Time-Space Fractional Cable Equation

• The time-space fractional cable equation is a mathematical model that generalizes classical cable equations by incorporating Riemann–Liouville time-fractional derivatives and fractional Laplacians to capture memory effects and spatial nonlocality.
• It integrates stochastic forcing via rough noise, effectively modeling anomalous diffusion phenomena in biological and neurological systems where traditional diffusion falls short.
• Analytical and numerical approaches, including operator theory, Wong–Zakai noise regularization, spectral Galerkin projection, and backward Euler convolution quadrature, provide provable error bounds and convergence rates dependent on fractional and stochastic parameters.

The time-space fractional cable equation generalizes classical cable equations by integrating nonlocality in both time and space, derived from a fractional generalization of Ohm’s law. This model is designed to describe anomalous diffusion phenomena, particularly in biological or neurological systems where classical diffusion fails to account for memory and spatial heterogeneity. The equation incorporates both Riemann–Liouville time-fractional derivatives and fractional Laplacians, allowing it to capture subdiffusive dynamics and spatial long-range interactions. Recent research introduces stochastic forcing via rough noise of fractional Brownian sheet type, necessitating the development of robust analytical and numerical techniques for existence, regularity, noise regularization, and discretization (He et al., 5 Jan 2026).

1. Governing Equation and Physical Rationale

The stochastic time-space fractional cable equation on a bounded one-dimensional domain $D=(0,\ell)$ with homogeneous Dirichlet boundary and initial datum $g$ is given by: $$\partial_t u(x,t) + \lambda\,\partial_t^{1-\beta} u(x,t) + \mu\,\partial_t^{1-\alpha} A^s u(x,t) = f(u(x,t)) + \gamma(t)\,\xi^{H_1,H_2}(x,t),$$ for $x \in D$, $0 < t \leq T$, along with

$u(0,t) = u(\ell,t) = 0, \quad 0 < t \leq T,$

$u(x,0) = g(x), \quad x \in D,$

where $A = -\Delta$ is the Dirichlet Laplacian, and $A^s$ its fractional spectral power. Time-fractional derivatives (Riemann–Liouville type) are defined for $0<\eta<1$ as

$$\partial_t^\eta y(t) = \frac{1}{\Gamma(1-\eta)} \frac{d}{dt} \int_0^t (t-\tau)^{-\eta} y(\tau)\,d\tau.$$

The stochastic term $$\xi^{H_1,H_2}(x,t) = \partial_x\partial_t W^{H_1,H_2}(x,t)$$ models rough noise with spatial and temporal Hurst indices $H_1$ and $H_2$ ($\leq 1/2$).

This framework is derived by replacing Ohm’s law with fractional constitutive relations. For instance, the macroscopic potential $V$ satisfies

$$r_m c_m\,\partial_t V = \frac{r_m d}{4 r_L}\,\partial_t^{1-\alpha}(-\Delta)^s V - D(\beta)\,\partial_t^{1-\beta}(V - V_{rest} - r_m i_e),$$

after normalization, variable changes, and the introduction of nonlinear and noisy terms.

2. Operator-Theoretic Analysis and Solution Properties

Existence, uniqueness, and regularity proofs are established through analytic resolvent families: $$S(t) = \frac{1}{2\pi i} \int_{\Gamma_{\kappa, \theta}} e^{z t} H(z)\,dz,$$ where $H(z) = \mu^{-1} z^{\alpha-1} (h(z) I + A^s)^{-1}$, $h(z) = \mu^{-1} z^\alpha (1 + \lambda z^{-\beta})$, with $\Gamma_{\kappa, \theta}$ a suitable contour.

Deterministic Smoothing

For initial data $g \in \dot H^p(D)$, $p \in [0,2]$, the homogeneous solution is $u(t) = S(t)g$, and for $m=0,1$ and $q\in [p,2]$,

$$\|A^{s(q-p)/2} S^{(m)}(t) g\|_{L^2(D)} \leq C t^{-m - \alpha(q-p)/2} \|g\|_{\dot H^p}.$$

Stochastic Regularity

If $g \in \dot H^\sigma(D)$ and $$2\sigma < \min\{2s + H_1 - 1, (2s H_2)/\alpha + H_1 - 1\}$$,

