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An Inverse Source Problem For a Time-Fractional Mixed Wave-Diffusion-Wave Equation in a Cylindrical Domain

This paper establishes the existence and uniform convergence of a solution to an inverse source problem for a time-dependent variable-order fractional wave-diffusion-wave equation in a cylindrical domain by constructing the solution via a Fourier-Bessel series and employing the method of separation of variables.

Original authors: Erkinjon Karimov, Muzaffar Toshpulatov

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Erkinjon Karimov, Muzaffar Toshpulatov

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

The Big Picture: A "Shape-Shifting" Gas Problem

Imagine you are trying to figure out where a gas leak is happening inside a long, hollow pipe (a cylinder). You can't see inside the pipe, but you can measure the pressure at the ends and at a specific moment in time. Your goal is to work backward from those measurements to find the source of the leak. This is called an Inverse Source Problem.

Usually, scientists use two different rules to describe how gas moves:

Wave Rules: Like a sound wave, the gas moves fast and hits a wall, bouncing back.
Diffusion Rules: Like a drop of ink spreading in water, the gas moves slowly and spreads out evenly over time.

In the real world, gas in tight underground rocks (like shale) doesn't just pick one rule. It starts moving like a fast wave, then slows down and spreads like ink, and might even speed up again later. This paper tackles a very complex mathematical model that combines both behaviors into one equation that can "switch" between them depending on the time of day.

The Mathematical "Swiss Army Knife"

To describe this switching behavior, the authors use a special mathematical tool called the Prabhakar-Caputo derivative.

The Analogy: Think of a standard stopwatch. It just counts seconds. But this special tool is like a "smart stopwatch" that remembers not just the current second, but also how the gas was behaving in the past. It has extra "knobs" (parameters) that allow it to mimic the complex, "sticky" memory of gas moving through rough, uneven rocks.

How They Solved It: The "Musical" Approach

The authors needed to find the exact formula for the gas source ( $f(r)$ ) and how the gas moves ( $u(t, r)$ ). To do this, they used a technique called Separation of Variables, which is like breaking a complex song into individual notes.

The Cylinder: They imagined the pipe as a drum. When you hit a drum, it vibrates in specific patterns called Bessel functions. These are the "notes" the pipe can naturally play.
The Series: Instead of trying to solve the whole messy equation at once, they assumed the solution was a sum of these "notes" (a Fourier-Bessel series).
The Switching Times: The problem is divided into three time zones:
- Zone 1: The gas acts like a wave.
- Zone 2: The gas acts like a diffuser (spreading ink).
- Zone 3: The gas acts like a wave again.
- The math ensures that when the gas moves from one zone to the next, it doesn't "jump" or break; it flows smoothly, like a car changing gears without stalling.

The "Puzzle" of the Coefficients

Once they broke the problem into "notes," they had to figure out the volume (amplitude) of each note. This turned into a giant algebraic puzzle.

They set up a system of equations based on the measurements they took at the start and at a specific time in the middle.
They used a method called Cramer's Rule (a way to solve puzzles with many variables) to find the missing pieces.
The Catch: To solve the puzzle, a specific number (called a determinant, $\tilde{\Delta}_k$ ) must not be zero. The authors proved that for very high-frequency "notes" (large numbers), this number stays positive and safe, meaning the puzzle always has a unique solution.

The "Safety Net": Proving the Math Works

In math, just writing down a formula isn't enough; you have to prove that if you add up all those infinite "notes," the result actually makes sense and doesn't explode into infinity.

The Analogy: Imagine stacking an infinite number of bricks. If the bricks get smaller fast enough, the tower stays stable. If they don't, the tower collapses.
The Proof: The authors used advanced estimates involving special functions (Mittag-Leffler functions) to show that the "bricks" (the terms in their series) get small very quickly. They proved that as long as the initial gas conditions are smooth enough (like a smooth curve rather than a jagged line), the infinite sum converges. This guarantees that their solution is real, stable, and mathematically valid.

Summary of Claims

The Problem: They modeled gas flow in a cylinder that switches between wave-like and diffusion-like behavior using a complex, memory-based fractional equation.
The Method: They broke the problem down into a series of Bessel function "notes" and solved for the unknown source using a system of linear equations.
The Result: They provided an explicit formula for the unknown gas source and proved that their infinite series solution is mathematically sound and converges uniformly (it works perfectly everywhere in the cylinder).
The Condition: This works as long as the input data (the initial gas shape and the final measurement) is smooth enough and a specific mathematical condition (the determinant) is not zero.

The paper does not claim to solve a specific real-world gas leak right now; rather, it provides the rigorous mathematical "blueprint" that proves such a problem can be solved under these specific, complex conditions.

