Analysis of solutions of stochastic evolution equations

Thesis Type: Doctorate

Institution Of The Thesis: Van Yüzüncü Yil University, Fen Bilimleri Enstitüsü, İstatistik (Dr), Turkey

Approval Date: 2020

Thesis Language: English


Consultant: Hatice Taşkesen


Nonlinear evolution equations are equations that contain the time variable t as an argument, appearing not only in many fields of mathematics but also in other branches of science such as physics, mechanics, and materials science. Since the deterministic evolution equations are insufficient in the modeling of physical phenomena, a term including the effect of uncertainty is usually added to the deterministic evolution equations. In this thesis, we investigate the effect of these terms, i.e. noise, on the solutions of some evolution equations. For this purpose, we use the Hermite transform and Galilean transform to obtain the stochastic equations deterministic counterparts and then use some analytical methods to obtain the solutions. The first chapter of the thesis includes a motivating example explaining why stochastic differential equations are needed. The second chapter summarizes the literature review. The third chapter contains the basic concepts, definitions, and preliminaries of the methods that are used in the thesis. In the fourth chapter, analytical solutions of stochastic KdV-Burgers, stochastic KdV, stochastic Burgers, stochastic Kuramoto-Sivashinsky and stochastic Kawahara equations are obtained by using Galilean transform and tanh, extended tanh methods. Moreover, the solutions of a stochastic Wick-type extended KdV equation are found by using Hermite transform and Jacobi elliptic functions. The illustrations of some solutions are given to see the effect of noise apparently.