The time-fractional radiative transport equation—Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion

Abstract

We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.

Fractional Schrödinger Equation in the Presence of the Linear Potential

“…Several years ago, the classical Schrödinger equation has been generalized to a fractional partial differential equation that takes into account the Riesz space-fractional derivative instead of the conventional Laplacian [1,2]. Apart from quantum mechanics, there are many other equations occurring in science that have been reconsidered in terms of fractional derivatives such as the diffusion-wave equation [3][4][5][6], the Langevin equation [7] or the radiative transport equation [8]. Analytical solutions to fractional differential equations are, in general, not available in terms of elementary functions.…”

Radiative transfer equation-based color prediction and color adjustment strategies

“…An important step when dealing with the source 𝑆(𝑧, 𝜇) via a particular solution of (1) is the removal of the ballistic or unscattered part as described by Liemert and Kienle [26], Gardener et al [34] or Machida et al [35] as well as using the delta-M approximation [36] for the phase function to improve the results for low to intermediate approximation orders 𝑁. Using the source term 𝑆(𝑧, 𝜇) = 𝛿(𝑧) 𝛿(𝜇 − 𝜇 0 ) with 𝜇 0 > 0, the unscattered radiance is given by…”

Inverse problems of determining coefficients of the fractional partial differential equations

“…where the double-signs correspond. We note that y(x, v, t) describes the density of some particles at the point x ∈ Ω and the time t with the velocity v. As for some physical backgrounds (see e.g., Machida [38]), and the equation where ∂ 1 2 t is replaced by ∂ t is called a radiative transport equation (e.g., Duderstadt and Martin [10]). By an idea similar to Xu, Cheng and Yamamoto [62], reducing (14) to ∂ t y − v 2 ∂ 2…”

“…where the double-signs correspond. We note that y(x, v, t) describes the density of some particles at the point x ∈ Ω and the time t with the velocity v. As for some physical backgrounds (see e.g., Machida [38]), and the equation where ∂…”