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A backward problem for the time-fractional pseudo-parabolic equation with a variable coefficient

This paper investigates the ill-posed backward problem for a time-fractional pseudo-parabolic equation with a variable coefficient by establishing theoretical existence and uniqueness results, developing a stable finite-difference scheme for the forward problem, and formulating a Tikhonov regularisation method to successfully reconstruct the unknown initial state from noisy final-time observations.

Original authors: Arshyn Altybay

Published 2026-03-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Arshyn Altybay

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 a detective trying to solve a mystery, but instead of finding clues at the crime scene, you are looking at the aftermath. You see a shattered vase on the floor (the final state), and your job is to figure out exactly what the vase looked like before it fell (the initial state).

This paper is about solving a very specific, difficult version of this detective story involving heat, fluids, and time.

The Setting: A Strange, "Memory-Keeping" Fluid

Usually, when we model how heat spreads or how a fluid moves, we use standard equations. But this paper deals with a Time-Fractional Pseudo-Parabolic Equation. That sounds like a mouthful, so let's break it down with an analogy:

Standard Heat Flow: Imagine pouring hot coffee into a cup. The heat spreads out smoothly and predictably.
The "Pseudo-Parabolic" Twist: Imagine the coffee is in a cup made of a special sponge. The sponge doesn't just let heat pass through; it remembers how fast the heat was moving a moment ago and resists sudden changes. It has "inertia."
The "Time-Fractional" Twist: Now, imagine the sponge is also a bit magical. It doesn't just remember the last second; it remembers everything that happened in the past, but the memory gets fuzzier the further back you go. This is called "memory" or "hereditary" behavior.
The Variable Coefficient: The sponge's properties change as time goes on (maybe it gets wetter or drier), making the rules of the game shift constantly.

The Problem: The Backward Mystery

In the real world, we usually know the starting point (the hot coffee) and want to predict the future (how cool it gets). This is the Forward Problem.

However, this paper tackles the Backward Problem:

The Scenario: We are given the temperature of the fluid at the very end of the experiment (Time $T$ ).
The Goal: We need to work backward to figure out what the temperature was at the very beginning (Time $0$).
The Catch: This is mathematically "ill-posed." In plain English, this means it's incredibly unstable. If you have even a tiny speck of dust (noise) on your final measurement, your guess for the starting temperature could be wildly wrong. It's like trying to guess the exact shape of a puzzle piece by looking at a blurry photo of the finished picture.

How the Authors Solved It

The authors, led by Arshyn Altybay, approached this in three main steps:

1. Proving It's Possible (The Theory)

First, they had to prove that a solution actually exists and is unique. They used a mathematical technique called spectral expansion.

The Analogy: Imagine the fluid's movement is like a song. Any complex sound can be broken down into simple musical notes (frequencies). The authors broke the complex fluid equation down into these simple "notes" (mathematical sine waves).
They proved that if you know the final "song" (the final state), you can mathematically reverse-engineer the exact "notes" that started it, provided the "sponge" (the material) behaves nicely. They showed that the math doesn't break down, as long as the material properties stay within reasonable limits.

2. Building a Digital Simulator (The Forward Solver)

To solve this on a computer, you can't use continuous math; you have to chop time and space into tiny little steps (pixels and frames).

The Analogy: Imagine taking a video of the fluid and freezing it into 1,000 tiny frames.
They created a Finite-Difference Scheme. This is a set of rules for calculating how the fluid moves from one frame to the next.
The Challenge: Because the fluid has "memory" (fractional derivative), calculating the next frame isn't just about the current frame; you have to look back at all previous frames. This is computationally heavy.
The Solution: They used a clever "graded mesh" strategy. Instead of taking equal steps in time, they took very small steps at the beginning (where things change fast) and larger steps later on. This made the simulation fast and accurate. They proved this digital simulator is stable—it won't crash or explode with errors.

3. Cracking the Code with "Tikhonov Regularisation" (The Inverse Solver)

This is the most critical part. Once they had a perfect simulator to move forward, they needed to move backward.

