Fractional 2D Heat Equation

• Fractional two-dimensional heat equation is a PDE that replaces classical time and spatial derivatives with fractional ones (e.g., Caputo and fractional Laplacian) to capture memory and nonlocal effects.
• It employs definitions such as the Caputo derivative and Riemann–Liouville formulations in time, combined with spatial fractional operators, to accurately model anomalous diffusion in heterogeneous media.
• Analytical and numerical techniques—including Laplace–Fourier transforms, spectral methods, and explicit kernel representations—provide avenues to understand solution properties and convergence behavior.

The fractional two-dimensional heat equation is a parabolic or pseudo-parabolic partial differential equation in which the classical time and/or spatial derivatives are replaced by non-integer (fractional) order operators. This generalization arises from the need to model memory, nonlocality, anomalous diffusion, and heterogeneity in physical, engineering, and stochastic systems. In this context, “fractional” refers both to derivatives of non-integer order (typically implemented via Caputo, Riemann–Liouville, or distributional definitions in time, and via the fractional Laplacian or related nonlocal operators in space) and to the possible use of distributed-order derivatives. In two dimensions, the construction and analysis of fractional heat equations involves unique challenges in boundary behavior, regularity, probabilistic representation, stochastic forcing, and the preservation of qualitative properties such as convexity, conservation, and Harnack inequalities.

1. Mathematical Formulations and Definitions

Fractional two-dimensional heat equations fall into several main classes depending on the type of fractional operator and the presence of stochastic or nonlinear effects. In the classical deterministic setting, a prototypical fractional time-space heat equation is

$$\begin{cases} \partial_t^\alpha u(x, y, t) = k [(-\Delta)^{\beta/2}u(x, y, t)] + f(x, y, t), & (x, y) \in \Omega \subset \mathbb{R}^2, \ t > 0, \ u(x, y, 0) = u_0(x, y), \end{cases}$$

where $\partial_t^\alpha$ denotes the Caputo or Riemann–Liouville time derivative of order $\alpha \in (0,2)$, $(-\Delta)^{\beta/2}$ is the fractional Laplacian of order $\beta \in (0,2]$, and $f$ is a source term. The two main fractional operators implemented are:

• Caputo time derivative:

$$\partial_t^\alpha f(t) = \frac{1}{\Gamma(n-\alpha)} \int_0^t (t-u)^{n-\alpha-1} f^{(n)}(u) du, \quad n-1 < \alpha < n.$$

• Fractional Laplacian:

$$(-\Delta)^{\beta/2} u(x) = C_{n,\beta} \, \text{PV} \int_{\mathbb{R}^n} \frac{u(x) - u(y)}{|x-y|^{n+\beta}} dy,$$

where $C_{n,\beta}$ is a normalization constant and PV denotes the principal value.

Boundary and initial conditions can be posed in several ways, including Dirichlet, Neumann, Robin, or more generally, dynamic free boundary conditions (as in the fractional Stefan problem (Roscani et al., 2013, Roscani et al., 2018)).

For stochastic and distributed-order models, the equations are further generalized by including:

• Additive or multiplicative noises (with white, fractional, or colored spatial/temporal structure) (Hachemi et al., 24 Feb 2024, Hachemi et al., 2022, Song et al., 2023).
• Nonlinear terms, e.g., quadratic nonlinearities or state-dependent noise (Schaeffer, 2021).
• Distributed-order time-fractional derivatives:

$$\int_0^1 \mu(\gamma) \partial_t^\gamma u(x, y, t) d\gamma,$$

with a suitable weighting measure $\mu$ (Želi et al., 2017, Kochubei et al., 2019).

Across these formulations, solutions can be considered in the classical, weak, mild (in $L^2(\mathbb{P})$), or distributional sense, depending on the context.

2. Regularity, Boundary Behavior, and Existence

A central theme in the analysis of the fractional two-dimensional heat equation is to establish existence, uniqueness, regularity, and precise boundary behavior of solutions. For basic initial value problems in the whole space, solutions are represented by convolution with fundamental solutions (“fractional heat kernels”) exhibiting self-similar scaling: $$u(x, y, t) = \int_{\mathbb{R}^2} P(x-y, t; s) u_0(y) dy, \quad P(\cdot, t; s) \sim t^{-1/s} F_s\left( \frac{|x|}{t^{1/(2s)}} \right),$$ with detailed heat kernel estimates and asymptotics available (Vázquez, 2017, Kochubei et al., 2019).