$E[\|u(t)\|^2_{\dot H^\sigma(D)}] \leq C.$

Temporal Hölder Continuity

For $0 < \xi < \min\{H_2, (H_2 + \alpha(H_1-1)/(2s))\}$,

$$E\left[\|u(t) - u(t-h)\|^2\right]^{1/2} \leq C (t-h)^{-\xi} h^\xi + C.$$

3. Noise Regularization via Wong–Zakai Approximation

Rough noise is regularized using a piecewise-constant Wong–Zakai approach. On a spatial and temporal mesh,

$$\xi_W(x,t) = \sum_{i,j} \left( \frac{1}{\tau h} \int_{t_i}^{t_{i+1}} \int_{x_j}^{x_{j+1}} \xi(dy d\tau) \right) \chi_{[t_i, t_{i+1})}(t) \chi_{[x_j, x_{j+1})}(x).$$

For solutions $u_W$ to this regularized problem, one obtains

$E[\|u_W(t)\|^2_{\dot H^\sigma(D)}] \leq C,$

and the error estimate

$$E[\|u(t) - u_W(t)\|^2] \leq C \left( h^{2\sigma + 2H_1 - 1} + h^{2H_1 - 1} \tau^{2H_2 - (\alpha/(2s)) - \varepsilon} \right).$$

Isometry estimates are derived for integrals involving fractional Brownian sheet noise and test functions, with bounds determined by the Hurst parameters.

4. Numerical Discretization Schemes

Spatial Discretization

Spectral Galerkin projection is applied to $L^2(D) \to H_N = \text{span}\{e_1,\ldots,e_N\}$ via

$P_N v = \sum_{k=1}^N (v, e_k) e_k,$

and fractional Laplacian as

$A_N^s v_N = \sum_{k=1}^N \rho_k^s (v_N, e_k) e_k.$

The semi-discrete system evolves in $H_N$: $$\partial_t u_N + \lambda\,\partial_t^{1-\beta} u_N + \mu\,\partial_t^{1-\alpha} A_N^s u_N = P_N f(u_N) + \gamma P_N \xi_W,$$ with $u_N(0)=0$. Solution representations and uniform boundedness mirror the continuous case.

Temporal Discretization

Backward Euler convolution quadrature is used for time-fractional derivatives. For the time mesh $\{t_n\}$,

$$\partial_t^{1-\alpha} v(t_n) \approx \sum_{i=0}^{n-1} d_i^{(1-\alpha)} v(t_{n-i}), \text{ with } \sum_{i\geq 0} d_i^{(1-\alpha)} \zeta^i = (\delta_\tau(\zeta))^{1-\alpha},\; \delta_\tau(\zeta) = \frac{1-\zeta}{\tau}.$$

The fully discrete scheme for $u_N^n$ is

$$\frac{u_N^n - u_N^{n-1}}{\tau} + \lambda \sum_{i=0}^{n-1} d_i^{(1-\beta)} u_N^{n-i} + \mu \sum_{i=0}^{n-1} d_i^{(1-\alpha)} A_N^s u_N^{n-i} = P_N f(u_N^{n-1}) + P_N \xi_{W,n}^{H_1,H_2},$$

with $u_N^0=0$.

5. Error Bounds and Convergence Rates

Spatial error (prior to time discretization) is quantified as

$$E[\|u(t) - u_N(t)\|^2] \leq C [(N+1)^{-4s\vartheta} + (N+1)^{-4\alpha\vartheta + 1} h^{2H_1 - 1}],$$

for $\vartheta \in (1/(4s), H_2/\alpha)$.

Temporal error (after space discretization) is

$$E[\|u_N(t_n) - u_N^n\|^2] \leq C \tau^{2H_2 + (H_1-1)\alpha/s - \varepsilon},$$

for any small $\varepsilon>0$.

A key finding is that all convergence rates depend crucially on the fractional orders $\alpha, s$ and the Hurst indices $H_1, H_2$, with temporal rate $2H_2 + (H_1-1)\alpha/s$ degraded by increased spatial roughness ($H_1<1/2$) and strong memory ($\alpha<1$).

6. Analytical and Numerical Significance

The time-space fractional cable equation provides a rigorous modeling tool for anomalous diffusion, capturing both spatial nonlocality and temporal memory. The stochastic extension with rough noise introduces significant analytical and numerical challenges, addressed via operator theory, resolvent analysis, Wong–Zakai noise regularization, and high-order discretization schemes. These methodologies guarantee well-posedness, regularity, and controlled approximations with provable error bounds, all quantifiably sensitive to the model’s fractional and stochastic parameters. A plausible implication is the suitability of these approaches for complex neurobiological modeling and related physical systems exhibiting non-Gaussian transport (He et al., 5 Jan 2026).