Research Specifications

Initial-boundary value problems including fractional partial dierential equa- tions are the mathematical models of physical problems and natural phenomena. In this paper, at rst we consider a fractional partial dierential equation which has no mixed term derivative with respect to spatial and time variable. We rst consider the spectral problem, then its eigenvalues and eigenfunctions are calculated. After that the eigen- values and eigenfunctions of the adjoint problem are calculated. By using these eigen- functions and Mittag-Leer functions the approximate solution is constructed. In second section, we consider dierential equation which has a mixed term derivative. In this case, by using Laplace transformation, the analytic solution and approximate solution are cal- culated as integral expression over suitable closed contours by contour integral method. At the end, some examples are presented for several cases of dierent distributions of eigenvalues in complex plane.