Variable-Exponent Subdiffusion Model

• Variable-exponent subdiffusion is a generalization of classical subdiffusion where the fractional order varies with time and space to capture heterogeneous transport phenomena.
• The model integrates fractional calculus with stochastic processes and numerical methods to analyze systems with nonuniform memory effects.
• It facilitates inversion and recovery of spatially dependent exponents using analytical reformulations, perturbation techniques, and discretization schemes applicable in physics, finance, and biophysics.

The variable-exponent subdiffusion model is a generalization of classical (single-exponent) subdiffusion which incorporates temporal and/or spatial variability in the order of the governing fractional operator. It describes anomalous transport where the degree of subdiffusivity, quantified by the fractional exponent $\alpha$, can depend on time, space, or both. This paradigm arises naturally from physical systems exhibiting inhomogeneous trapping, multiscale heterogeneity, or continuously varying nonlocal memory. It sits at the intersection of fractional calculus, stochastic processes (especially continuous-time random walks with heterogeneous waiting laws), and numerical analysis for noncoercive integro-differential equations.

1. Mathematical Formulations

Several forms of the variable-exponent subdiffusion model have been investigated.

(a) Time-Variable Exponent (Caputo Type):

$$\partial_t^{\alpha(t)} u(x,t) - \Delta u(x,t) = f(x,t), \quad 0<\alpha(t)<1,$$

with the Caputo derivative

$$\partial_t^{\alpha(t)} v(x,t) = \int_0^t \frac{(t-s)^{-\alpha(t)}}{\Gamma(1-\alpha(t))} \partial_s v(x,s) \, ds.$$

(b) Spatially Variable Exponent:

$$\partial_t^{\alpha(x)} u(x,t) - \Delta u(x,t) = f(x,t), \quad 0<\alpha(x)<1,$$

with the Caputo derivative in $t$ of order $\alpha(x)$.

(c) General Space-Time Variable Exponent:

$$\partial_t^{\alpha(x,t)} u(x,t) - \Delta u(x,t) = f(x,t)$$

with $\alpha: \Omega\times[0,T]\rightarrow(0,1)$.

(d) Variable-Exponent Subdiffusive Fokker-Planck/Black-Scholes:

For financial modeling,

$${}^R_0 D_t^{\alpha(t)} v(S,t) + \frac{1}{2}\sigma^2 S^2 \partial_{SS}v + rS \partial_S v - r v = 0,$$

with a Riemann–Liouville or Caputo-type memory kernel of variable order (Zhang et al., 2024).

(e) Distributed-Order and Generalized Time-Change Models:

Distributed-order and $g$-Caputo derivatives describe crossover between different subdiffusive regimes: $${}^{C}D_g^{\alpha} f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (g(t)-g(u))^{-\alpha} f'(u)du,$$ where $g(t)$ controls the dynamics, enabling continuous crossover between exponents (Kosztołowicz et al., 2022, Eab et al., 2010).

These formulations arise via continuous limits of lattice CTRW schemes with variable or distributed waiting-time exponents (Roth et al., 2020).

2. Analytical Reformulations and Well-posedness

Analysis of variable-exponent subdiffusion presents significant challenges due to the loss of semigroup properties and non-positivity of associated kernels. Two dominant methodologies address these issues:

(a) Convolution (Kernel-Balancing) Method:

Convolving the fractional equation against a judiciously chosen power-law kernel transforms the variable-order equation into a constant-order equation with an additional convolution-memory perturbation (Zheng, 2024): ${}^c_t^{\alpha(t)} u = g' * u - \Delta u + f,$ where $g'$ is derived by convolution of Abel kernels and encodes the residual variable exponent effects.

(b) Perturbation Method:

Expanding the kernel: $$k(t) = \frac{t^{-\alpha(t)}}{\Gamma(1-\alpha(t))} = \frac{t^{-\alpha_0}}{\Gamma(1-\alpha_0)} + \widetilde{g}(t),$$ where $\alpha_0 = \alpha(0)$, recasts the problem as a constant-order Caputo equation plus a perturbative convolution-memory term (Li et al., 30 Jan 2025, Qiu et al., 11 Jan 2026).

Well-posedness Results:

For $f$ in $W^{1,1}(0,T;L^2(\Omega))$ and $u_0\in \check H^2(\Omega)$, the transformed models admit unique solutions in $$W^{1,p}(0,T;L^2(\Omega)) \cap L^p(0,T;\check H^2(\Omega))$$, $1<p<1/(1-\alpha_0)$, with explicit a priori bounds (Zheng, 2024, Li et al., 30 Jan 2025).

3. Model Variants: Space-Dependence and Inverse Theory

Variable-exponent subdiffusion with spatially dependent order emerges as the continuum limit of heterogeneous (space-inhomogeneous) lattice CTRWs (Roth et al., 2020). The resulting PDE has the form: $$\partial_t^{\alpha(x)} u(x,t) - \Delta u(x,t) = f(x,t)$$ and is tightly coupled to local physical microstructure.

Inverse Problems:

Reconstruction of unknown space-dependent exponent functions from boundary or internal data is well-posed under monotonicity and piecewise constancy assumptions:

• Asymptotic Laplace expansions of boundary flux observables allow recovery of the exponents and partition interfaces from a single time series measurement, both in 1D (Hong et al., 2024) and in multidimensional domains (Hong et al., 9 Jul 2025).
• For general $L^r$-measurable $\alpha(x)$ with $\mathrm{ess\,inf}\,\alpha(x) > 0$, unique recovery and Lipschitz-type stability can be established from boundary data, leveraging resolvent estimates and Neumann series in the Laplace domain.

