Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

1. Problem Statement

The paper addresses the first initial-boundary value problem (IBVP) for a sub-diffusion equation defined on a rectangular domain $\Omega = \{(t, x) : 0 < t < T, 0 < x < a\}$. The governing equation involves the regularized Prabhakar fractional derivative (a Caputo-type derivative) and is given by:

$$PCD_{0t}^{\alpha, \beta, \gamma, \delta} u(t, x) - u_{xx}(t, x) = f(t, x)$$

Key Components:

• Operator: $PCD_{0t}^{\alpha, \beta, \gamma, \delta}$ is the Caputo-type Prabhakar fractional derivative, defined via the Prabhakar fractional integral operator involving the three-parameter Mittag-Leffler function $E_{\alpha, \beta}^\gamma$.
• Parameters: $\alpha > 0$, $0 < \beta \leq 1$, and $\gamma, \delta$ are real numbers.
• Conditions:
  • Boundary Conditions: $u(t, 0) = \phi_0(t)$ and $u(t, a) = \phi_1(t)$.
  • Initial Condition: $u(0, x) = \tau(x)$.
• Goal: To find a unique regular solution $u(t, x)$ and construct an explicit representation using a Green's function.

2. Methodology

The authors employ a rigorous analytical approach combining the superposition principle with operational calculus techniques specific to fractional derivatives.

• Decomposition (Superposition): The problem is split into two sub-problems:
  1. Problem A (Non-homogeneous equation, homogeneous initial condition): Solves for the effect of the source term $f(t, x)$ and non-homogeneous boundary conditions, assuming $u(0, x) = 0$.
  2. Problem B (Homogeneous equation, non-homogeneous initial condition): Solves for the effect of the initial data $\tau(x)$, assuming zero boundary conditions and zero source.
• Conversion to Riemann-Liouville Form: The authors utilize the relationship between Caputo and Riemann-Liouville Prabhakar derivatives to handle the initial conditions. Specifically, for a function $y(t, x)$ with $y(0, x)=0$, the Caputo derivative equals the Riemann-Liouville derivative. For the initial value problem, the non-zero initial condition introduces a specific source term involving the Mittag-Leffler function.
• Green's Function Construction:
  • The solution is expressed as an integral involving a Green's function $G(t, x, \eta, \xi)$.
  • The Green's function is derived using the method of images (infinite series of images) to satisfy the boundary conditions on the finite interval $[0, a]$.
  • The core of the Green's function is expressed in terms of a bivariate Mittag-Leffler-type function, denoted as $E_{12}$, which generalizes the standard Mittag-Leffler function to two variables.
• Verification: The authors provide a detailed proof that the constructed integral representation satisfies:
  • The governing differential equation (by differentiating under the integral sign and utilizing properties of the Wright-type function and Mittag-Leffler series).
  • The boundary conditions (by taking limits as $x \to 0$ and $x \to a$).
  • The initial condition (by taking the limit as $t \to 0$).

3. Key Contributions

• Explicit Green's Function: The primary contribution is the derivation of an explicit closed-form expression for the Green's function $G(t, x, \eta, \xi)$ and the associated kernel $\tilde{G}$ for the initial condition. These are expressed using the bivariate Mittag-Leffler function $E_{12}$, defined as a double series:
  $$E_{12}\left( \dots \middle| \frac{x}{y} \right) = \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{\Gamma(\alpha_1 n + \beta_1 m + \delta_1) x^n y^m}{\Gamma(\alpha_2 n + \beta_2 m + \delta_2) \dots}$$
• Regularized Derivative Framework: The work specifically addresses the regularized (Caputo-type) Prabhakar derivative, which is crucial for physical modeling as it allows for standard initial conditions (unlike the Riemann-Liouville type which requires fractional initial conditions).
• Rigorous Existence and Uniqueness: The paper proves the existence and uniqueness of the regular solution under specific continuity and Hölder conditions on the input functions ($\phi_0, \phi_1, \tau, f$).
• Generalization: This framework generalizes previous results on Riemann-Liouville and Caputo derivatives, as the Prabhakar operator encompasses them as special cases (by setting specific parameters).

4. Results

• Solution Representation: The final solution $u(t, x)$ is presented as a sum of four integral terms:
  1. Boundary contribution from $\phi_0(t)$.
  2. Boundary contribution from $\phi_1(t)$.
  3. Initial condition contribution from $\tau(x)$.
  4. Source term contribution from $f(t, x)$.
• Numerical Illustrations:
  • Example 1 (Initial Data Influence): Simulates sub-diffusion with zero external force and initial profile $\sin(x)$. Results show that lower fractional orders ($\beta$) lead to extremely slow decay (deep sub-diffusion), while higher orders approach classical diffusion behavior.
  • Example 2 (External Force Influence): Simulates a system with zero initial data and a time-dependent source $f(t, x) = t \sin(x)$. Results demonstrate that systems with higher fractional orders respond more rapidly to external forcing, accumulating amplitude faster than those with low fractional orders.
• Parameter Sensitivity: The study highlights how parameters $\alpha, \beta, \gamma, \delta$ modulate the memory effects and the rate of diffusion/relaxation.

5. Significance

• Modeling Complex Memory: The Prabhakar derivative offers a more flexible framework for modeling memory effects in physical systems (e.g., anomalous diffusion, viscoelasticity) than standard fractional derivatives. The inclusion of the three-parameter Mittag-Leffler kernel allows for a "nuanced" description of relaxation processes.
• Analytical Tool: The derivation of the explicit Green's function provides a powerful analytical tool for solving linear sub-diffusion equations in bounded domains, which is often difficult with purely numerical methods due to the non-local nature of fractional operators.
• Foundation for Future Research: The paper establishes a basis for investigating inverse problems, stability analysis, and multi-dimensional extensions of Prabhakar-type fractional differential equations. It bridges the gap between abstract fractional calculus and concrete boundary value problems in mathematical physics.

In summary, this paper successfully extends the theory of Green's functions to the regularized Prabhakar fractional derivative, providing explicit solutions and rigorous proofs that enhance the toolkit available for modeling complex systems with memory and hereditary properties.