For bounded, smooth domains with Dirichlet conditions, sharp boundary regularity is proven: for the equation $\partial_t u + (-\Delta)^s u = 0$ in a $C^{1,1}$ domain $\Omega$, solutions satisfy $u(\cdot, t) \in C^s(\mathbb{R}^2)$, and the normalized function $u(\cdot, t)/\delta^s$ is $C^{s-\epsilon}$ up to $\overline \Omega$ for any $\epsilon > 0$, where $$\delta(x) = \operatorname{dist}(x, \partial \Omega)$$ (Fernández-Real et al., 2014, Grubb, 2017). It is shown that, even in the presence of nonlocal operators of Lévy type, the same rate holds for the behavior at the boundary: $-c \delta^s(x) \leq u(x, t) \leq c \delta^s(x)$ and the optimal spatial regularity at the boundary is limited by the transmission property.

Existence and uniqueness are typically established via spectral methods (eigenfunction expansions), semigroup theory, variational approaches (for $L_p$ or Sobolev/Besov spaces), or stochastic representation formulas (e.g., Feynman–Kac) (Greco et al., 2016, Grothaus et al., 2015, Grubb, 2017).

Preservation of convexity by the fractional heat flow is also established: if the initial datum is convex and satisfies natural polynomial growth bounds, then the solution remains convex (or “ruled”) for all positive times (Greco et al., 2016).

3. Analytical and Numerical Techniques; Explicit Solution Representations

Analytical solution techniques for the fractional two-dimensional heat equation utilize:

• Laplace and Fourier transforms in time and space to obtain closed forms for the transformed solution; inversion is achieved via explicit representations involving the Mittag–Leffler function ($E_{\alpha, \beta}$) and/or Wright functions (Singh et al., 2017, Grothaus et al., 2015, Želi et al., 2017).
• Mittag–Leffler analysis for stochastic and non-Gaussian formulations, enabling explicit Green’s function constructions and extension of classical Feynman–Kac formulas using generalized grey Brownian motion (Grothaus et al., 2015).
• Representation of self-similar solutions, and explicit similarity solutions for problems with moving (free) boundaries; such solutions involve a combination of scaling laws $s(t) = \sigma t^{\alpha/2}$ and special functions (Wright, Mainardi, error function in the $\alpha \to 1$ limit), especially for the (fractional) Stefan problem (Roscani et al., 2013, Roscani et al., 2018).

Spectral and operational matrix approaches (e.g., via Bernstein polynomials) provide high-order, matrix-based numerical schemes for time- and space-fractional heat equations, ensuring efficient computation under physically realistic boundary and initial conditions (Bakhshandeh-Chamazkoti et al., 2022). For distributed-order models with memory effects, both analytical (transform inversion) and numerical (finite difference: Adams–Bashforth, Grunwald–Letnikov, leap frog in space) solutions facilitate detailed paper of wave-like or anomalous transient behavior (Želi et al., 2017, Kochubei et al., 2019).

4. Probabilistic Representations and Stochastic Formulations

Fractional heat equations admit rich probabilistic interpretations:

• Subordination: Solutions to time-fractional equations are constructed via subordination of the classical heat kernel by random time changes (e.g., stable subordinators generating Caputo derivatives, distributed order subordinators for more general kernel types) (Kochubei et al., 2019, Vázquez, 2017).
• Feynman–Kac formulas, both in the classical and fractional frameworks, enable pathwise and expectation representations involving Brownian motion or generalized grey Brownian motion (ggBm) (Grothaus et al., 2015). In the fractional case, fundamental solutions involve the Mittag–Leffler function in the characteristic function/skew kernel of the subordinated process.
• In stochastic settings, both additive and multiplicative (Wick/Skorohod) noise inputs are considered, the latter requiring Hida–Malliavin calculus and Wiener chaos expansions to achieve well-posedness (Song et al., 2023, Hachemi et al., 2022, Hachemi et al., 24 Feb 2024). Explicit representation of the solution is obtained in the distributional sense, often as a stochastic Volterra integral. The well-posedness of solutions (e.g., the existence of a mild solution—i.e., being $L^2(\mathbb{P})$-valued) depends sharply on both the fractional order and the spatial dimension:
  • For the classical heat equation ($\alpha = 1$), mild solutions exist only for $d=1$.
  • For $\alpha \in (1,2)$ (“superdiffusive”), mild solutions exist in $d=1,2$ (Hachemi et al., 2022, Hachemi et al., 24 Feb 2024).
  • For $\alpha < 1$ (“subdiffusive”), mild solutions fail to exist in any dimension.

In the stochastic case, the covariance/spectral structure of the driving noise (e.g., based on the heat kernel, on fractional Brownian motion with Hurst parameter $H$) and the use of renormalization (Wick products, time-dependent subtraction of diverging terms due to roughness of the noise) are crucial technical aspects (Schaeffer, 2021, Hachemi et al., 24 Feb 2024).

