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Solutions of time fractional anomalous diffusion equations with coefficients depending on both time and space variables

This paper derives explicit solutions for time-fractional anomalous diffusion equations with space- and time-dependent diffusivity coefficients, expressing them in terms of Fox-H and generalized Wright functions to advance the understanding of anomalous diffusion across various fields.

Original authors: Ganbileg Bat-Ochir, Khongorzul Dorjgotov, Uuganbayar Zunderiya

Published 2026-07-31

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CC BY 4.0

Original authors: Ganbileg Bat-Ochir, Khongorzul Dorjgotov, Uuganbayar Zunderiya

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world as a giant, bustling city where particles are like tiny commuters trying to get from point A to point B. In a perfect, calm city, these commuters move in a predictable, straight line, spreading out evenly like ink dropped in a glass of still water. This is "normal" diffusion, a rule that physics has understood for a long time. But the real world is rarely that calm. Sometimes the streets are clogged, the traffic lights are broken, or the terrain is made of jagged, fractal mountains. In these chaotic environments, particles don't move in straight lines; they get stuck, jump unexpectedly, or wander aimlessly. This messy, unpredictable spreading is called "anomalous diffusion."

To describe this chaos, scientists use special math tools called "fractional equations." Think of these as a camera with a slow shutter speed that captures not just where a particle is, but also how its past movements influence its future. Usually, scientists have only been able to solve these equations when the "traffic rules" (the diffusion coefficients) changed based on where you were in the city (space). But what if the traffic rules also changed depending on what time of day it was? What if the streets were jammed in the morning but empty at night, and this pattern shifted as you moved across town? Until now, no one had figured out the exact math to describe this double-changing chaos. This paper steps into that gap, tackling the messy reality where the rules of movement depend on both space and time simultaneously.

The authors of this study, Ganbileg Bat-Ochir, Khongorzul Dorjgotov, and Uuganbayar Zunderiya, have successfully derived explicit solutions for these complex time-fractional anomalous diffusion equations. They focused on a specific, generalized equation where the "diffusivity coefficient"—the number that tells us how fast particles spread—is a function of both space ( $x$ ) and time ( $t$ ). While earlier research had proposed a specific form like $K(x, t) = cx^at^b$ to describe diffusion in turbulent media, this study goes further. They generalized the problem to a broader equation where the coefficient depends on both variables in a more complex way, specifically involving terms like $tm(Ax^d \partial^2/\partial x^2 + Bx^{d-1} \partial/\partial x + Cx^{d-2})$ . This allows them to model processes that were previously mathematically inaccessible.

The main finding is that they found exact mathematical formulas to describe how particles move in this double-variable chaos. However, these aren't simple formulas you'd find in a high school algebra book. Instead, the solutions are expressed using advanced, specialized mathematical functions known as Fox-H functions and generalized Wright functions. You can think of these functions as "super-tools" or "master keys" that the authors used to unlock the door to the solution. Just as a regular key opens a simple lock, these complex functions are necessary to open the intricate lock of equations where time and space are tangled together.

The paper provides three distinct solutions depending on the specific numbers involved in the equation:

If the "fractional order" (a number $\alpha$ that dictates how "weird" the diffusion is) is between 0 and 2, the solution is written using the Fox-H function.
If that number is greater than 2, the solution is a sum of terms involving the generalized Wright function.
There is also a special case where the space variable behaves in a specific way (when a certain parameter $d$ equals 2), which leads to a solution involving exponential functions, a much simpler and more familiar shape.

The authors didn't just guess these answers; they proved them rigorously. They started with a known class of linear fractional differential equations and used a clever substitution (changing variables to simplify the problem) to transform their complex diffusion equation into a form they could solve. They then verified their answers by plugging them back into the original equation to ensure everything balanced perfectly.

It is important to note what this paper doesn't do. It doesn't simulate a specific real-world event like oil spreading in the ocean or a virus moving through a crowd. Instead, it provides the theoretical "blueprint" or the exact mathematical map for a broad class of these chaotic systems. The authors explicitly state that previous research had only solved cases where the diffusion coefficient depended on space alone. By extending this to include time, they have opened the door to modeling processes that were previously mathematically inaccessible. While the paper mentions that these results have potential applications in a wide range of fields, it stops short of detailing specific real-world uses, focusing instead on the mathematical breakthrough itself.

In essence, this paper is a significant step forward in understanding the "grammar" of chaotic movement. By providing exact solutions for equations where the rules of diffusion change with both time and space, the authors have given scientists a new, powerful lens to view the world's most unpredictable spreading processes. Whether it's heat moving through a turbulent fluid or particles navigating a fractal structure, we now have a more precise mathematical language to describe the dance.

