Search

In this paper a new definitionn of fractional derivative and fractional integral in the sense ofconformable derivative type is presented. This form of definition shows that it is more compatible withclassical natural definition of derivative and is more convinient fractional derivative one. We will definethis for 0 ≤ α < 1 and n − 1 ≤ α < n and further, if α = 1 the definition coincides with the classicaldefinition of derivative of first order.

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the asymptotic expansion of the roots is used. Finally, by suitable choice of these asymptotic expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical solutions to the main initial-boundary value problem are given.