Numerical methods for solving oscillatory and fuzzy differential equations
In this thesis we develop ¯ve numerical schemes for solving ordinary di®erential equations. These include exponentially-¯tted Runge-Kutta method, trigonometri- cally ¯tted hybrid method, Legendre wavelet method on large intervals as well as an iterative spectral collocation method, exponentially-...
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my-upm-ir.704972019-10-30T03:01:47Z Numerical methods for solving oscillatory and fuzzy differential equations 2014-06 Dizicheh, Ali Karimi In this thesis we develop ¯ve numerical schemes for solving ordinary di®erential equations. These include exponentially-¯tted Runge-Kutta method, trigonometri- cally ¯tted hybrid method, Legendre wavelet method on large intervals as well as an iterative spectral collocation method, exponentially-¯tted fuzzy Runge-Kutta method, exponentially-¯tted system of fuzzy Runge-Kutta method. The stability analysis, estimation of local truncation errors and the e±ciency of the methods' implementation in computer programs are discussed. An exponentially-¯tted explicit Runge-Kutta method of algebraic order 4 is for- mulated for the ¯rst-order ordinary di®erential equations y0 = f(x; y); y(x0) = y0: It integrates exactly the ¯rst-order systems where their solutions are expressed as linear combinations of fexp (wx); exp (¡wx)g or fcos (¸x); sin(¸x)g where w = ¸i. Stability analysis of our approach as well as a good estimation for the local trunca- tion errors are presented. The e±ciency of the exponentially-¯tted Runge-Kutta method is tested via some numerical experiments and a comparison with other existing methods. A trigonometrically ¯tted explicit hybrid three-stage method is derived for the second-order initial value problems with oscillatory solutions. We compare our results with the classical hybrid method and the trigonometrically ¯tted explicit Runge-Kutta method through several examples. Our results indicate that trigono- metrically ¯tted explicit hybrid method is more e±cient than the classical hybrid method. we analyze the stability, phase- lag (dispersion) and dissipation. An iterative spectral collocation method are introduced for solving initial value problems de¯ned on large intervals. Indeed, the Legendre wavelet method is ex- tended and proved valid for large interval. Then, the Legendre-Guass collocation points of the Legendre wavelets are computed. By employing an interpolation based on Legendre wavelet, we ¯nd approximate solution for any order (¯rst-order and second-order) di®erential equations. Using this strategy the iterative spectral method converts the di®erential equation to a set of algebraic equations. Solving this set of algebraic equations yields an approximate solution. Using exponentially-¯tted Runge-Kutta (EFRK) method, we develop a method for numerically solving fuzzy ¯rst order linear and nonlinear di®erential equations under generalized di®erentiability. In addition, this method is applied for the sys- tem of ¯rst order fuzzy di®erential equations with uncertainty. The generalized Hukuhara di®erentiability are applied to estimate the solutions. For solving the fuzzy problems, the exponentially-¯tted Runge-Kutta method is applied. Finally, some examples are solved to illustrate our proposed approaches. The results are compared with those in the literature. We show that our proposed methods are simple and more accurate than the other existing methods. Differential equations - Numerical solutions Fuzzy sets Numerical analysis 2014-06 Thesis http://psasir.upm.edu.my/id/eprint/70497/ http://psasir.upm.edu.my/id/eprint/70497/1/FS%202014%2063%20-%20IR.pdf text en public doctoral Universiti Putra Malaysia Differential equations - Numerical solutions Fuzzy sets Numerical analysis |
| institution |
Universiti Putra Malaysia |
| collection |
PSAS Institutional Repository |
| language |
English |
| topic |
Differential equations - Numerical solutions Fuzzy sets Numerical analysis |
| spellingShingle |
Differential equations - Numerical solutions Fuzzy sets Numerical analysis Dizicheh, Ali Karimi Numerical methods for solving oscillatory and fuzzy differential equations |
| description |
In this thesis we develop ¯ve numerical schemes for solving ordinary di®erential
equations. These include exponentially-¯tted Runge-Kutta method, trigonometri-
cally ¯tted hybrid method, Legendre wavelet method on large intervals as well as
an iterative spectral collocation method, exponentially-¯tted fuzzy Runge-Kutta
method, exponentially-¯tted system of fuzzy Runge-Kutta method. The stability
analysis, estimation of local truncation errors and the e±ciency of the methods'
implementation in computer programs are discussed.
An exponentially-¯tted explicit Runge-Kutta method of algebraic order 4 is for-
mulated for the ¯rst-order ordinary di®erential equations
y0 = f(x; y); y(x0) = y0:
It integrates exactly the ¯rst-order systems where their solutions are expressed as
linear combinations of fexp (wx); exp (¡wx)g or fcos (¸x); sin(¸x)g where w = ¸i.
Stability analysis of our approach as well as a good estimation for the local trunca-
tion errors are presented. The e±ciency of the exponentially-¯tted Runge-Kutta
method is tested via some numerical experiments and a comparison with other
existing methods.
A trigonometrically ¯tted explicit hybrid three-stage method is derived for the
second-order initial value problems with oscillatory solutions. We compare our
results with the classical hybrid method and the trigonometrically ¯tted explicit
Runge-Kutta method through several examples. Our results indicate that trigono- metrically ¯tted explicit hybrid method is more e±cient than the classical hybrid
method. we analyze the stability, phase- lag (dispersion) and dissipation.
An iterative spectral collocation method are introduced for solving initial value
problems de¯ned on large intervals. Indeed, the Legendre wavelet method is ex-
tended and proved valid for large interval. Then, the Legendre-Guass collocation
points of the Legendre wavelets are computed. By employing an interpolation
based on Legendre wavelet, we ¯nd approximate solution for any order (¯rst-order
and second-order) di®erential equations. Using this strategy the iterative spectral
method converts the di®erential equation to a set of algebraic equations. Solving
this set of algebraic equations yields an approximate solution.
Using exponentially-¯tted Runge-Kutta (EFRK) method, we develop a method
for numerically solving fuzzy ¯rst order linear and nonlinear di®erential equations
under generalized di®erentiability. In addition, this method is applied for the sys-
tem of ¯rst order fuzzy di®erential equations with uncertainty. The generalized
Hukuhara di®erentiability are applied to estimate the solutions. For solving the
fuzzy problems, the exponentially-¯tted Runge-Kutta method is applied.
Finally, some examples are solved to illustrate our proposed approaches. The
results are compared with those in the literature. We show that our proposed
methods are simple and more accurate than the other existing methods. |
| format |
Thesis |
| qualification_level |
Doctorate |
| author |
Dizicheh, Ali Karimi |
| author_facet |
Dizicheh, Ali Karimi |
| author_sort |
Dizicheh, Ali Karimi |
| title |
Numerical methods for solving oscillatory and fuzzy differential equations |
| title_short |
Numerical methods for solving oscillatory and fuzzy differential equations |
| title_full |
Numerical methods for solving oscillatory and fuzzy differential equations |
| title_fullStr |
Numerical methods for solving oscillatory and fuzzy differential equations |
| title_full_unstemmed |
Numerical methods for solving oscillatory and fuzzy differential equations |
| title_sort |
numerical methods for solving oscillatory and fuzzy differential equations |
| granting_institution |
Universiti Putra Malaysia |
| publishDate |
2014 |
| url |
http://psasir.upm.edu.my/id/eprint/70497/1/FS%202014%2063%20-%20IR.pdf |
| _version_ |
1747812852099448832 |