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Uniqueness of simultaneous reconstruction of general space- and time-dependent sources and initial states in fractional diffusion equations and systems from single boundary measurements

This paper establishes the uniqueness of simultaneously reconstructing general space- and time-dependent sources and initial states in fractional diffusion equations and coupled systems from single boundary measurements, provided the fractional derivative order is irrational.

Original authors: Jaan Janno

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jaan Janno

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 you are in a dark room trying to figure out what happened inside a complex machine just by listening to the sound coming from a single small hole in the wall. This is essentially what the paper "Uniqueness of simultaneous reconstruction..." by Jaan Janno is about, but instead of a machine and sound, it deals with heat (or diffusion) and mathematical equations.

Here is a breakdown of the paper's core ideas using simple analogies.

The Setting: A Strange Diffusion Process

Usually, when we think of something spreading out—like a drop of ink in water or heat moving through a metal rod—we use standard math (calculus) that assumes the process moves forward in time step-by-step.

However, this paper deals with Fractional Diffusion Equations. Think of these as a "memory-heavy" version of diffusion.

Standard Diffusion: Like a person walking down a hallway; where they are now depends only on where they were a second ago.
Fractional Diffusion: Like a person walking down a hallway who is constantly looking back at their entire history of steps. Their current movement depends on everything they did since they started. This makes the math much more complex but also more accurate for certain real-world phenomena (like how pollutants move through soil or how drugs spread in biological tissue).

The Mystery: The "Black Box" Problem

The author sets up a scenario with two main unknowns hidden inside a room (a mathematical domain):

The Initial State: How the system started (e.g., was the room hot or cold at the very beginning?).
The Source: Something that was added to the system over time (e.g., a heater turned on and off, or a chemical injected).

The Catch:

The "Source" stops working after a certain time ( $t_0$ ).
We cannot see inside the room.
We cannot measure the whole room.
The only clue we have: We can measure the "flow" (the rate of change) coming out of one small section of the wall for a short period after the source has stopped.

The question is: Can we figure out exactly what the initial state was AND what the source was, just from that tiny bit of data on the wall?

The Big Discovery: The "Irrational Key"

The paper proves that yes, you can solve this mystery, but there is a very specific condition required: The "memory" of the system must be "irrational."

In math terms, the "order" of the fractional derivative (let's call it $\alpha$ ) must be an irrational number (like $\sqrt{2}$ or $\pi$ , not a simple fraction like $1/2$ or $3/4$ ).

The Analogy of the "Irrational Key":
Imagine the system is a lock with many tumblers.

If the memory order is a simple fraction (rational), the tumblers might line up in a repeating pattern that creates "ghost" solutions. You might think you found the source, but it could actually be a different source that just happens to look the same from the outside. The lock is ambiguous.
If the memory order is irrational, the tumblers never line up in a repeating pattern. The "ghost" solutions disappear. The lock has only one unique key that fits.

The author shows that because the order is irrational, the "echo" of the source and the "echo" of the initial state are distinct enough that, given enough time to listen to the wall, you can mathematically separate them and reconstruct the entire history of the room.

The System: Two Interacting Rooms

The paper doesn't just stop at one room; it looks at two rooms connected together (a coupled system).

Imagine two adjacent rooms where heat can flow between them.
Both rooms have their own hidden sources and starting temperatures.
You can only measure the flow coming out of Room A's wall.
The Result: Even though you can't see Room B, the fact that the two rooms are connected, combined with the "irrational memory" rule, allows you to figure out the secrets of both rooms simultaneously.

How They Did It (The Magic Trick)

The author uses a mathematical tool called the Laplace Transform.

Think of the Laplace Transform as a prism that takes a messy signal (the flow out of the wall) and splits it into different "colors" (frequencies).
The author takes this signal and looks at it from many different angles (mathematically, extending it into the complex plane).
Because the order is irrational, the signal behaves in a way that allows the author to "rotate" the view infinitely many times.
By rotating the view enough, they can isolate the specific "signature" of the initial state and the specific "signature" of the source, proving that no other combination could have produced the same result.

The Bottom Line

This paper is a theoretical proof. It doesn't tell you how to build a better heater or diagnose a disease. Instead, it answers a fundamental question: "Is the information we have enough to solve the puzzle?"

The answer is: Yes, but only if the physics of the system involves an irrational "memory" parameter. If that condition is met, the puzzle has exactly one solution, and we can uniquely reconstruct both the starting conditions and the hidden events that caused them, even with very limited data.

1. Problem Statement

The paper addresses a class of inverse problems associated with Fractional Diffusion Equations (FDEs). Specifically, the goal is to simultaneously reconstruct:

A general space- and time-dependent source function $f(t, x)$ (and $\chi(t, x)$ in coupled systems).
The initial state of the system ( $\phi(x)$ and $\psi(x)$ ).

Key Constraints and Conditions:

Equation Type: The governing equations involve the Riemann-Liouville (or Caputo) fractional time derivative of order $\alpha$ ( $0 < \alpha < 1$ ).
Source Behavior: The source functions are assumed to vanish after a certain time $t_0$ (i.e., $f(t, x) = 0$ for $t > t_0$ ). This creates a "gap" between the active source period and the observation period.
Observation Data: The reconstruction relies on single boundary measurements. Specifically, the normal derivative of the state variable, $\frac{\partial u}{\partial \nu}$ , is measured on a portion of the boundary $\Gamma \subset \partial \Omega$ over a time interval $(t_0, t_1)$ .
Critical Assumption: The order of the fractional derivative, $\alpha$ , must be irrational.

