analytical-numerical solution for a third order space-time conformable fractional pde with mixed derivative by spectral and asymptotic methods

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.