Analytical Solution of Generalized Space-Time Fractional Cable Equation

Abstract

Abstract:In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.

Generalized Langevin Equation and the Prabhakar Derivative

“…For the free particle case ω = 0, the Prabhakar integral corresponds to the R-L fractional integral, therefore, from Equation (43) one finds the previously obtained result for free particle, Equation (30).…”

“…Here we note that different fractional equations have been used for modeling anomalous diffusion in various systems, including fractional reaction-diffusion equations [27,28] and their application [29], fractional relaxation and diffusion equations [5,6,9,10,[24][25][26], fractional cable equation [30], etc.…”

Reformulating Fick’s Law of Diffusion into a Half-Order Version, with Applications in the Physical Sciences

“…Comparison with Equation 1:2 shows that this is precisely Fick's second law of diffusion, with 1/(  ) playing the role of diffusivity. Electricity can truly be said to diffuse along a resistivecapacitive transmission line [22].…”

Generalized distributed order diffusion equations with composite time fractional derivative

“…where we apply the properties of the completely monotone and Bernstein functions of the Laplace transform of x 0 (t) (see for example [14,53,57]). The case with one composite time fractional derivative (b = 0) yields the known result x 0 (t) = t −(1−ν 1 )(1−µ 1 ) [43].…”