Exact solution methods for nonlinear fractional differential equations

Fractional calculus is the extension of integer order derivative and integral to non-integer order derivative and integral, respectively. By authorities, it is commonly expressed that fractional order differential equations has more realistic mathematical modelling of nonlinear world rather than integer order differential equations. In rapidly developing science and technology, especially in engineering, it has been very important to find exact solutions of fractional differential equations. In recent times, many scientific workers have published many papers on fractional order differential equations and its exact solutions. Upon these papers, different and efficient methods arose. The purpose of this dissertation is to obtain new exact solutions of fractional differential equations and systems of fractional differential equations via different methods that can be used in many research areas such as physics, biology, engineering. Since these solutions contain travelling vawe, solitary wave, periodic, exponential, rational, hyperbolic and soliton type solutions, this research will be a pioneer work to develop many physical models used in these days. This thesis consists of six sections. In the first section, fractional calculus concept, historical evolution and employment areas of fractional calculus are given. In the second section, some special functions that will be used in next sections are presented. In the third section, definitions and properties of some special fractional order differential equations in literature are given. In the fourth section, fractional complex transformation which is used to convert fractional order differential equations to ordinary differential equations is introduced. Besides, sub-equation, functional variable, modified simple equation, Kudryashov and modified trial equation that are used to get exact solutions of fractional differential equations are discussed. In the next section, via transformations and methods given in fourth section, solutions of some special fractional order differential equations and systems of fractional order differential equations have been investigated, different kinds of solutions have been found. In the last section of the thesis, obtained results from the application part in the previous section are interpreted and suggestions are presented.