Ghent University Academic Bibliography

Advanced

Application of exponential fitting techniques to numerical methods for solving differential equations

Davy Hollevoet (2013)
abstract
Ever since the work of Isaac Newton and Gottfried Leibniz in the late 17th century, differential equations (DEs) have been an important concept in many branches of science. Differential equations arise spontaneously in i.a. physics, engineering, chemistry, biology, economics and a lot of fields in between. From the motion of a pendulum, studied by high-school students, to the wave functions of a quantum system, studied by brave scientists: differential equations are common and unavoidable. It is therefore no surprise that a large number of mathematicians have studied, and still study these equations. The better the techniques for solving DEs, the faster the fields where they appear, can advance. Sadly, however, mathematicians have yet to find a technique (or a combination of techniques) that can solve all DEs analytically. Luckily, in the meanwhile, for a lot of applications, approximate solutions are also sufficient. The numerical methods studied in this work compute such approximations. Instead of providing the hypothetical scientist with an explicit, continuous recipe for the solution to their problem, these methods give them an approximation of the solution at a number of discrete points. Numerical methods of this type have been the topic of research since the days of Leonhard Euler, and still are. Nowadays, however, the computations are performed by digital processors, which are well-suited for these methods, even though many of the ideas predate the modern digital computer by almost a few centuries. The ever increasing power of even the smallest processor allows us to devise newer and more elaborate methods. In this work, we will look at a few well-known numerical methods for the solution of differential equations. These methods are combined with a technique called exponential fitting, which produces exponentially fitted methods: classical methods with modified coefficients. The original idea behind this technique is to improve the performance on problems with oscillatory solutions.
Please use this url to cite or link to this publication:
author
promoter
UGent
organization
year
type
dissertation
publication status
published
subject
keyword
numerical analysis, differential equations, boundary value problems, exponential fitting, parameter selection
pages
244 pages
publisher
Ghent University. Faculty of Sciences
place of publication
Ghent, Belgium
defense location
Gent : Campus Sterre (S9, auditorium A3)
defense date
2013-06-14 17:00
language
English
UGent publication?
yes
classification
D1
copyright statement
I have retained and own the full copyright for this publication
id
4107125
handle
http://hdl.handle.net/1854/LU-4107125
date created
2013-07-30 10:22:12
date last changed
2017-01-16 10:43:41
@phdthesis{4107125,
  abstract     = {Ever since the work of Isaac Newton and Gottfried Leibniz in the late 17th century, differential equations (DEs) have been an important concept in many branches of science. Differential equations arise spontaneously in i.a. physics, engineering, chemistry, biology, economics and a lot of fields in between. From the motion of a pendulum, studied by high-school students, to the wave functions of a quantum system, studied by brave scientists: differential equations are common and unavoidable. It is therefore no surprise that a large number of mathematicians have studied, and still study these equations. The better the techniques for solving DEs, the faster the fields where they appear, can advance.
Sadly, however, mathematicians have yet to find a technique (or a combination of techniques) that can solve all DEs analytically. Luckily, in the meanwhile, for a lot of applications, approximate solutions are also sufficient. The numerical methods studied in this work compute such approximations. Instead of providing the hypothetical scientist with an explicit, continuous recipe for the solution to their problem, these methods give them an approximation of the solution at a number of discrete points. Numerical methods of this type have been the topic of research since the days of Leonhard Euler, and still are. Nowadays, however, the computations are performed by digital processors, which are well-suited for these methods, even though many of the ideas predate the modern digital computer by almost a few centuries. The ever increasing power of even the smallest processor allows us to devise newer and more elaborate methods.
In this work, we will look at a few well-known numerical methods for the solution of differential equations. These methods are combined with a technique called exponential fitting, which produces exponentially fitted methods: classical methods with modified coefficients. The original idea behind this technique is to improve the performance on problems with oscillatory solutions.},
  author       = {Hollevoet, Davy},
  keyword      = {numerical analysis,differential equations,boundary value problems,exponential fitting,parameter selection},
  language     = {eng},
  pages        = {244},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {Application of exponential fitting techniques to numerical methods for solving differential equations},
  year         = {2013},
}

Chicago
Hollevoet, Davy. 2013. “Application of Exponential Fitting Techniques to Numerical Methods for Solving Differential Equations”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Hollevoet, D. (2013). Application of exponential fitting techniques to numerical methods for solving differential equations. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Hollevoet D. Application of exponential fitting techniques to numerical methods for solving differential equations. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2013.
MLA
Hollevoet, Davy. “Application of Exponential Fitting Techniques to Numerical Methods for Solving Differential Equations.” 2013 : n. pag. Print.