second order initial value problem (IVP) directly are developed. These methods are in the scheme of a multi-step method which is known as the two-step method. The two-step method has an advantage as it can estimate the solution with less function evaluations compared to the one-step method. The selection of step size is also important in obtaining more accurate and efficient results. Smaller step sizes will produce a more accurate result, but it lengthens the execution time. Two-Step Runge-Kutta (TSRK) method were derived to solve first-order Ordinary Differential Equations (ODE). The order conditions of TSRK method were obtained by using Taylor series expansion. The explicit TSRK method was derived and its stability were investigated. It was then analyzed experimentally. The numerical results obtained were analyzed by making comparisons with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The explicit Two-Step Runge-Kutta-Nyström (TSRKN) method was derived with reference to the technique of deriving the TSRK method. The order conditions of TSRKN method were also obtained by using Taylor series expansion. The strategies in choosing the free parameters were also discussed. The stability of the methods derived were also investigated. The explicit TSRKN method was then analyzed experimentally and comparisons of the numerical results obtained were made with the existing methods in terms of maximum global error, number of steps taken and function evaluations. Next, we discussed the derivation of an embedded pair of the TSRKN (ETSRKN) methods for solving second order ODE. Variable step size codes were developed and numerical results were compared with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The ETSRKN were then used to solve second-order Fuzzy Differential Equation (FDE). We observe that ETSRKN gives better accuracy at the end point of fuzzy interval compared to other existing methods. In conclusion, the methods developed in this thesis are able to solve the system of second-order differential equation (DE) which consists of ODE and FDE directly.
format	Thesis
qualification_level	Doctorate
author	Md Ariffin, Latifah
author_facet	Md Ariffin, Latifah
author_sort	Md Ariffin, Latifah
title	Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
title_short	Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
title_full	Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
title_fullStr	Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
title_full_unstemmed	Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations
title_sort	two step runge-kutta-nyström method for solving second order ordinary differential equations
granting_institution	Universiti Putra Malaysia
publishDate	2016
url	http://psasir.upm.edu.my/id/eprint/69295/1/FS%202016%2081%20UPM%20IR.pdf
_version_	1747812680814559232

Two Step Runge-Kutta-Nyström method for solving second order ordinary differential equations

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