Analysis of periodic wave solitions of fractional derivative mathematical models

In this thesis, the periodic wave solutions and the numerical solutions of the fractional order derivative mathematical models were analyzed. First of all, general information about the nonlinear fractional order partial differential equations and the nonlinear partial differential equations is given. The solution methods for these equations and the purpose of the study are mentioned. The modified exponential function method (MEFM) and the conformable fractional q-Shehu homotopy analysis transform method (CFq-SHATM) are explained in detail. First, MEFM is applied to obtain the exact solutions of the nonlinear conformable time-fractional Date-Jimbo-Kashiwara-Miwa equation, the nonlinear (2+1)-dimensional generalized Hirota–Satsuma–Ito equation system and the nonlinear Dullin-Gottwald-Holm equation. Secondly, CFq-SHATM is applied to obtain the numerical solutions of the nonlinear conformable fractional two-dimensional Navier-Stokes equations, the nonlinear conformable fractional Cahn-Hilliard equation and the nonlinear conformable fractional Rosenau–Hyman equation. The two-dimensional, three-dimensional, contour and density graphs of the obtained the exact solutions of the nonlinear conformable fractional Date-Jimbo-Kashiwara-Miwa, the nonlinear (2+1)-dimensional generalized Hirota–Satsuma–Ito and the nonlinear Dullin-Gottwald-Holm equations were plotted by using the Mathematica software program. In addition, the two-dimensional and three-dimensional graphs of the obtained numerical solutions of the nonlinear conformable fractional Navier-Stokes, nonlinear conformable fractional Cahn-Hilliard and nonlinear conformable fractional Rosenau–Hyman equations were plotted by using the Maple software program. As a result, the findings obtained in the study showed that the methods used were effective and successful.