General linear methods for solving ordinary and fuzzy differential equations
In this thesis, a third order General Linear Method (GLM) is proposed for solving ordinary and fuzzy differential equations as well as second kind of fuzzy Volterra integro-differential equations (FVIDEs). Also, the fuzzy version of Improved Runge-Kutta (IRK) method is adapted to solve the sec...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2017
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/69423/1/IPM%202018%204%20IR.pdf |
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| Summary: | In this thesis, a third order General Linear Method (GLM) is proposed for solving
ordinary and fuzzy differential equations as well as second kind of fuzzy Volterra
integro-differential equations (FVIDEs). Also, the fuzzy version of Improved
Runge-Kutta (IRK) method is adapted to solve the second kind of FVIDEs.
Third order General Linear Method is derived using the technique of rooted
trees and B-series. The algebraic order conditions of the method are obtained up
to trees of order four. Using the order conditions, we obtained the different sets of
coefficients for GLM of order three. Stability region is discussed and numerical
results of GLMs for solving first order ordinary differential equations (ODEs) are
compared with the existing method.
Subsequently, numerical solutions of first order fuzzy differential equations
(FDEs) are proposed using fuzzy version of obtained GLM. The approach of
generalized Hukuhara differentiability is used to define the FDEs. Based on this
approach, the characterization theorem which converted the FDEs into systems of
ODEs is explored. Then, a fuzzy version of third order GLM for solving FDEs
using the generalized Hukuhara differentiability is developed. The convergence of
the method is given and numerical results compared with different existing methods
are presented.The study of FDEs is then extended to the first order fuzzy Volterra integrodifferential
equations. Unlike FDEs, a differential and integral operators appear
simultaneously in FVIDEs. Therefore, suitable numerical quadrature rules which are
the combination of composite Simpson’s rule together with Lagrange interpolation
polynomial and Trapezoidal rule are used to solve the integral part whereas the third
order GLM is considered for the differential part. Again, the generalized Hukuhara
differentiability is applied to develop the GLM combined with given numerical
quadrature rules for solving FVIDEs. Numerical results are tabulated to illustrate
the performance of the proposed method.
Finally, Improved Runge-Kutta method of order four with three stages is proposed
to obtain the numerical solutions of FVIDEs. A similar strategy is used
to develop the Improved Runge-Kutta method by adapting the same numerical
quadrature rules as used for General Linear Method and is based on generalized
Hukuhara differentiability. The performance of Improved Runge-Kutta method is
demonstrated by comparing the numerical results with the existing method of same
order. |
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