General solution of a fractional diffusion–advection equation for solar cosmic-ray transport

Abstract

In this effort we exactly solve the fractional diffusion-advection equation for solar cosmic-ray transport proposed in [15] and give its general solution in terms of hypergeometric distributions. Also, we regain all the results and approximations given in [15] as particular cases of our general solution.

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

“…Numerous models for fractional advection-dispersion equation(FADE) has been introduced by many researchers (see [2,3,8,10,11] and references therein). In this work we introduce the fractional advection-dispersion equation(FADE) in the general fractional form…”

“…Case I is concerned with an exponential function as initial condition and for arbitrary fractional order 0 < α ≤ 1 and β = 1, γ = 2. Considering γ = 2β and any function f (x) for the initial condition (9), we get the solution for equation (8) in case II. Finally, case III is dealt with the space-time fractional (ADE) (8) for any arbitrary fractional order 0 < α, β ≤ 1 and 1 < γ ≤ 2 under the general initial condition u(x, 0) = f (x).…”

“…Some special forms of the fractional (ADE) were solved analytically in [7]. Rocca et al [8] considered the fractional diffusion-advection equation for solar cosmicray transport and gave its general solution. The authors in [9] presented a level IV fugacity model coupled to a dispersion-advection equation to simulate the environmental concentration of a pesticide in rice fields.…”

New Solution of Diffusion–Advection Equation for Cosmic-Ray Transport Using Ultradistributions

“…A more conventional treatment of this equation is given in [9]. The present treatment is of a much more general character.…”

“…Fractional derivatives have recently been used in regards to many physical problems [for a small sample, see for instance [3,4,5,6]] and to hydrology [7]. Fractional derivatives have been recently applied to model super-diffusion of particles in astrophysical scenarios [8,9]. There is a considerable evidence emerging from data gathered by spacecrafts showing that the transport of energetic particles in the turbulent heliospheric medium is super-diffusive [11,12].…”