Eur. Phys. J. Special Topics 222, 1847–1856 (2013) c  EDP Sciences, Springer-Verlag 2013 DOI: 10.1140/epjst/e2013-01968-x T HE EUROPEAN P HYSICAL JOURNAL SPECIAL TOPICS Regular Article Fractional Fokker-Planck equation for anomalous diffusion in a potential: Exact matrix continued fraction solutions W.T. Coffey 1 , Y.P. Kalmykov 2, a , and S.V. Titov 3 1 Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland 2 Univ. Perpignan, via Domitia, Laboratoire de Mathmatiques et de Physique, EA 4217, 66860 Perpignan, France 3 Kotelnikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region 141120, Russia Received 3 June 2013 / Received in final form 6 August 2013 Published online 7 October 2013 Abstract. Methods for the exact solution of fractional Fokker-Planck equations for anomalous diffusion in an external potential are discussed using both ordinary and matrix continued fractions, whereby the scalar multi-term recurrence relations generated by such fractional diffusion equations are reduced to three-term matrix ones. The procedure is il- lustrated by solving various problems concerning the anomalous trans- lational diffusion in both periodic and double-well potentials. 1 Introduction Both the free Brownian motion and that in a field of force are indispensable in the discussion of relaxation and resonance phenomena in stochastic systems [1, 2]. The best known example is the translational diffusion of noninteracting Brownian particles treated by Einstein [3] with a host of applications in physics, chemistry, biology, finance, etc. Now, Einstein’s theory relies on a discrete time random walk. Here the random walker or particle which has no memory of the previous positions makes a jump of a fixed mean square length in a fixed time. In the noninertial limit and in one dimension, such a process, which is local both in space and time, can be modeled by the diffusion equation ∂f ∂t = K ∂ 2 f ∂x 2 (1) for the distribution function f (x, t) in configuration space, where x specifies the position of the walker (particle) at time t, and K is the diffusion coefficient. Equation (1) in the presence of an external potential V (x) (e.g., the gravita- tional field of the earth) becomes the noninertial Fokker-Planck (Smoluchowski) a e-mail: kalmykov@univ-perp.fr