Document Type : Research Paper


Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, P.O.Box 53714-161, Tabriz, Iran.


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.


[1] A. Ahmadkhanlu and M. Jahanshahi, On The Existence and Uniqueness of Solution of Initial Value Problem for Fractional Order Differential Equations On Time Scales, Bull. Iran. Math. Soc., 38(1), (2012), pp. 241-252.
[2] N.M. Aslanova, M. Bayramoglu and K.M. Aslanova, Numerical solution of fractional partial differential equations by using radial basis functions combined with Legendre wavelets, J. Math. Model., 8(4), (2020), pp. 435-454.
[3] A. Demir and M.A. Bayrak, A New Approach for the Solution of Space-Time Fractional Order Heat-Like Partial Differential Equations by Residual Power Series Method, Commun. Math. Appl., 10, (2019), pp. 585-597.
[4] J.J. Feng and Y.S. Li, Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives, Axioms, 7, (2018), pp. 1-18.
[5] R. Hosseini, M. Jahanshahi, A.A. Pashavand and N. Aliyev, The Study of Some Boundary Value Problems Including Including Fractional Partial Differential Equations with non-Local Boundary Conditions, Iran. J. Math. Sci. Inform., 14(2), (2019), pp. 69-77.
[6] O.A. İlhan, S.G. Kasimov, S.Q. Otaev and H.M. Baskonus, On the Solvability of a Mixed Problem for a High-Order Partial Differential Equation with Fractional Derivatives with Respect to Time, with Laplace Operators with Spatial Variables and Nonlocal Boundary Conditions in Sobolev Classes, Mathematics, 7, (2019), pp. 1-20.
[7] M. Jahanshahi, N. Aliyev and F. Jahanshahi , Solving Two Initial-Boundary Value Problems Including Fractional Partial Differential Equations By Spectral and Contour Integral Methods, Azerb. J. Math., 10, (2020), pp. 31-48.
[8] M. Jahanshahi and H. Kazemidemneh, Contrast of Homotopy and Adomian Decomposition Methods with Mittag-leffer Function for Solving Some Nonlinear Fractional Partial Differential Equations, Int. J. Industrial Mathematics, 12, (2020), pp. 263-271.
[9] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264, (2014), pp. 65-70.
[10] H. Kheiri, A. Mojaver, and S. Shahi, Analytical Solutions For The Fractional Fisher's Equation, Sahand Commun. Math. Anal., 2(1), (2015), pp. 27-49.
[11] R.H. Komlae and M. Jahanshahi, Invariant Functions for Solving Multiplicative Discrete and Continuous Ordinary Differential Equations, Comput. Methods Differ. Equ., 264, (2018), pp. 271-279.
[12] A.S. Mohammeda and H. Shathera, Mixed fractional partial differential equations by the base method, Int. J. Nonlinear Anal. Appl., 12, (2021), pp. 1687-1697.
[13] S. Nemati, A Spectral Method Based On The Second Kind Cebyshev Polynomials For Solving a Class Of Fractional Optimal Control Problems, Sahand Commun. Math. Anal., 4(1), (2016), pp. 15-27.
[14] Q. Yang, F. Liu and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Modelling, 34, (2010), pp. 200-2018.