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Operational solutions for the generalized Fokker-Planck and generalized diffusion-wave equations

K. Górska

TL;DR

This work develops an operator-based framework to solve generalized Fokker-Planck and generalized diffusion–wave equations in 1+1 dimensions with memory kernels. By introducing evolution operators and integral kernels, the authors establish a subordination structure for memory kernels that are Stieltjes functions, yielding parent processes and nonlocal time dynamics; diffusion-like initial conditions are essential to preserve positivity and normalization. They derive explicit MSD and moment expressions for power-law and distributed-order memory, showing how drift and memory shape long-time scaling, and connect diffusion-like and wave-like regimes via diffusion–diffusion–wave interpolation. The approach extends to arbitrary memory kernels and highlights composition properties via Efros' theorem, offering a unified tool for anomalous diffusion and diffusion–wave phenomena with memory.

Abstract

The evolution operator method is used to solve the generalized Fokker-Planck equations and the generalized diffusion-wave equations in the (1+1) dimensional space in which $x\in\mathbb{R}$ and $t\in\mathbb{R}_+$. These equations contain either the first- or the second-time derivatives smeared by memory functions, each of which forms an integral kernel (denoted by $f(ξ, t)$, $ξ\in\mathbb{R}_+$) of suitable evolution operators. If memory functions in the Laplace space are Stieltjes functions, then $f(ξ, t)$ satisfy normalization, non-negativity, and infinite divisibility to be considered a probability density function. The evolution operators also contain exponential-like operators whose action on initial condition $p_0(x) > 0$ leads to the parent process distribution functions. This makes the results fully analogous to those obtained within the standard subordination approach. The above conclusion is satisfied by the solution of the generalized Fokker-Planck equation. In the case of the generalized diffusion-wave equation, to get this property, we should employ a special class, namely "diffusion-like" initial conditions. The key models of the operator method involve power-law memory functions. It leads to the characterization of $f(ξ, t)$ by applying one-sided stable Lévy distributions. The article also examines the properties of evolution operators in terms of evolution and self-reproduction.

Operational solutions for the generalized Fokker-Planck and generalized diffusion-wave equations

TL;DR

This work develops an operator-based framework to solve generalized Fokker-Planck and generalized diffusion–wave equations in 1+1 dimensions with memory kernels. By introducing evolution operators and integral kernels, the authors establish a subordination structure for memory kernels that are Stieltjes functions, yielding parent processes and nonlocal time dynamics; diffusion-like initial conditions are essential to preserve positivity and normalization. They derive explicit MSD and moment expressions for power-law and distributed-order memory, showing how drift and memory shape long-time scaling, and connect diffusion-like and wave-like regimes via diffusion–diffusion–wave interpolation. The approach extends to arbitrary memory kernels and highlights composition properties via Efros' theorem, offering a unified tool for anomalous diffusion and diffusion–wave phenomena with memory.

Abstract

The evolution operator method is used to solve the generalized Fokker-Planck equations and the generalized diffusion-wave equations in the (1+1) dimensional space in which and . These equations contain either the first- or the second-time derivatives smeared by memory functions, each of which forms an integral kernel (denoted by , ) of suitable evolution operators. If memory functions in the Laplace space are Stieltjes functions, then satisfy normalization, non-negativity, and infinite divisibility to be considered a probability density function. The evolution operators also contain exponential-like operators whose action on initial condition leads to the parent process distribution functions. This makes the results fully analogous to those obtained within the standard subordination approach. The above conclusion is satisfied by the solution of the generalized Fokker-Planck equation. In the case of the generalized diffusion-wave equation, to get this property, we should employ a special class, namely "diffusion-like" initial conditions. The key models of the operator method involve power-law memory functions. It leads to the characterization of by applying one-sided stable Lévy distributions. The article also examines the properties of evolution operators in terms of evolution and self-reproduction.
Paper Structure (17 sections, 6 theorems, 97 equations, 4 figures)

This paper contains 17 sections, 6 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The evolution operator method for generalized Fokker-Planck equation (GFPE)
  3. MSD calculated for $q_{s\hat{M}}(x, t)$
  4. Toy models: the diffusion-wave equation
  5. Solution of DWE-I
  6. Solution of DWE-II
  7. Solution of GDWE via the evolution operator
  8. Discussion of initial conditions
  9. Properties of the evolution operators: ${\cal E}_{s \hat{M}}(t)$ and ${\cal U}_{s \hat{k}}(t)$
  10. Conclusion
  11. Useful formulaes
  12. One-sided Lévy stable distribution
  13. Efros' theorem
  14. Definitions and proofs of Sect. \ref{['sect1']} statements
  15. Proofs in Sect. \ref{['sect3']}
  16. ...and 2 more sections

Key Result

Theorem 1

If $\hat{G}_{1}(s)$ and $\hat{G}_{2}(s)$ are analytic functions, and $\mathscr{L}[h(x, \xi); s] = \hat{h}(x, s)$ as well as $\mathscr{L}[f(\xi, t); s] = \int_{0}^{\infty} f(\xi, t) \mathop{\mathrm{e}}\nolimits^{-z t} \mathop{\mathrm{d\!}}\nolimits t = \hat{G}_{1}(s) \mathop{\mathrm{e}}\nolimits^{-\x

Figures (4)

  • Figure 1: Plot of Eq. \ref{['28/06/24-1']} for $\alpha=1/2$ and different initial conditions $q_0(x)$. The black curve no. I is for $q_0(x) = \delta(x)$, the blue curve no. II is for the box (rectangular) initial condition \ref{['13/06/24-1']}, and the red curve no. III is for the Gaussian initial condition \ref{['22/01/24-1']}. We take $t = 1$, $\mu = 1$, $B=1$, $\epsilon = 1$, and $\sigma_x = 1$.
  • Figure 2: Plot of $p_1(2\beta; x, t)$ given by Eq. \ref{['6/11/24-1']} for $\beta = 3/4$ and diffusion-like initial conditions in which $v_0(x) = 0$ and $p_0(x)$ is changing as follows: the black curve no. I is for $p_0(x) = \delta(x)$, the blue curve no. II is for the box (rectangular) function \ref{['13/06/24-1']}, and the red curve no. III is for the Gaussian \ref{['22/01/24-1']}. We take $t = 1$, $a=1$, $\epsilon = 1/2$, and $\sigma_x = 1$.
  • Figure 3: Plot of $p_1(2\beta; x, t)$ given by Eq. \ref{['6/11/24-1']} for $\beta = 3/4$ and diffusion-like initial conditions in which $v_0(x) = 0$ and $p_0(x)$ is given by the Gaussian \ref{['22/01/24-1']}. We take $a=1$, $\sigma_x = 1$, and $t = 1$ (the black curve), $t = 2$ (the blue curve), and $t = 3$ (the red curve).
  • Figure 4: Plot of $p_1(2\beta; x, t)$ given by Eq. \ref{['5/07/24-1']} for $\beta = 3/4$, $t = 1$ and $a = 1$. The initial $p_0(x)$ is the Gaussian \ref{['22/01/24-1']} in which $\sigma_x = 1$, whereas $v_0(x)$ is given by $v_{0, 2}(x)$ given by \ref{['20/11/24-5']} in which we are changing the variance $\sigma$. The blue curve no. I is for $\sigma = 0.5$, the black curve no. II is for $\sigma = 1$, and the red curve no. III is for $\sigma = 1.5$.

Theorems & Definitions (12)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Lemma 3
  • ...and 2 more