The paper considers two scenarios:

IP1 (Single Equation): Reconstruction of $f(t,x)$ and $\phi(x)$ for a single FDE where the coupling term is zero ( $a=0$ ).
IP2 (Coupled System): Reconstruction of two sources ( $f, \chi$ ) and two initial states ( $\phi, \psi$ ) for a coupled system of FDEs, using only the boundary measurement of the first component $u$ .

2. Methodology

The author employs a sophisticated analytical approach combining spectral theory, complex analysis, and properties of fractional calculus. The core methodology proceeds as follows:

A. Spectral Decomposition and Laplace Transform

The problem is analyzed using the eigenfunctions of the Laplacian operator on the domain $\Omega$ . The solution is expanded into a Fourier series with respect to these eigenfunctions.
The Laplace transform is applied to the time-domain equations. This converts the fractional differential equations into algebraic equations in the complex frequency domain ( $s$ -domain).
The solution in the $s$ -domain is expressed as a series involving the Laplace transforms of the unknown sources and initial states, divided by a characteristic denominator dependent on the eigenvalues and the fractional order.

B. Analytic Continuation and Branch Cuts

The paper exploits the fact that for $t > t_0$ , the source is zero. This implies that the Laplace transforms of the sources are entire functions (analytic everywhere in the complex plane).
The observation data (boundary flux) is shown to be analytic in a specific sector of the complex plane.
By taking the limit as the argument of the complex variable approaches the negative real axis from above and below ( $\theta \to \pm \pi$ ), the author calculates the jump across the branch cut of the fractional power $s^\alpha$ .

C. The "Irrational Order" Argument

This is the most novel part of the methodology. The author defines a series $Q(n, z)$ involving the jump across the branch cut and a rotation factor $e^{i 2\pi n / \alpha}$ .
Lemma 5 & 6: The proof relies on the fact that if $\alpha$ is irrational, the sequence $e^{i 2\pi n / \alpha}$ is dense on the unit circle.
By using the density of this sequence, the author introduces an additional continuous variable (an angle $y$ ) into the discrete series. This allows the transformation of the problem from a discrete sum over eigenvalues into a continuous functional equation.
By differentiating with respect to this new variable and evaluating at specific points, the author isolates the coefficients corresponding to specific eigenvalues.

D. Uniqueness via Linear Independence

The method isolates the Fourier coefficients of the unknowns ( $f_{kl}, \phi_{kl}$ , etc.) for each eigenmode $k$ .
Using the linear independence of the normal derivatives of the eigenfunctions on the boundary $\Gamma$ (a known result from elliptic theory), the author proves that if the boundary measurement is zero, then all Fourier coefficients of the unknowns must be zero.

3. Key Contributions

Simultaneous Reconstruction: Unlike many previous works that assume the initial state is known or the source has a separable form ( $g(t)f(x)$ ), this paper proves uniqueness for general space-time dependent sources and unknown initial states simultaneously.
Coupled Systems: The results are extended to coupled systems of fractional diffusion equations, where only the boundary measurement of one component is available to reconstruct both sources and both initial states.
Minimal Data Requirements: The uniqueness is proved even if the measurement interval $(t_0, t_1)$ and the boundary portion $\Gamma$ are arbitrarily small (provided they have non-zero measure).
The Role of Irrationality: The paper establishes that the irrationality of the fractional order $\alpha$ is a sufficient condition for uniqueness in this general setting. This highlights a fundamental difference between fractional and classical (integer-order) diffusion equations, where such uniqueness for general sources is often an open problem or requires stronger data.
First of its Kind: The author claims this is the first paper to prove uniqueness for determining a general function in a higher-dimensional domain (space-time) from data in a lower-dimensional domain (boundary-time) for this specific class of fractional inverse problems.

4. Main Results

Theorem 1 (Direct Problem): Establishes the existence, uniqueness, and regularity of the solution to the forward problem (coupled FDEs) in appropriate functional spaces ( $W^{\alpha, p}$ ). It also proves the analytic extension of the solution into a sector of the complex plane for $t > t_0$ .
Theorem 2 (Uniqueness for IP1): If $\alpha$ is irrational, and the boundary flux $\frac{\partial u}{\partial \nu}$ is zero on $\Gamma \times (t_0, t_1)$ , then the source $f \equiv 0$ and the initial state $\phi \equiv 0$ .
Theorem 3 (Uniqueness for IP2): For the coupled system, if $\alpha$ is irrational and the boundary flux of the first component is zero, then both sources ( $f, \chi$ ) and both initial states ( $\phi, \psi$ ) are identically zero, provided the eigenvalues satisfy a non-resonance condition (Condition 49).

5. Significance

Theoretical Advancement: The paper bridges a gap in the theory of inverse problems for fractional PDEs. It moves beyond simple separable sources to general functions, which are more realistic for physical modeling.
Physical Interpretation: The requirement of an irrational order $\alpha$ suggests that in physical systems modeled by FDEs (often used for anomalous diffusion), the "memory" effect (history dependence) combined with the irrationality of the order provides enough information to uniquely identify both the starting state and the forcing function from limited boundary data.
Methodological Innovation: The technique of using the density of $e^{i 2\pi n / \alpha}$ to introduce a continuous variable into a spectral series is a powerful new tool for solving inverse problems where standard Carleman estimate methods might be difficult to apply or where data is sparse.
Practical Implications: The results suggest that in experimental settings involving anomalous diffusion (e.g., in porous media, biological tissues, or complex fluids), one does not necessarily need full-field measurements or knowledge of the initial state to identify the source dynamics, provided the fractional order is known and irrational.