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

Regularization Methods for Inverse Source Problems in Time-Fractional Convection-Diffusion-Reaction Equations

Problem Formulation and Analytical Solution

The paper addresses the identification of an unknown, time-dependent source term $f(t)$ in the one-dimensional time-fractional convection-diffusion-reaction equation: $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$ where $\partial_{0^+}^{\alpha}$ denotes the Caputo fractional derivative for $\alpha\in(0,1]$, with homogeneous zero initial and boundary conditions, and boundedness is assumed.

Measurements are available in the form $u(x_0, t)=y(t)$, contaminated by bounded noise $y_\delta(t)$, with $\|y - y_\delta\|_{L^2}\leq\delta$. The aim is to recover $f(t)$ from $y_\delta(t)$ given knowledge of model parameters $\omega$, $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$0, $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$1, $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$2.

By employing the Fourier transform in time, an explicit analytical solution for the source in the frequency domain is obtained: $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$3 where the transfer function $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$4 is given by

$$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$5

The explicit expression enables the analysis of the underlying inverse problem's stability properties.

Ill-Posedness and Regularization

The key technical result is the demonstration that the inverse mapping defined via $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$6 is unbounded. As $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$7, $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$8 grows rapidly, resulting in strong amplification of high-frequency noise in $$\partial_{0^+}^{\alpha} u(x,t)=\omega \,u_{xx}(x,t)-\beta \, u_{x}(x,t)-\nu \, u(x,t)+f(t), \quad x>0,\, t>0,$$9. The solution fails to depend continuously on the data in the sense of Hadamard, establishing the inverse problem as severely ill-posed.

To mitigate this, three one-parameter families of regularization operators $\partial_{0^+}^{\alpha}$0 ($\partial_{0^+}^{\alpha}$1) are introduced in the frequency domain: $\partial_{0^+}^{\alpha}$2 where $\partial_{0^+}^{\alpha}$3 is the regularization parameter. The regularizers effectively damp high frequencies, converting the ill-posed problem into a family of well-posed ones parameterized by $\partial_{0^+}^{\alpha}$4.

Parameter Choice and Error Estimates

A central contribution lies in the derivation of a parameter choice rule based solely on the noise level $\partial_{0^+}^{\alpha}$5 and a user-specified (or estimated) maximum tolerated noise $\partial_{0^+}^{\alpha}$6: $\partial_{0^+}^{\alpha}$7 where $\partial_{0^+}^{\alpha}$8 corresponds to the regularity of $\partial_{0^+}^{\alpha}$9 in the Sobolev space $\alpha\in(0,1]$0.

Sharp upper bounds for the $\alpha\in(0,1]$1-error between the true source and the regularized estimate are proven: $\alpha\in(0,1]$2 where $\alpha\in(0,1]$3 depends on the unknown source's a priori smoothness and problem parameters but not on $\alpha\in(0,1]$4.

Importantly, the derivation of this error bound does not require a prior estimate of $\alpha\in(0,1]$5—the parameter choice is independent of the unknown source norm and only uses the noise level, which is a practical advantage.

Numerical Experiments

Two numerical examples, one with a discontinuous and one with a smooth source, validate the proposed regularization schemes:

• Discontinuous source: Piecewise constant $\alpha\in(0,1]$6, outside $\alpha\in(0,1]$7 for $\alpha\in(0,1]$8. Despite the lack of theoretical convergence guarantees, regularization improves approximation quality, with the relative error decreasing as noise is reduced.
• Continuous source: $\alpha\in(0,1]$9 on $u(x_0, t)=y(t)$0. All three regularizers yield low relative errors that diminish as $u(x_0, t)=y(t)$1 decreases, affirming the practical stability and efficiency of the methods even for $u(x_0, t)=y(t)$2.

For both cases, the first operator $u(x_0, t)=y(t)$3 (Tikhonov/Fourier-like) exhibits the best empirical accuracy. The regularization schemes robustly handle both smooth and non-smooth target functions. The results empirically confirm that, for decreasing data noise, the regularized solutions $u(x_0, t)=y(t)$4 stably converge to the true source.

Implications and Future Directions

The proposed regularization framework significantly advances the practical solvability of inverse source problems governed by time-fractional parabolic equations with convection and reaction terms. These models are critical in describing anomalous transport phenomena in physics, biology, and engineering, where classical integer-order models are inadequate.

The main implications include:

• Extension and unification of regularization techniques to time-fractional models with drift and reaction, which previously lacked dedicated algorithms.
• An explicit, rigorously justified strategy for a priori selection of regularization parameters based solely on data noise characteristics.
• Provision of analytic error bounds that are optimal with respect to smoothness assumptions on the unknown source.

Potential future research includes the extension of these regularization operators to multidimensional domains, nonlinear problems, the analysis of more general measurement functionals, and the integration with data-driven parameter selection schemes (e.g., discrepancy principle, L-curve). Analysis of optimality conditions for the regularizer form and systematic comparison with iterative and Bayesian techniques in the fractional scenario are also natural continuations.

Conclusion

This work rigorously formulates and solves the ill-posed inverse source problem associated with the time-fractional convection-diffusion-reaction equation using a frequency-domain approach and three distinct families of regularization operators. The operators are proved to yield stable, convergent solutions with explicit error bounds under minimal a priori assumptions. The parameter choice rule, depending solely on measured noise, makes the approach practical for real-world applications. Numerical results demonstrate robust performance for both smooth and discontinuous sources, confirming the theoretical development. This framework establishes a reference methodology for regularized inversion in fractional parabolic models.