On invariant solutions of linear time-fractional diffusion-wave equations with variable coefficients

This paper employs Lie symmetry analysis to determine infinitesimal symmetries and derive exact invariant solutions for a class of time-fractional diffusion-wave equations with variable coefficients, expressing the results in terms of Mittag-Leffler, generalized Wright, and Fox H-functions.

The Gist

Imagine you are trying to predict how a drop of ink spreads in a glass of water, or how a sound wave travels through a strange, stretchy jelly. In the real world, things don't always move in perfect, predictable lines. Sometimes they spread slowly and erratically (like ink in thick honey), and sometimes they bounce and wobble in complex ways (like sound in a rubber band).

Mathematicians call these messy, unpredictable movements "fractional diffusion-wave equations." The word "fractional" is the key here. Instead of measuring change in whole steps (like 1 second, 2 seconds), these equations measure change in "fractional" steps, capturing the memory and history of how the material behaved in the past.

This paper is like a master key that unlocks the secrets of these complex equations, specifically when the "jelly" or "honey" isn't uniform (i.e., it has variable coefficients).

Here is the breakdown of what the authors did, using some everyday analogies:

1. The Problem: A Shifting Landscape

Imagine you are trying to drive a car, but the road keeps changing its texture. Sometimes it's smooth asphalt, sometimes it's gravel, and sometimes it's mud.

• The Equation: This is the map of the road.
• The Variable Coefficients: These are the changing textures of the road (represented by $a(x)$ and $b(x)$ in the paper).
• The Goal: The authors wanted to find exact paths (solutions) that a car could take that remain "invariant" (unchanged in their pattern) despite the road shifting.

2. The Tool: The Lie Symmetry "Magic Mirror"

To solve this, the authors used a mathematical technique called Lie Symmetry Analysis.

• The Analogy: Imagine looking at a kaleidoscope. No matter how you twist the tube, the pattern inside stays symmetrical. Lie Symmetry Analysis is like finding the specific angles you can twist the "mathematical kaleidoscope" so that the equation still looks the same.
• What they found: They identified specific "twists" (called infinitesimal symmetries) that work for different types of roads (different variable coefficients). They created a Table 1, which is essentially a menu of all the possible road textures and the specific "twists" that work for each one.

3. The Result: New Languages for Old Problems

Once they found these "twists," they could simplify the complex, multi-dimensional problem into a simpler one-dimensional problem. Solving this simpler problem gave them the exact paths (solutions).

However, these paths are written in a very special, advanced language. The authors didn't just give you a simple line; they expressed the solutions using three "super-functions":

• Mittag-Leffler Functions: Think of these as the "Swiss Army Knives" of fractional math. They are the general version of the famous exponential function ($e^x$). Just as $e^x$ describes standard growth, Mittag-Leffler describes "fractional" growth (growth with memory).
• Generalized Wright Functions: These are like a more complex version of the Swiss Army Knife, capable of handling even weirder shapes of growth and decay.
• Fox H-Functions: This is the "Master Key" of them all. It is a massive, all-encompassing function that can turn into any of the other functions (or even standard math functions like sine, cosine, or exponentials) depending on how you set the dials.

4. Why This Matters

The authors showed that their new solutions are universal.

• If you turn the "fractional dial" to 1, their complex Fox H-functions magically turn into the standard solutions for diffusion (like ink spreading in water).
• If you turn the dial to 2, they turn into the standard solutions for waves (like sound or light).
• But for any number in between (like 1.5), they provide the only correct way to describe the physics of those "in-between" states.

The Bottom Line

This paper is a comprehensive instruction manual for a very specific, difficult type of physics problem.

1. They mapped out every possible shape of the "road" (variable coefficients).
2. They found the specific "symmetry keys" to unlock the math for each shape.
3. They wrote down the exact answers using powerful, generalized mathematical tools (Fox H-functions) that cover everything from slow diffusion to fast waves and everything in between.

In short, they took a chaotic, shifting mathematical landscape and found the hidden patterns that allow us to predict exactly how things will move, even when the rules of the road keep changing.

Technical Summary

1. Problem Statement

The paper addresses the mathematical challenge of finding exact invariant solutions for a specific class of linear time-fractional diffusion-wave equations (FDWEs) with variable coefficients. These equations model complex physical phenomena such as anomalous diffusion, wave propagation in visco-elastic media, and electromagnetic processes.

The specific equation under investigation is:
$$\partial_t^\alpha u(x, t) = a(x)^2 u_{xx}(x, t) + b(x) u_x(x, t)$$
where:

• $u(x, t)$ is the unknown function.
• $\partial_t^\alpha$ denotes the Riemann-Liouville fractional derivative of order $\alpha > 0$.
• $a(x)$ and $b(x)$ are sufficiently differentiable variable coefficients, with $a(x) \neq 0$.

The core difficulty lies in the combination of fractional time derivatives (which introduce non-locality and memory effects) and spatially varying coefficients, making standard analytical methods difficult to apply.