Technical Summary: An Inverse Source Problem for a Time-Fractional Mixed Wave-Diffusion-Wave Equation in a Cylindrical Domain

Problem Formulation
This paper investigates an inverse source problem for a mixed-type fractional wave-diffusion-wave equation defined within a cylindrical domain $\Omega$ . The governing equation models physical processes where the system transitions between wave-like and diffusive behaviors over distinct time intervals. Specifically, the domain is partitioned into three temporal regions ( $\Omega_1, \Omega_2, \Omega_3$ ) separated by time points $T_1$ and $T_2$ .

The governing equation involves the Prabhakar-Caputo fractional derivative, denoted as $P CD^{\alpha, \beta, \gamma, \delta}_{at}$ , which generalizes the classical Caputo derivative by incorporating a three-parameter Mittag-Leffler kernel. This operator allows for the modeling of complex memory effects and multi-scale relaxation phenomena inherent in anomalous transport processes, such as gas flow in porous media. The equation takes the form:
$P CD^{\alpha_i, \beta_i, \gamma_i, \delta}_{t_{i-1}t} u(t, r) - \Delta u(t, r) = f(r)$
where the fractional order $\beta$ varies across the intervals: $1 < \beta_1 \leq 2$ (wave-like), $0 < \beta_2 \leq 1$ (diffusive), and $1 < \beta_3 \leq 2$ (wave-like). The spatial operator $\Delta$ is the Laplacian in polar coordinates.

The inverse problem seeks to determine the pair of functions $\{u(t, r), f(r)\}$ , where the source term $f(r)$ is unknown and depends only on the spatial variable. The problem is supplemented with:

Initial conditions at $t=0$ .
Boundary conditions at $r=0$ and $r=1$ .
Transmission conditions ensuring continuity of the solution and specific limits of the fractional derivatives at the transition times $T_1$ and $T_2$ .
An additional final-time measurement condition $u(\xi, r) = \psi(r)$ for a fixed time $\xi \in [T_1, T_2]$ .

Methodology
The authors employ the method of separation of variables to construct the solution. The spatial component is solved using a Sturm-Liouville system, yielding eigenfunctions in the form of zero-order Bessel functions of the first kind, $J_0(\mu_k r)$ , where $\mu_k$ are the roots of the boundary conditions.

The temporal components are reduced to a sequence of fractional differential equations for each mode $k$ . The general solutions for these equations are expressed using bivariate Mittag-Leffler-type functions ( $E_2$ ), which arise naturally from the Prabhakar fractional integral and derivative operators.

To solve the inverse problem, the authors:

Express the unknown solution $u(t, r)$ and the unknown source $f(r)$ as Fourier-Bessel series expansions.
Derive a system of linear algebraic equations for the unknown coefficients of the series by applying the initial, boundary, transmission, and final-time measurement conditions.
Utilize Cramer's rule to explicitly solve for the unknown coefficients ( $A_{k}$ ) and the Fourier-Bessel coefficients of the source function ( $f_k$ ).
Analyze the determinant of the system ( $\tilde{\Delta}_k$ ) to ensure the uniqueness of the solution.

Key Results

Explicit Analytical Representation: The paper derives explicit analytical formulas for the unknown source function $f(r)$ and the solution $u(t, r)$ in terms of infinite series involving Bessel functions and bivariate Mittag-Leffler functions.
Existence and Uniqueness: The authors prove that the system of equations for the coefficients has a unique solution provided that the main determinant $\tilde{\Delta}_k$ is non-zero. Lemma 4.1 demonstrates that $\lim_{k \to \infty} \tilde{\Delta}_k = T_1 > 0$ , ensuring the determinant is non-zero for sufficiently large $k$ .
Uniform Convergence: A rigorous proof of the uniform convergence of the infinite series solution is provided. By utilizing the asymptotic properties of the bivariate Mittag-Leffler functions and the decay rates of the Fourier-Bessel coefficients (based on the smoothness of the input data $\phi(r)$ and $\psi(r)$ ), the authors establish that the series converges absolutely and uniformly under specific regularity conditions (specifically, if the input functions are differentiable at least 10 times and satisfy certain boundary vanishing conditions).

Significance and Claims
The paper claims to provide a rigorous mathematical framework for solving inverse source problems in mixed-type fractional evolution equations within cylindrical domains. The primary contribution lies in the extension of these problems to the Prabhakar fractional derivative, which offers greater flexibility in modeling memory effects compared to standard Caputo or Riemann-Liouville derivatives.

The authors emphasize that their approach successfully handles the "time-switched" nature of the equation, where the system transitions between wave and diffusion regimes. The work generalizes previous studies on classical parabolic-hyperbolic equations and fractional wave-diffusion systems by incorporating the more complex Prabhakar kernel and establishing the convergence of the resulting series solutions in a cylindrical geometry. The study does not propose new experimental setups or specific numerical simulations but focuses on the theoretical existence, uniqueness, and convergence of the analytical solution.