The Problem: If you just try to reverse the math directly, the tiny errors in your final measurement get amplified into huge, nonsensical errors in your initial guess. It's like trying to un-mix a smoothie; without help, you just get a mess.
The Solution (Tikhonov Regularisation): They added a "stabilizer" or a "smoothness filter."
- The Analogy: Imagine you are trying to guess the starting shape of a crumpled piece of paper based on its crumpled state. If you just guess randomly, you might guess a shape that is jagged and impossible. Tikhonov regularisation says, "Okay, let's find the starting shape that fits the final data best, but also looks the most 'natural' and smooth."
- It balances two goals: matching the final data and keeping the initial guess from being too crazy.

The Results: Does It Work?

The authors tested their method with computer simulations:

Perfect Data: When they gave the computer perfect, noise-free data, the method recovered the initial state with extremely high precision.
Noisy Data: In the real world, measurements are never perfect. They added "noise" (random errors) to the final data to simulate real-world imperfections.
- The Result: Even with 5% noise (which is a lot in math terms), the method still recovered the general shape of the initial state. The "smoothness filter" prevented the errors from blowing up. The reconstructed image looked like the original, just slightly blurred, rather than a chaotic mess.

Summary

In short, this paper is about reverse-engineering the past for a complex, memory-having fluid system.

They proved mathematically that the past can be found.
They built a robust digital engine to simulate the fluid.
They applied a "stabilizing filter" (Tikhonov) to make the reverse-engineering process robust against real-world measurement errors.

It's a powerful toolkit for engineers and scientists who need to figure out what happened in the past (like pollution sources in groundwater or heat sources in materials) based only on what they can measure today.

1. Problem Statement

The paper addresses an inverse problem (specifically, a backward-in-time problem) for a time-fractional pseudo-parabolic equation with a time-dependent coefficient. The governing equation is:

$\partial^\alpha_t u(x, t) - u_{xx}(x, t) - \mu(t) u_{xxt}(x, t) = f(x, t), \quad (x, t) \in (0, l) \times (0, T)$

Key Components:

Operator: $\partial^\alpha_t$ is the Caputo fractional derivative of order $\alpha \in (0, 1)$ .
Coefficient: $\mu(t)$ is a time-varying pseudo-parabolic coefficient.
Boundary Conditions: Homogeneous Dirichlet conditions $u(0, t) = u(l, t) = 0$ .
Data: The source term $f(x, t)$ and the final-time measurement $u(x, T) = \psi(x)$ are known.
Goal: Reconstruct the unknown initial state $u_0(x) = u(x, 0)$ and the full trajectory $u(x, t)$ on $[0, T]$ .

This problem models anomalous transport phenomena with memory and relaxation effects, common in porous media filtration and non-Newtonian fluid dynamics. The backward problem is inherently ill-posed, meaning small errors in the final data $\psi$ can lead to large deviations in the reconstructed initial state.

2. Methodology

The authors employ a three-pronged approach: theoretical analysis, numerical discretization of the forward problem, and a regularized reconstruction algorithm for the inverse problem.

A. Theoretical Analysis (Existence and Uniqueness)

Spectral Decomposition: The problem is analyzed using the eigenfunctions of the spatial operator $-\partial_{xx}$ (sine series expansion).
Volterra Reduction: By projecting the PDE onto the eigenbasis, the problem reduces to a system of fractional ordinary differential equations (ODEs). These are transformed into Volterra integral equations of the second kind with weakly singular kernels.
Explicit Reconstruction: The authors derive an explicit series representation for the solution $u(x, t)$ and the initial data $u_0(x)$ .
Positivity Lemma: A critical step involves proving that the coefficient function $A_k(T)$ in the reconstruction formula is strictly positive, ensuring the solution is well-defined.
Stability: Continuous dependence of the initial state on the final data is established, providing a stability estimate in $L^2$ norms.

