Nonlinear Dyn (2013) 71:671–683 DOI 10.1007/s11071-012-0710-x ORIGINAL PAPER Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions Živorad Tomovski · Trifce Sandev Received: 20 June 2012 / Accepted: 8 December 2012 / Published online: 3 January 2013 © Springer Science+Business Media Dordrecht 2013 Abstract We give an analytical treatment of a time fractional diffusion equation with Caputo time-frac- tional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler- type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experi- ment in limiting diffusion space. Keywords Fractional diffusion equation · Mittag-Leffler function · Generalized integral operator · Sturm–Liouville problem 1 Introduction It is shown that the time-fractional diffusion equation of order 0 <μ< 1 gives the same results in the de- scription of anomalous subdiffusion as those obtained Ž. Tomovski ( ) Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Saints Cyril and Methodius University, 1000 Skopje, Macedonia e-mail: tomovski@pmf.ukim.mk T. Sandev Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia e-mail: trifce.sandev@drs.gov.mk by the continuous-time random walk (CTRW) theory for a diffusive process characterized by a distribution of jump lengths with finite variance 〈δx 2 〉 and broad distribution of waiting times τ of the form ψ(τ) ≃ (τ ∗ ) μ /τ 1+μ with 0 <μ< 1[35]. Two equivalent formulations in terms of Riemann–Liouville (R–L) and Caputo sense were given [35]. As it is known, fractional derivatives represent infinitesimal genera- tors of time-fractional evolutions arising in the tran- sition from microscopic to macroscopic time scales [15, 17, 18] (see also [19]). It was shown that frac- tional derivatives arise in the description of differ- ent physical processes, such as nonexponential relax- ation processes of proteins [7], anequilibrium phase transitions [14], etc. Thus, the investigation of frac- tional differential equations has started to attract more and more attention [4, 9, 10, 54, 64]. The space frac- tional diffusion equation can be used to describe Lévy flights, which can be described by the CTRW theory as processes with finite characteristic waiting times and long-tailed distribution of jump lengths of the form λ(x) ≃|x | −1−α , where 0 <α< 2[35]. The Caputo or R–L time-fractional derivatives are usu- ally used for modeling anomalous diffusive processes. As a generalization, one can use fractional diffusion equations with composite fractional time derivative [54, 64], which combines the R–L and Caputo no- tation. We showed that such models represent very flexible frameworks for the description of complex processes, and we reported for the numerical scheme for solving a space–time fractional diffusion equation