SPECTRAL PROBLEM FOR AN INITIAL-BOUNDARY VALUE PROBLEM INVOLVING FIRST ORDER TWO DIMENSIONAL GENERALIZED NONHOMOGENOUS CAUCHY-RIEMANN EQUATION WITH GENERAL NON-LOCAL BOUNDARY CONDITIONS

In this paper, we consider an initial boundary value problem which contains a fractional differential equation by some initial and boundary conditions by spectral method. This problem is considered in two cases. At first, the given fractional equation is whit out mixed term derivative. We get its spectral problem and the eigenvalues and eigenfunctions. In the second part, we consider this equation whit a mixed term derivative of time variable and space variable. In this case, we cannot use the spectral method. Its spectral problem is obtained by Laplace transformation. The analytic solution of this problem is written as integral form over a suitable countour that its Green function is the solution of related spectral problem.