Homotopy-based methods for fractional differential equations
Başlık çevirisi mevcut değil.
- Tez No: 403363
- Danışmanlar: Dr. PAUL ZEGELING
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2017
- Dil: İngilizce
- Üniversite: Universiteit Utrecht
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 99
Özet
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Özet (Çeviri)
This thesis discusses numerical methods for models with fractional order derivatives. These methods make use of infinite series, which are based on the notion of homotopy with an additional method parameter to accelerate the convergence properties. In Chapter 1, we start with a brief motivation and history of fractional calculus. Next, we introduce the Gamma function and the Mittag-Leffler function, which play an important role in the theory of fractional derivatives. There exist many definitions of fractional order derivatives. Among the most important ones, we present four definitions: the Riesz, the Riemann-Liouville, the Caputo, and the Grünwald-Letnikov derivative. Properties, relations and differences between these are mentioned to show the different aspects and consequences. Furthermore, the importance of fractional calculus is emphasized and the background for using fractional differential equations is briefly described. The widespread application of fractional order derivatives is discussed and a series of fractional order models from many different application areas is given. At the end of chapter 1, an outline of the new contribution to this area is shortly indicated. In Chapter 2, we focus on a fractional order advection-diffusion-reaction model. We introduce the homotopy perturbation method to solve the model. First, we analyze the convergence of the method and compare it with the Adomian decomposition method to understand its effectivity for different models. Some theoretical results are given and are explained in terms of convergence tables and graphics. Numerical experiments illustrate the performance of this method, when applied to a test set of boundary-value problems. Both convergent and divergent numerical solutions are presented. In Chapter 3, we deal with traveling wave solutions in time-fractional partial differential equations. This is done by using the homotopy analysis method which is an extension of the homotopy perturbation method. Primarily, we explain the importance of the convergence parameter ~ in the so-called ~-curve. This curve enables us, somehow, to control the convergence region of the method. Especially, for this curve, the role of an optimal ~ value is emphasized. Numerical results for a special partial differential equation are given: the time-fractional Fisher equation. The effect of changing the fractional order on the solution behavior is shown. In Chapter 4, we study Padé approximations to improve the numerical solutions which were obtained by the homotopy analysis method in the previous chapter. A Padé approximation has the potential to produce more accurate numerical solutions not only for higher time values in the differential equation, but also speeds up the computation time of the series. First, we introduce the rational homotopy perturbation method which makes use of a Padé approximation for stationary problems. For example, it is applied to a convective-radiative equation and to Troesch'smodel. Finally, we apply a Padé approximation with the homotopy analysis method to a timefractional Fisher partial differential equation and show its usefulness in several experiments
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