Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X42220160401An analytic solution for a non-local initial-boundary value problem including a partial differential equation with variable coefficients315326761ENM.JahanshahiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.M.DarabadiDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranJournal Article20140909This paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficients. At first, the non-self-adjoint spectral problem is derived. Then its adjoint problem is calculated. After that, for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined. Finally the convergence of series solution and the uniqueness of this solution will be proved.http://bims.iranjournals.ir/article_761_706ebf130c6496e7db6e47caf0549442.pdf