Yüksek mertebeden boussinesq denklemi i̇çin fourier spektral yöntemi

The higher-order Boussinesq equation (HBq) is given by utt = uxx+eta1uxxtt-eta2uxxxxtt +(f(u))xx where f(u)=u^p; p>1 is an integer. Here eta1 and eta2 are real positive constants. The HBq equation models the longitudinal vibrations of a dense lattice. In this thesis study, we propose a Fourier pseudo-spectral method for the HBq equation. Thesis study is organized as follows: Chapter 1 is devoted to the preliminaries. We briefly review some basic definitions related to linear algebra, some special function spaces and weak derivative. We also introduce continuous and discrete Fourier transforms. We then consider three examples to understand discrete and continuous Fourier expansions and differentiation. In Chapter 2, we first give a brief introduction to the HBq equation and its properties such as conserved quantities. We then derive solitary wave solutions of the HBq equation by using ansatz method which is one of the most effective direct methods to construct the solitary wave solutions of the nonlinear evolution equation. In Chapter 3, we introduce the Fourier pseudo-spectral method for the HBq equation. We first prove the convergence of the semi-discrete scheme in the appropriate energy space. We then define fully-discrete scheme for the HBq equation. Solution steps are (i) constituting the grid points in space, (ii) transforming the equation into the Fourier space and obtaining an ordinary differential equation in terms of Fourier coefficients, (iii) solving the resulting ordinary differential equation by using the fourth-order Runge-Kutta method (RK4), iv) forming the numerical solution from Fourier coefficients by using the inverse Fourier transform. To see the validation of the proposed scheme, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. In these problems, we consider various power type nonlinearities. Our numerical results show that the Fourier pseudo-spectral method exhibits fourth-order convergence in time and provides spectral accuracy in space. As far as we know, the present study is the first numerical study in literature for the HBq equation. Therefore, we couldn't compare our numerical results with the results in literature.