Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation

1. Problem Statement

The paper addresses the inverse source identification problem for a time-fractional convection-diffusion-reaction equation on an unbounded domain. The goal is to recover a time-dependent source term, $f(t)$, based on noisy measurements of the solution $u(x, t)$ taken at a single arbitrary spatial position $x_0$.

The governing equation is:
$$\partial^\alpha_{0+}u(x, t) = \omega u_{xx}(x, t) - \beta u_x(x, t) - \nu u(x, t) + f(t)$$
where:

• $\partial^\alpha_{0+}$ is the Caputo fractional derivative of order $0 < \alpha \leq 1$.
• $\omega, \beta, \nu$ are positive constants representing diffusion, convection, and reaction coefficients, respectively.
• The initial and boundary conditions are zero ($u(x,0)=0, u(0,t)=0$), and the solution is bounded as $x \to \infty$.

The Challenge: The problem is ill-posed in the sense of Hadamard. Specifically, the solution does not depend continuously on the data. High-frequency components of arbitrarily small measurement noise are amplified exponentially by the inverse operator, leading to unstable and non-physical solutions.

2. Methodology

A. Analytical Formulation

The authors first solve the problem analytically using the Fourier transform with respect to time.

1. Transforming the PDE into the frequency domain yields an ordinary differential equation in space.
2. Solving this ODE with the given boundary conditions allows the derivation of the source term in the frequency domain, $\hat{f}(\xi)$, as a function of the measured data $\hat{y}(\xi)$:
   $$\hat{f}(\xi) = \Lambda(\xi)\hat{y}(\xi)$$
   where $\Lambda(\xi)$ is the inverse operator.
3. Instability Analysis: The authors prove that $\Lambda(\xi)$ is unbounded as $|\xi| \to \infty$. This unboundedness confirms the ill-posed nature of the problem, as noise in the high-frequency components of the data leads to infinite errors in the reconstructed source.

B. Regularization Strategy

To stabilize the solution, the authors propose three one-parameter families of regularization operators ($R^i_\mu$ for $i=1, 2, 3$) designed to counteract the unbounded growth of $\Lambda(\xi)$. These operators modify the inverse operator by introducing a denominator that suppresses high frequencies:

1. Operator 1 ($R^1_\mu$): Based on a quadratic penalty in frequency.
   $$(R^1_\mu \hat{y})(\xi) = \frac{\Lambda(\xi)}{1 + \mu^2 \xi^2} \hat{y}(\xi)$$
2. Operator 2 ($R^2_\mu$): Based on a quartic penalty in frequency.
   $$(R^2_\mu \hat{y})(\xi) = \frac{\Lambda(\xi)}{1 + \mu^2 \xi^4} \hat{y}(\xi)$$
3. Operator 3 ($R^3_\mu$): Based on an exponential (Gaussian-like) penalty.
   $$(R^3_\mu \hat{y})(\xi) = \frac{\Lambda(\xi)}{e^{\mu^2 \xi^2/4}} \hat{y}(\xi)$$

Here, $\mu > 0$ is the regularization parameter.

C. Parameter Selection Rule

A key contribution is the proposal of a new a priori parameter selection rule. Unlike traditional methods that require knowledge of the norm of the unknown source (which is often unavailable), this rule depends solely on the noise level ($\delta$) and the maximum tolerated noise ($\delta_M$):
$$\mu^2 = \left( \frac{\delta}{\delta_M} \right)^{\frac{2}{p+2}}$$
where $p$ relates to the smoothness of the source in the Sobolev space $H^p$.

3. Key Contributions

1. Extension to Fractional Convection-Diffusion-Reaction: While previous works focused on fractional diffusion or classical parabolic equations, this paper generalizes the source identification problem to the more complex time-fractional convection-diffusion-reaction equation.
2. Three Regularization Families: The introduction and comparative analysis of three distinct regularization operators tailored to the specific structure of the fractional inverse operator.
3. Noise-Dependent Parameter Rule: The derivation of a parameter selection rule that does not require prior knowledge of the source's norm, making the method more practical for real-world applications where the source is unknown.
4. Rigorous Error Bounds: The authors derive an optimal error bound for the estimation error in the $L^2$ norm. They prove that the regularized solutions converge to the true source as the noise level $\delta \to 0$.
   $$\|f - f^i_{\delta, \mu}\|_{L^2} \leq K_i \max \left\{ \left(\frac{\delta}{\delta_M}\right)^{\frac{2}{p+2}}, \left(\frac{\delta}{\delta_M}\right)^{\frac{p}{p+2}}, \left(\frac{\delta}{\delta_M}\right)^{\frac{p-2}{p+2}} \right\}$$

4. Numerical Results and Discussion

The authors validated the theoretical findings using two numerical examples with different source characteristics:

• Example 1 (Discontinuous Source): A piecewise constant function (step-like). This tests the method's ability to handle the Gibbs phenomenon and lack of smoothness.
• Example 2 (Continuous Source): An exponentially decaying function ($e^{-t}$).

Findings:

• Stability: All three regularization operators successfully stabilized the solution, preventing the explosion of errors seen in the non-regularized case.
• Convergence: The relative error decreased as the noise level ($\epsilon$) decreased, confirming the theoretical convergence.
• Performance Comparison:
  • $R^1_\mu$ (quadratic denominator) generally yielded the lowest relative errors across different noise levels.
  • The methods performed well even for the discontinuous source, despite the theoretical error bounds assuming the source belongs to a specific Sobolev space ($H^p$).
  • The results suggest that smoother high-frequency filtering (as provided by the exponential operator $R^3_\mu$) is effective, but the quadratic filter ($R^1_\mu$) offered the best balance for the tested cases.

5. Significance

This work is significant because it bridges a gap in the literature regarding inverse problems in fractional transport processes.

• Modeling Realism: Fractional convection-diffusion-reaction equations are more realistic models for anomalous transport in complex media (e.g., porous media, biological tissues) than simple diffusion equations.
• Practical Applicability: By providing a parameter selection rule independent of the unknown source's norm, the proposed method is more robust and applicable to experimental data where the "ground truth" is unavailable.
• Theoretical Foundation: The rigorous derivation of error bounds and the proof of convergence provide a solid mathematical foundation for future numerical implementations in engineering and physics applications involving fractional calculus.