4. Numerical Methods and Error Analysis

The numerical analysis of variable-exponent subdiffusion is nontrivial due to the non-monotonic, non-positivity, and lack of semigroup structure of the memory kernels.

(a) Nonuniform L1-type Temporal Discretization:

The nonuniform L1 method on graded meshes achieves optimal order $O(N^{-\min\{2-\alpha_0, r\alpha_0\}})$, with $r$ the grading parameter, resolving initial singularities (Qiu et al., 11 Jan 2026).

Key tools:

• Discrete weights $a^{(n)}_{n-k}$ satisfy stability properties and explicit error estimates.
• Complementary discrete convolution kernel (CDCK) techniques provide stability and convergence under non-positive, weakly singular kernels (Qiu et al., 8 May 2025).

(b) Second-Order L2-1$_\sigma$ and Superconvergence:

Second-order accuracy with respect to the Caputo derivative is attained by optimizing the location “superconvergence shift” at each step. The superconvergence criterion can be relaxed to admit analytic or minimal cost determinations without loss of accuracy, achieving near-optimal $O(N^{-2})$ temporal rates under mesh grading (Huang et al., 2024).

(c) Fully Discrete Finite Element Schemes:

Piecewise linear spatial FEM yields $O(h^2)$ spatial error for sufficiently regular solutions (Zheng, 2024, Qiu et al., 8 May 2025, Zhang et al., 2024).

(d) Black–Scholes and Non-Coercive Operators:

The subdiffusive Black–Scholes model with variable exponent can be regularized via a sequence of log-asset and exponential gauge transformations, yielding a coercive, convolution-type PDE. Error estimates demonstrate temporal $O(\tau^{1+\alpha_0})$ and spatial $O(h^2)$ convergence, even in the presence of non-monotonic, sign-changing kernels (Zhang et al., 2024).

5. Physical and Stochastic Origins

Variable-exponent subdiffusion emerges in physical systems where trapping statistics or memory effects are heterogeneous or evolve dynamically:

• Disordered Hubbard Chains: Singular power-law distributions of exchange couplings yield variable exponents determined by the localization length and filling, with $\alpha = \frac{\mu}{1+\mu}$, $\mu\sim n\xi$ (Kozarzewski et al., 2018).
• CTRW Models: Anomalous exponent varies spatially when the local waiting-time PDF exhibits power-law behavior with location-dependent exponent (Roth et al., 2020).
• Distributed Order, $g$-Subdiffusion: These frameworks interpolate exponents in time, modeling retarding or crossover subdiffusion (e.g., from $\alpha$ to $\beta$), with MSDs crossing over between power-laws (Kosztołowicz et al., 2022, Eab et al., 2010).

6. Inverse Problems and Conditional Stability

Inverse source recovery and exponent identification are tractable for variable-exponent subdiffusion under analytic extension and unique continuation properties:

• Analytical regularity in time, established via kernel splitting and Laplace techniques, underpins identification theory (Li et al., 30 Jan 2025, Hong et al., 9 Jul 2025).
• Variational identities, weak-norm reconstructions, and Tikhonov/TV regularization strategies yield stable and accurate source identification even in the presence of noise (Li et al., 30 Jan 2025).
• Lipschitz stability of recovery in global $L^1$ or local “weak” norms is achievable given sufficient boundary or internal data (Hong et al., 9 Jul 2025).

7. Applications and Implications

Variable-exponent subdiffusion models are relevant in:

• Physics: Spin and charge transport in strongly disordered or glassy systems, anomalous diffusion in complex media, porous materials, and crowds.
• Finance: Option pricing with changing memory effects or stochastic volatility (subdiffusive Black–Scholes).
• Biophysics: Drug release and mass transport in living tissues, biofilms, or composite environments where effective transport exponents evolve due to environmental response or structural transformation (Kosztołowicz et al., 2022).

The framework enables characterization of systems where the anomalous transport exponent is not universal but is governed by local microstructure or nonstationary processes, bridging microscopic stochastic kinetics and macroscopic transport PDEs.

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References by arXiv id:

• Analytical reformulations, well-posedness: (Zheng, 2024, Li et al., 30 Jan 2025, Qiu et al., 8 May 2025, Qiu et al., 11 Jan 2026)
• Numerical methods and error analysis: (Huang et al., 2024, Li et al., 23 May 2025, Zhang et al., 2024, Qiu et al., 8 May 2025, Qiu et al., 11 Jan 2026)
• Space-dependent exponents and inversion: (Hong et al., 2024, Hong et al., 9 Jul 2025, Roth et al., 2020)
• Physical and stochastic aspects: (Kozarzewski et al., 2018, Eab et al., 2010, Kosztołowicz et al., 2022, Roth et al., 2020)
• Applications and model variants: (Zhang et al., 2024, Li et al., 30 Jan 2025, Li et al., 23 May 2025, Kosztołowicz et al., 2022)

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The variable-exponent subdiffusion model provides a unifying mathematical and computational framework for complex anomalous transport with non-uniform or evolving memory, striking a balance between physical realism, analytical tractability, and numerical implementability.