LEE Matematik MühendisliğiDoktora
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ÖgeThe generalized fractional Benjamin Bona Mahony equation: Analytical and numerical results(Lisansüstü Eğitim Enstitüsü, 2021) Oruç, Göksu ; Mihriye Muslu, Gülçin ; Borluk, Handan ; 692763 ; Matematik MühendisliğiIn this thesis study we consider the generalized fractional BenjaminBonaMahony (gfBBM) equation u_t+ u_x + \frac{1}{2}(u^{p+1})_x+ \frac{3}{4}D^{\alpha} u_{x}+ \frac{5}{4}D^{\alpha} u_{t}=0, where $x$ and $t$ represents spatial coordinate and time, respectively. This equation is derived to model the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The gfBBM equation has a general powertype nonlinearity and two fractionaltype terms. Thanks to these properties, the gfBBM equation is noticed as a satisfactory and interesting model in the literature. The aim of this thesis study is to perform various mathematical and numerical analyses for the gfBBM equation and to understand the influence of nonlinearity and fractional dispersion on the dynamics of solutions. The thesis study is organized in the following way: In the first chapter, we briefly introduce the general background on the fractional type nonlinear partial differential equations with lower dispersion such as fractional Korteweg de Vries (fKdV) and fractional BenjaminBonaMahony (fBBM) and gfBBM equations. Then, we propose derivation and some properties of the gfBBM equation. We also state the analytical and numerical methods used to solve this equation. Furthermore, the literature overview on gfBBM and related equations is given in this chapter. The second chapter is devoted to the analytical results for the gfBBM equation. In the first section of this chapter we recall the preliminaries. This section contains useful definitions related to functional analysis, lemmas and theorems used in the thesis. In the second section, we derive conserved quantities of the gfBBM equation. We also find constraints on the order $\alpha$ of the fractional term. The aim of the third section is to prove the local wellposedness of the Cauchy problem for the gfBBM equation together with the initial condition u(x,0)=u_0 (x). For the case $1 \leq \alpha \leq 2$, we prove the local wellposedness of the solutions by using contraction mapping principle. On the other hand, for the case $0 < \alpha < 1$, we use the approaches given for the fBBM equation by He and Mammeri (2018). Therefore, we consider the regularization of the Cauchy problem for the gfBBM equation and then use the convergence of regularized solutions to the solutions of main problem. The section 4 presents the conditions for the nonexistence of solitary wave solutions to the gfBBM equation. Existence and uniqueness of solitary wave solutions are obtained by using the result of Frank and Lenzmann (2013). We also consider the restrictions on the $\alpha$ and speed of wave $c$ so that the gfBBM equation admits positive or negative solitary waves. Finally, we derive exact solitary wave solutions to the gfBBM equation for the special cases $\alpha=1$ and $\alpha=2$ when $p=1$. In the last section of this chapter we discuss the stability properties of solitary wave solutions associated to the gfBBM equation. We first give the Hamiltonian formulation of the equation. Then, we prove the orbital stability of solitary wave solutions by using approach given by Grillakis Shatah Strauss (GSS) (1987) and for the stability we obtain following conditions when $1 \leq p \leq 4$: 1) $\frac{p}{p+2}<\alpha < \frac{p}{2}$ and $c>c_{1,p}>1$, 2) $\frac{p}{2}<\alpha < 2$ and $c>1$ or $\frac{3}{5}>c>c_{2,p}$, with $c_{1,p}=\frac{6\alpha + 2p + 3 \alpha p + \sqrt 2 p \sqrt{2 \alpha  p + \alpha p} }{5(2 \alpha + \alpha p)}$ and $c_{2,p}=\frac{6\alpha + 2p + 3 \alpha p  \sqrt 2 p \sqrt{2 \alpha  p + \alpha p} }{5(2 \alpha + \alpha p)}$. In the last chapter, we present the numerical results for the gfBBM equation. We first state efficient numerical algorithms for gfBBM equation and then carry out various numerical experiments. The Petviashvili method is proposed for the generation of the solitary wave solutions that cannot be obtained analytically. We numerically investigate the effects of the relation between the nonlinearity and the dispersion on the solutions. The evolution of generated wave profiles in time is investigated numerically by Fourier pseudospectral method. The efficiency of the methods will be demonstrated by various numerical simulations.