The space-time fractional diffusion equation with Caputo derivatives

Abstract

Abstract. We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or RiemannLiouville) derivative of order β ∈ (0, 2] and the first-order time derivative with Caputo derivative of order α ∈ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

Analytical Solution of Generalized Space-Time Fractional Cable Equation

“…Fractional diffusion equations in presence of external force were also of interest in many recent papers [12][13][14][15]. Analytical solutions of fractional differential equations and numerical methods of fractional cable equation have been considered by many authors [16][17][18][19][20][21][22][23][24], to name but a few. In our work, we further generalize Equation (2) by introducing time fractional derivative of Caputo form of order 0 1    , defined by [8,9] …”

An approach via fractional analysis to non-linearity induced by coarse-graining in space

“…All this material is necessary to fully expand the fractional probability calculus outlined below. For an extended bibliography on fractional calculus, one can consult for instance [20,26,25,21,22,4,[27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

A solution to the problem of time-fractional heat conduction in a multi-layer slab

“…The theory of the fractional calculus is contained in the books [3][4][5]. The methods to solutions of the fractional differential equations are presented in papers [6][7][8][9]. In Huang and Liu [6] to solve the Cauchy problem for the time-space fractional diffusion equation, temporal Laplace and spatial Fourier transforms have been applied.…”

“…The methods to solutions of the fractional differential equations are presented in papers [6][7][8][9]. In Huang and Liu [6] to solve the Cauchy problem for the time-space fractional diffusion equation, temporal Laplace and spatial Fourier transforms have been applied. Demirci and Ozalp [7] use a transformation of the considered fractional differential equation in the equivalent fractional Volterra integral equation.…”