B. Numerical Discretization (Forward Solver)

To solve the direct (forward) problem numerically, a finite-difference scheme is developed:

Spatial Discretization: Standard second-order central differences are used for the Laplacian ( $u_{xx}$ ) and the mixed derivative term ( $u_{xxt}$ ).
Temporal Discretization: The Caputo derivative is approximated using the L1 scheme on a graded temporal mesh ( $t_k = T(k/M)^r$ ). Graded meshes are essential to handle the initial singularity typical of fractional derivatives ( $\partial_t u \sim t^{\alpha-1}$ ).
Linear System: The discretization yields a tridiagonal linear system at each time step, solved efficiently via the Thomas algorithm.
Analysis: The scheme is proven to be unconditionally stable and convergent. The convergence rate is optimal for the graded mesh, limited primarily by the first-order accuracy of the pseudo-parabolic term discretization.

C. Inverse Reconstruction Algorithm

The inverse problem is solved using an operator construction + Tikhonov regularization approach:

Forward Map Construction: The discrete forward solution operator $F_h$ is constructed by running the forward solver $N-1$ times with impulse initial conditions (canonical basis vectors) and zero source term.
Affine Splitting: The final state is decomposed into a homogeneous part (dependent on $u_0$ ) and a forced part (dependent on $f$ ).
Regularization: The reconstruction is formulated as a minimization problem using zeroth-order Tikhonov regularization:
$J_\lambda(u_{0,h}) = \frac{1}{2} \| F_h u_{0,h} - d_h \|_2^2 + \frac{\lambda}{2} \| u_{0,h} \|_2^2$
where $d_h$ is the difference between the measured final data and the forced contribution. The parameter $\lambda$ stabilizes the ill-posed system.

3. Key Contributions

Theoretical Rigor: The paper provides the first rigorous proof of existence and uniqueness for the backward time-fractional pseudo-parabolic equation with a time-dependent coefficient $\mu(t)$ . Previous works often focused on constant coefficients or lacked full numerical analysis.
Stable Numerical Scheme: Development of a stable, convergent finite-difference scheme on graded meshes for the direct problem, which serves as the foundation for the inverse solver.
Robust Inverse Algorithm: A practical reconstruction algorithm combining the forward solver with Tikhonov regularization, specifically designed to handle noisy final-time data.
Comprehensive Validation: Extensive numerical experiments validating the method under both noise-free and noisy conditions, demonstrating the method's resilience.

4. Numerical Results

Noise-Free Validation: The method successfully recovers the initial state with high accuracy. Error analysis shows expected convergence rates as the spatial ( $N$ ) and temporal ( $M$ ) grids are refined. The reconstructed initial state and the forward-propagated final state match the exact solutions closely.
Noisy Data Robustness:
- Experiments were conducted with additive Gaussian noise at levels of 1%, 3%, and 5%.
- Results: As noise levels increased, the error in the reconstructed initial state increased, but the growth was controlled and proportional (stable dependence), rather than exploding.
- Visuals: Even with 5% noise, the Tikhonov regularization preserved the main shape of the initial state profile, preventing spurious oscillations.
- Fractional Order Sensitivity: The accuracy was found to be relatively insensitive to the fractional order $\alpha$ , with the noise level being the dominant factor affecting error magnitude.

5. Significance

This work bridges a significant gap in the literature regarding inverse problems for time-fractional pseudo-parabolic equations.

Physical Relevance: It addresses models with time-varying material properties ( $\mu(t)$ ), which are more realistic for engineering applications like fluid transport in fractured rocks or filtration processes.
Ill-Posedness Handling: It demonstrates a viable computational framework for solving backward problems in fractional diffusion-wave models, which are notoriously unstable.
Future Impact: The methodology provides a template for extending these analyses to multi-dimensional domains and nonlinear source terms, which are critical for more complex physical simulations.

In summary, the paper establishes a solid theoretical foundation and a robust numerical framework for identifying initial conditions in complex fractional diffusion systems, proving that accurate reconstruction is possible even when data is corrupted by noise.