Technical Summary: Solutions of Time Fractional Anomalous Diffusion Equations with Coefficients Depending on Both Time and Space Variables

Problem Statement
Anomalous diffusion processes are frequently modeled using fractional differential equations. While previous research has successfully derived exact solutions for anomalous diffusion equations where the diffusivity coefficient depends solely on the spatial variable, a gap remains in the literature regarding cases where the diffusivity depends on both space and time. Specifically, there were no published explicit solutions for time-fractional anomalous diffusion equations with coefficients of the form $K(x, t) = cx^at^b$ . This work addresses the generalized time-fractional anomalous diffusion equation:
$\frac{\partial^\alpha}{\partial t^\alpha}u(x, t) = t^m \left( A x^d \frac{\partial^2}{\partial x^2}u(x, t) + B x^{d-1} \frac{\partial}{\partial x}u(x, t) + C x^{d-2}u(x, t) \right)$
where $x > 0, t > 0$ , $\alpha$ is the fractional order, $m$ is a non-negative integer, $d$ is a real number, and $A, B, C$ are real constants. This equation generalizes models proposed for turbulent media and fractal structures by incorporating time-dependent diffusivity.

Methodology
The authors employ a method of invariant solutions combined with the theory of special functions. The core strategy involves:

Variable Transformation: A substitution of the form $u(x, t) = x^a \phi(z)$ , where $z = x^{\frac{d-2}{\alpha+m}}t$ , is applied to reduce the partial differential equation (PDE) into a linear fractional ordinary differential equation (ODE) with variable coefficients.
Factorization of Operators: The resulting ODE is analyzed by constructing a characteristic polynomial associated with the differential operator. The operator is factorized into a product of first-order operators involving the roots of this polynomial.
Special Function Representation: The reduced fractional ODE is solved using Fox H-functions and generalized Wright functions. The authors utilize specific identities and differentiation properties of these functions (referenced from literature [10, 16, 17, 18, 19, 20]) to verify that the proposed forms satisfy the differential equation.
Case Analysis: The solutions are derived and categorized based on the relationship between the fractional order $\alpha$ and the spatial exponent $d$ , as well as the value of $d$ (specifically distinguishing $d \neq 2$ and $d = 2$ ).

Key Contributions and Results
The primary contribution of this work is the derivation of explicit exact solutions for the generalized equation (1) in terms of Fox H-functions and generalized Wright functions. The main results are presented in Theorem 1, which provides solutions under three distinct conditions:

Case $0 < \alpha < 2$ and $d \neq 2$ : The solution is expressed as a Fox H-function:
$u(x, t) = c_1 x^a H_{1, m+2}^{m+2, 0} \left[ \frac{x^{2-d}}{A(d-2)^2(\alpha+m)^m t^{\alpha+m}} \Bigg| \begin{matrix} (1, \alpha+m) \\ (-\frac{s_1}{\alpha+m}, 1), (-\frac{s_2}{\alpha+m}, 1), (\frac{j}{\alpha+m}, 1)_{1,m} \end{matrix} \right]$
where $s_1, s_2$ are roots of the characteristic equation derived from the transformation.
Case $\alpha > 2$ and $d \neq 2$ : The solution is expressed as a finite sum of generalized Wright functions ( $\Psi$ ):
$u(x, t) = x^a \sum_{k=1}^{[\alpha]+1} c_k x^{\frac{(d-2)(\alpha-k)}{\alpha+m}} t^{\alpha-k} \times {}_{m+3}\Psi_1 \left[ \dots \right]$
Case $d = 2$ : The solution is also given in terms of generalized Wright functions, with the argument depending on the constant $K = Aa^2 - Aa + Ba + C$ .

Additionally, Corollary 1 demonstrates that for the specific case where $\alpha = 1$ and $d \neq 2$ , and with a specific choice of the parameter $a$ , the solution simplifies to an exponential function form. This connects the generalized fractional results to classical diffusion behaviors in specific limits.

Significance and Claims
The authors state that this work represents a "significant advancement" in the understanding of anomalous diffusion. By extending previous exact solution methods (which were limited to space-dependent coefficients) to include time-dependent coefficients, the study enables the modeling of processes that were previously inaccessible to analytical treatment. The paper claims that these explicit solutions, expressed in Fox H and generalized Wright functions, provide a rigorous mathematical foundation for analyzing anomalous diffusion in turbulent media and fractal structures where diffusivity varies with both space and time. The work builds directly upon the framework established in references [1–14] regarding space-dependent coefficients and the specific turbulent media model in [15].

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