Solutions and conservation laws of some systems of fractional nonlinear partial differential equations with the help of lie symmetry method

Recently, most complex problems in science and engineering have been formulated by nonlinear partial differential equations. Therefore, it has become more important to obtain analytical solutions for analysing the behavioral properties of these equations. The Lie symmetry analysis method, which was introduced by Sophus Lie at the beginning of the 19th century and later named after him, brought a new perspective to the solution of nonlinear partial differential equations. The main purpose of this method is to construct similarity transformations that leave the differential equations invariant form. These similarity transformations reduce a partial differential equation into a ordinary differential equation of fractional order and make it easy to obtain exact solutions. Also, using the connection between Lie symmetries and Noether operators, conservation laws of differential equations can be obtained. In this thesis, the Lie symmetry method is applied to the nonlinear time fractional generalized Drienfeld Sokolov and Super KdV systems in order to acquire similarity transformations for systems with the help of the Riemann-Liouville fractional derivative. These two systems are reduced into fractional ordinary differential equation systems using the similarity transformations. Then, the conserved vectors of the systems are obtained by using the conservation laws that Ibragimov put forward by using the connection between the Lie symmetries and Noether operators. Finally, we obtain explicit power series solutions of the reduced fractional ordinary differential equations and show that these solutions are convergence.