Cent. Eur. J. Phys. • 1-15 Author version Central European Journal of Physics Exact solution for the fractional cable equation with nonlocal boundary conditions Research Article Emilia G. Bazhlekova * , Ivan H. Dimovski † Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria Abstract: The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied. PACS (2008): 02.30.Gp, 02.30.Vv, 02.60.Lj, 87.10.Ed, 87.19.L- Keywords: cable equation • anomalous diffusion • fractional derivative • three-parameter Mittag-Leffler function Versita sp. z o.o. 1. Introduction The partial differential equations of fractional order play an important role in modeling the so-called anomalous transport phenomena, see e.g [1–3]. Recently, the fractional cable equation (FCE) was derived from the fractional Nernst-Planck equations for modeling the anomalous electrodiffusion in nerve cells [4–6]. The FCE can be written in dimensionless variables as follows ut = D 1−α t (uxx) − cD 1−β t u, t> 0, 0 ≤ x ≤ 1, (1) where 0 <α,β ≤ 1, c> 0, and D δ t denotes the Riemann-Liouville fractional partial derivative of order δ ∈ (0, 1) D δ t u(x,t) := 1 Γ(1 − δ) ∂ ∂t  t 0 u(x,τ ) (t − τ ) δ dτ, (2) ∗ E-mail: e.bazhlekova@math.bas.bg † E-mail: dimovski@math.bas.bg