Examination of the numerical solutions with the homotopy analysis method of partial differential equation with fractional powers

This study has been probed in six sections. In Section 1 history of fractional calculus and the studies that have been done in this area have been given. In addition to that Homotopy analysis method (HAM) that will be used in this thesis has been discussed in this section. In Section 2 some basic definitions and theorems that will be used in other sections have been stated. In Section 3 definitions and properties of special functions that has an important place in this studies that related to fractional analysis have been mentioned. Moreover, basic definitions, theorems and properties of fractional analysis have been discussed. In Section 4 the basis of Homotopy analysis method that we will use and application of the method have been applied on an example. In Section 5 general structure of the fractional cable equation has been represented and numerical solutions of the equation have been obtained by applying the HAM to the fractional cable equation. Sensitivity of the HAM for the fractional cable equation that is examined numerically has been interpreted. In Section 6 some information about Time-Fractional Hirota-Satsuma Coupled KdV Equation and concept of soliton has been given. Sensitivity of the method for Time-Fractional Hirota-Satsuma Coupled KdV equation, which is obtained with numerical solutions, has been interpreted.