Kesirli gear-grimshaw modelinin kesin ve nümerik çözümleri

Fractional differential equations, i.e. differential equations containing fractional derivatives, have been the focus of attention of the scientific world due to their extraordinary abilities in modeling real world problems such as engineering, physics and biology. Many definitions of the fractional derivative have been made so far, some of them are: Riemann-Liouville, Caputo, modified Riemann-Liouville, Jumarie, Hadamard, Atangana–Baleanu, Caputo–Fabrizio, Riesz fractional operators. In this thesis, the Gear-Grimshaw equation system with time fractional derivatives, which is from the family of KdV equations, is discussed. And with this equation , the dynamics of two - layer shallow water waves in oceans and rivers are modeled . Also covers many shallow water wave models with special values taken to parameters. The excat solutions of the fractional GG equation, which was reduced to the ordinary differential equation by using the fractional transformation, were obtained in terms of hyperbolic functions by using the Sine-Gordon expansion method. 3D and contour graphs of some of the solutions found are drawn. In addition, approximate solutions of the equation were obtained for three different values of α by using the residual power series method, which is one of the numerical solution methods. The absolute values of the difference between the exact solutions and the approximate solutions are given as a table.