Fractional derivatives and related numerical methods

The notions of arbitrary order derivative and integral are extended form non-integer-order derivative and integral. In the recent years, it has been applied extensively in mathematics, physics, biology and engineering branches. Since the derivative determines the rate of change of physical events, fractional-order derivatives close the openings that cannot be calculated by integer-order derivatives. There are many definitions of fractional derivative in literature. Having more than one definition allows to use the most suitable one according to the type of problem and thus to obtain the best solution of the problem. Some of these definitions are Grünwald-Letnikov, Riemann-Liouville and Caputo fractional derivatives. In this study, the Caputo fractional derivative, which is more advantageous since it contains initial values, have been used. This derivative is applied to some known functions and the results have been given in table form. Then, this derivative, with use of Algorithm 1 and Gauss-Jacobi rule have been examined. A new method has been created with the transformation we applied to Gauss-Jacobi and with the help of Jacobian matrix. This method has been applied to selected functions and very efficient results have been obtained.