Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations

An introduction to Runge-Kutta methods for the solution of ordinary differential equations (ODEs) is introduced. The technique of using Singly Diagonally Implicit Runge-Kutta (SDIRK) method for the integration of stiff and non-stiff ODEs has been widely accepted, this is because SDIRK method is comp...

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Main Author:	Alkhasawneh, Rae’d Ali Ahmed
Format:	Thesis
Language:	English English
Published:	2006
Subjects:	Runge-Kutta formulas Delay differential equations Differential equations
Online Access:	http://psasir.upm.edu.my/id/eprint/4977/1/FS_2006_63.pdf
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spelling	my-upm-ir.49772013-05-27T07:19:31Z Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations 2006 Alkhasawneh, Rae’d Ali Ahmed An introduction to Runge-Kutta methods for the solution of ordinary differential equations (ODEs) is introduced. The technique of using Singly Diagonally Implicit Runge-Kutta (SDIRK) method for the integration of stiff and non-stiff ODEs has been widely accepted, this is because SDIRK method is computationally efficient and stiffly stable. Consequently embedded SDIRK method of fourth-order six stage in fifth-order seven stage which has the property that the first row of the coefficient matrix is equal to zero and the last row of the coefficient matrix is equal to the vector output value is constructed. The stability region of the method when applied to linear ODE is given. Numerical results when stiff and non-stiff first order ODEs are solved using the method are tabulated and compared with the method in current use. Introduction to delay differential equations (DDEs) and the areas where they arise are given. A brief discussion on Runge-Kutta method when adapted to delay differential equation is introduced. SDIRK method which has been derived previously is used to solve delay differential equations; the delay term is approximated using divided difference interpolation. Numerical results are tabulated and compared with the existing methods. The stability aspects of SDIRK method when applied to DDEs using Lagrange interpolation are investigated and the region of stability is presented. Runge-Kutta-Nystróm (RKN) method for the solution of special second-order ordinary differential equations of the form ),(yxfy=′′ is discussed. Consequently, Singly Diagonally Implicit Runge-Kutta Nystróm (SDIRKN) method of third-order three stage embedded in fourth-order four stage with small error coefficients is constructed. The stability region of the new method is presented. The method is then used to solve both stiff and non-stiff special second order ODEs and the numerical results suggest that the new method is more efficient compared to the current methods in use. Finally, introduction to general Runge-Kutta-Nystrom (RKNG) method for the solution of second-order ordinary differential equations of the form ),,(yyxfy′=′′ is given. A new embedded Singly Diagonally Implicit Runge-Kutta-Nystróm General (SDIRKNG) method of third-order four stage embedded in fourth-order five stage is derived. Analysis on the stability aspects of the new method is given and numerical results when the method is used to solve both stiff and non-stiff second order ODEs are presented. The results indicate the superiority of the new method compared to the existing method. Runge-Kutta formulas Delay differential equations Differential equations 2006 Thesis http://psasir.upm.edu.my/id/eprint/4977/ http://psasir.upm.edu.my/id/eprint/4977/1/FS_2006_63.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Runge-Kutta formulas Delay differential equations Differential equations Science English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Runge-Kutta formulas Delay differential equations Differential equations
spellingShingle	Runge-Kutta formulas Delay differential equations Differential equations Alkhasawneh, Rae’d Ali Ahmed Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
description	An introduction to Runge-Kutta methods for the solution of ordinary differential equations (ODEs) is introduced. The technique of using Singly Diagonally Implicit Runge-Kutta (SDIRK) method for the integration of stiff and non-stiff ODEs has been widely accepted, this is because SDIRK method is computationally efficient and stiffly stable. Consequently embedded SDIRK method of fourth-order six stage in fifth-order seven stage which has the property that the first row of the coefficient matrix is equal to zero and the last row of the coefficient matrix is equal to the vector output value is constructed. The stability region of the method when applied to linear ODE is given. Numerical results when stiff and non-stiff first order ODEs are solved using the method are tabulated and compared with the method in current use. Introduction to delay differential equations (DDEs) and the areas where they arise are given. A brief discussion on Runge-Kutta method when adapted to delay differential equation is introduced. SDIRK method which has been derived previously is used to solve delay differential equations; the delay term is approximated using divided difference interpolation. Numerical results are tabulated and compared with the existing methods. The stability aspects of SDIRK method when applied to DDEs using Lagrange interpolation are investigated and the region of stability is presented. Runge-Kutta-Nystróm (RKN) method for the solution of special second-order ordinary differential equations of the form ),(yxfy=′′ is discussed. Consequently, Singly Diagonally Implicit Runge-Kutta Nystróm (SDIRKN) method of third-order three stage embedded in fourth-order four stage with small error coefficients is constructed. The stability region of the new method is presented. The method is then used to solve both stiff and non-stiff special second order ODEs and the numerical results suggest that the new method is more efficient compared to the current methods in use. Finally, introduction to general Runge-Kutta-Nystrom (RKNG) method for the solution of second-order ordinary differential equations of the form ),,(yyxfy′=′′ is given. A new embedded Singly Diagonally Implicit Runge-Kutta-Nystróm General (SDIRKNG) method of third-order four stage embedded in fourth-order five stage is derived. Analysis on the stability aspects of the new method is given and numerical results when the method is used to solve both stiff and non-stiff second order ODEs are presented. The results indicate the superiority of the new method compared to the existing method.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Alkhasawneh, Rae’d Ali Ahmed
author_facet	Alkhasawneh, Rae’d Ali Ahmed
author_sort	Alkhasawneh, Rae’d Ali Ahmed
title	Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
title_short	Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
title_full	Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
title_fullStr	Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
title_full_unstemmed	Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
title_sort	runge-kutta methods for solving ordinary and delay differential equations
granting_institution	Universiti Putra Malaysia
granting_department	Science
publishDate	2006
url	http://psasir.upm.edu.my/id/eprint/4977/1/FS_2006_63.pdf
_version_	1747810322404605952

Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations

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