Fifth-stage stochastic runge-kutta method for stochastic differential equations

Most of the physical systems around us are subjected to uncontrollable factors. Hence, models for these systems are required via stochastic differential equations (SDEs). However, it is often difficult to find analytical solutions of SDEs. In such a case, a numerical method provides an alternative w...

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Main Author:	Noor Amalina Nisa, Ariffin
Format:	Thesis
Language:	English
Published:	2018
Subjects:	QA Mathematics
Online Access:	http://umpir.ump.edu.my/id/eprint/23433/1/Fifth-stage%20stochastic%20runge-kutta%20method%20for%20stochastic%20differential%20equations.pdf
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spelling	my-ump-ir.234332021-11-10T02:24:35Z Fifth-stage stochastic runge-kutta method for stochastic differential equations 2018-07 Noor Amalina Nisa, Ariffin QA Mathematics Most of the physical systems around us are subjected to uncontrollable factors. Hence, models for these systems are required via stochastic differential equations (SDEs). However, it is often difficult to find analytical solutions of SDEs. In such a case, a numerical method provides an alternative way to solve problems with such systems. The development of numerical methods for SDEs is far from complete. Conversely, numerical methods for their deterministic counterparts are well-developed. The relative paucity of numerical methods in SDEs is due to the complexity of approximating high-order multiple stochastic integrals. A stochastic integral provides information of the Wiener process, which then contributes to the order of the methods. Motivated by the development of high-order Runge-Kutta methods for solving ordinary differential equations (ODEs), this research was aimed to develop a new fifth-stage stochastic Runge-Kutta (SRK5) method for SDEs with a strong order of 2.0. The derivation of this derivative-free method was based on the stochastic Taylor series expansion. The Taylor series expansion for both Taylor series and numerical solutions up to 2.0 order of convergence have been expanded. The analysis of the order conditions for the SRK5 was performed by evaluating the local truncation error in terms of the mean square in MAPLE. The difference between Taylor series solution and numerical solution was evaluated. In order to analyze the order conditions, the local truncation error between both solutions was minimized. All equations arise in order conditions analysis have been solved simultaneously by using MATLAB, and three newly developed SRK5 schemes were presented. A mean-square stability analysis was then performed on the SRK5 scheme in order to ensure the efficiency of the newly-developed numerical scheme. The stability function for each scheme was derived and the change of variables have been applied for stability region plotting purposed. Stability region have been plotted on the uv-plane to visualize the stability property of each scheme. In addition, the simple numerical experiments have been performed to check on the stability property. In order to validate the efficiency of the newly develop numerical schemes, all schemes have been used to solve both linear and non-linear stochastic models respectively in C++. The performances of SRK5 schemes in solving linear SDEs have been measured by comparing the root mean-square error and the global error obtained by solving linear SDE via SRK5 schemes, SRK2.0, SRK4, Milstein and Euler-Maruyama methods. Besides, three different models of fermentation process were solved by using SRK5 schemes, SRK2.0 and SRK4. The errors obtained have been compared. The SRK5 is proved to be a more efficient tool for the numerical approximation of solutions to SDEs. 2018-07 Thesis http://umpir.ump.edu.my/id/eprint/23433/ http://umpir.ump.edu.my/id/eprint/23433/1/Fifth-stage%20stochastic%20runge-kutta%20method%20for%20stochastic%20differential%20equations.pdf pdf en public phd doctoral Universiti Malaysia Pahang Faculty of Industrial Sciences and Technology
institution	Universiti Malaysia Pahang Al-Sultan Abdullah
collection	UMPSA Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Noor Amalina Nisa, Ariffin Fifth-stage stochastic runge-kutta method for stochastic differential equations
description	Most of the physical systems around us are subjected to uncontrollable factors. Hence, models for these systems are required via stochastic differential equations (SDEs). However, it is often difficult to find analytical solutions of SDEs. In such a case, a numerical method provides an alternative way to solve problems with such systems. The development of numerical methods for SDEs is far from complete. Conversely, numerical methods for their deterministic counterparts are well-developed. The relative paucity of numerical methods in SDEs is due to the complexity of approximating high-order multiple stochastic integrals. A stochastic integral provides information of the Wiener process, which then contributes to the order of the methods. Motivated by the development of high-order Runge-Kutta methods for solving ordinary differential equations (ODEs), this research was aimed to develop a new fifth-stage stochastic Runge-Kutta (SRK5) method for SDEs with a strong order of 2.0. The derivation of this derivative-free method was based on the stochastic Taylor series expansion. The Taylor series expansion for both Taylor series and numerical solutions up to 2.0 order of convergence have been expanded. The analysis of the order conditions for the SRK5 was performed by evaluating the local truncation error in terms of the mean square in MAPLE. The difference between Taylor series solution and numerical solution was evaluated. In order to analyze the order conditions, the local truncation error between both solutions was minimized. All equations arise in order conditions analysis have been solved simultaneously by using MATLAB, and three newly developed SRK5 schemes were presented. A mean-square stability analysis was then performed on the SRK5 scheme in order to ensure the efficiency of the newly-developed numerical scheme. The stability function for each scheme was derived and the change of variables have been applied for stability region plotting purposed. Stability region have been plotted on the uv-plane to visualize the stability property of each scheme. In addition, the simple numerical experiments have been performed to check on the stability property. In order to validate the efficiency of the newly develop numerical schemes, all schemes have been used to solve both linear and non-linear stochastic models respectively in C++. The performances of SRK5 schemes in solving linear SDEs have been measured by comparing the root mean-square error and the global error obtained by solving linear SDE via SRK5 schemes, SRK2.0, SRK4, Milstein and Euler-Maruyama methods. Besides, three different models of fermentation process were solved by using SRK5 schemes, SRK2.0 and SRK4. The errors obtained have been compared. The SRK5 is proved to be a more efficient tool for the numerical approximation of solutions to SDEs.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Noor Amalina Nisa, Ariffin
author_facet	Noor Amalina Nisa, Ariffin
author_sort	Noor Amalina Nisa, Ariffin
title	Fifth-stage stochastic runge-kutta method for stochastic differential equations
title_short	Fifth-stage stochastic runge-kutta method for stochastic differential equations
title_full	Fifth-stage stochastic runge-kutta method for stochastic differential equations
title_fullStr	Fifth-stage stochastic runge-kutta method for stochastic differential equations
title_full_unstemmed	Fifth-stage stochastic runge-kutta method for stochastic differential equations
title_sort	fifth-stage stochastic runge-kutta method for stochastic differential equations
granting_institution	Universiti Malaysia Pahang
granting_department	Faculty of Industrial Sciences and Technology
publishDate	2018
url	http://umpir.ump.edu.my/id/eprint/23433/1/Fifth-stage%20stochastic%20runge-kutta%20method%20for%20stochastic%20differential%20equations.pdf
_version_	1783732067031318528

Fifth-stage stochastic runge-kutta method for stochastic differential equations

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