5. Qualitative Behavior: Harnack Inequality, Conservation Laws, and Asymptotics

Significant qualitative properties are established for the fractional two-dimensional heat equation:

• Harnack inequalities and quantitative bounds: Double-sided, sharp Harnack estimates for positive solutions of a nonlocal heat equation are derived by exploiting explicit kernel representations and optimal comparison arguments, generalizing the Li–Yau inequality to the nonlocal setting. The structure of these bounds is inherently tied to the heavy tails and nonlocality induced by the fractional operators; in two dimensions, these results carry over provided sharp kernel asymptotics are available (Dembny et al., 2023).
• Asymptotic behavior: Weak solutions with integrable (or finite measure) initial data converge to self-similar fundamental profiles as $t \rightarrow \infty$, with explicit decay rates determined by the order of the fractional operators (e.g., $t^{-n/(2s)}$ decay, with $s$ the order of the spatial fractional Laplacian); relative error convergence and precise rates under higher moment assumptions are proved (Vázquez, 2017).
• Conservation laws and symmetry: The Lie algebra of symmetries in the fractional heat equation is severely restricted relative to the classical case; for example, two-dimensional time-fractional equations have only six symmetry generators (lacking time translation invariance), impacting the available similarity reductions and conservation laws. Conservation laws, including those of energy and momentum types, are constructed, but their structure is altered by the nonlocality of the operators (Halder et al., 2020, Bakhshandeh-Chamazkoti et al., 2022).

6. Fractional Free-Boundary and Two-Phase Problems

A key application of the fractional two-dimensional heat equation is in the fractional Stefan (free-boundary) problem. Here, both the phase-change interface and the temperature (or concentration) evolve according to time-fractional diffusion:

• The Caputo derivative of order $\alpha \in (0,1)$ modifies both the heat equation and the classical Stefan condition, leading to an interface $s(t)$ progressing as $t^{\alpha/2}$, with temperature and moving boundary expressed in terms of Wright and Mainardi functions. Detailed relationships between temperature and flux boundary conditions are shown to produce equivalent moving boundary evolutions, and explicit inequalities for minimum heat flux (instantaneous melting) are obtained (Roscani et al., 2013, Roscani et al., 2018).
• In the limit $\alpha \to 1$ (classical case), solutions converge to the known Neumann (error function) solutions of the two-phase Stefan problem, demonstrating the continuous transition between memory-dominated and memoryless heat/phase-change phenomena.

7. Applications, Implications, and Open Problems

Fractional two-dimensional heat equations are indispensable in the modeling of systems where anomalous diffusion (subdiffusion, superdiffusion), memory effects, heterogeneous media, and stochastic fluctuations are dominant. Applications span:

• Phase-change phenomena with memory and latent heat effects (e.g., solidification/melting, porous media) (Roscani et al., 2013, Roscani et al., 2018).
• Transport in materials and composites with fractal or multi-scale structure.
• Price evolution in random or fractal environments in finance.
• Modeling of transport and reactions in biological systems and tissues.
• Stochastic modeling of field fluctuations, random walks in random environments, and directed polymers (Song et al., 2023, Hachemi et al., 24 Feb 2024).

Despite extensive recent progress, open problems remain regarding:

• Sharp regularity up to the boundary for rough domains or with more singular initial data.
• Strong maximum principles, blow-up phenomena, and fine asymptotics for nonlinear or degenerate fractional heat equations.
• Numerical schemes capable of treating multi-dimensional, non-smooth domains with complex fractional operator structure with guaranteed accuracy and stability.

---

Summary Table: Core Mathematical Structures

Feature	Deterministic	Stochastic/Random Environment
Time Derivative	Caputo: $\partial_t^\alpha$	Caputo in distributional/mild
Space Derivative	$(-\Delta)^{\beta/2}$	$(-\Delta)^{\beta/2}$
Initial/Boundary Data	e.g., $u_0(x,y)$, Dirichlet, etc.	Same, often with distributional
Solution Class	$C^{s}$, $L_p$, mild, etc.	$\mathcal{S}'$, $L^2(\mathbb{P})$
Representation Formula	Convolution with $P(x,t;s)$	Stochastic Feynman–Kac, chaos
Qualitative Properties	Convexity, asymptotics, conservation	Mildness, well-posedness case-by-case

This synthesis incorporates and interrelates the principal analytical, probabilistic, and computational directions in the theory of fractional two-dimensional heat equations, as established in the cited literature.