2. Methodology: Lie Symmetry Analysis

The authors employ Lie symmetry analysis, a powerful group-theoretic method for solving differential equations. The methodology proceeds in three main stages:

1. Infinitesimal Generator Determination:
   The authors construct the infinitesimal generator $X = \xi \partial_x + \tau \partial_t + \eta \partial_u$ and its extended version $X^*$, which includes terms for fractional derivatives ($\mu^{(\alpha)}$). They apply the invariance condition $X^*(\text{Eq.})|_{\text{Eq.}=0} = 0$ to derive a system of determining equations.

2. Group Classification:
   By solving the determining equations, the authors classify the possible forms of the coefficient functions $a(x)$ and $b(x)$ that admit non-trivial symmetries.

   • They introduce a transformation variable $\omega(x) = \int \frac{dr}{a(r)} + \lambda_1$ to simplify the equation.
   • They define a reduced coefficient function $c(\omega) = \frac{b(x)}{a(x)} - a'(x)$.
   • The equation is reduced to a canonical form: $\partial_t^\alpha u = u_{\omega\omega} + c(\omega)u_\omega$.
   • Table 1 and Theorem 2.1 present a complete classification of 8 distinct cases based on the functional form of $c(\omega)$ (e.g., logarithmic, power-law, trigonometric, exponential, or zero).

3. Invariant Surface Condition:
   For each identified symmetry generator, the authors solve the characteristic equation (invariant surface condition) $\xi u_x + \tau u_t - \eta u = 0$ to find the similarity variables and the form of the similarity solution.

3. Key Contributions

• Complete Group Classification: The paper provides a comprehensive classification of linear time-FDWEs with variable coefficients, identifying exactly which coefficient structures allow for Lie point symmetries. This generalizes previous work which was limited to constant coefficients or specific integer orders ($\alpha=1, 2$).
• Reduction to Fractional ODEs: The authors successfully reduce the partial differential equations (PDEs) to fractional ordinary differential equations (FODEs) using similarity transformations derived from the symmetries.
• Exact Solutions via Special Functions: Unlike classical cases where solutions are often elementary or standard hypergeometric functions, the fractional reduced equations are solved using advanced special functions:
  • Fox H-functions ($H_{p,q}^{m,n}$)
  • Generalized Wright functions ($p\Psi_q$)
  • Mittag-Leffler functions ($E_{\alpha, \beta}$)
• Unification of Limits: The derived solutions are shown to consistently reduce to known solutions for classical diffusion ($\alpha=1$) and classical wave equations ($\alpha=2$), recovering results involving exponential, hyperbolic, and Gauss hypergeometric functions.

4. Key Results

The paper derives explicit invariant solutions for the 8 classified cases. Notable results include:

• General Case (Arbitrary $c(\omega)$): Solutions are expressed in terms of the Fox H-function. For example, for specific generators, the solution takes the form:
  $$u(\omega, t) \propto \omega^s H_{1,2}^{2,0} \left[ \frac{\omega^2}{4t^\alpha} \middle| \dots \right]$$
• Specific Coefficient Cases:
  • Logarithmic/Power cases: Solutions involve Generalized Wright functions ($3\Psi_1$).
  • Exponential/Trigonometric cases ($c(\omega) \propto \tanh, \tan$): Solutions involve Mittag-Leffler functions $E_{\alpha, \beta}(z)$.
  • Zero Coefficient ($c(\omega)=0$): Solutions simplify to forms involving $e^{\epsilon \omega} E_{\alpha, \beta}(t^\alpha)$.
• Limiting Behavior: The authors explicitly demonstrate that as $\alpha \to 1$ or $\alpha \to 2$, the complex Fox H and Wright functions collapse into standard exponential, hyperbolic, and Gauss hypergeometric functions ($_2F_1$), validating the generality of their approach.

5. Significance

• Theoretical Advancement: This work bridges the gap between fractional calculus and variable-coefficient PDEs. It demonstrates that Lie symmetry analysis remains a viable and powerful tool even when dealing with the non-local nature of fractional derivatives.
• Physical Modeling: The exact solutions provided offer new analytical tools for modeling physical systems with anomalous diffusion (where mean squared displacement scales non-linearly with time) and wave propagation in inhomogeneous media (e.g., seismology, acoustics in visco-elastic materials).
• Generalization of Special Functions: The paper highlights the necessity of generalized special functions (Fox H and Wright) in fractional calculus, positioning them as the natural generalizations of hypergeometric functions found in integer-order differential equations.
• Foundation for Future Research: By providing a complete group classification, the paper sets a foundation for further studies on boundary value problems, numerical approximations, and applications in electrodynamics and mechanics involving fractional derivatives.

In conclusion, the paper successfully extends the theory of invariant solutions to a broad class of time-fractional diffusion-wave equations with variable coefficients, providing a rich set of exact analytical solutions expressed through advanced special functions.