On the Cauchy problem for the Langevin-type fractional equation

1. Problem Statement

The paper investigates the Cauchy problem for a specific class of time-fractional differential equations of Langevin type in a separable Hilbert space $H$. The equation is given by:

$$D^\beta_t (D^\alpha_t u(t)) + D^\beta_t (Au(t)) = f(t), \quad 0 < t \le T$$

Key Components:

• Operators: $D^\alpha_t$ and $D^\beta_t$ denote the Caputo fractional derivatives of orders $\alpha$ and $\beta$, respectively, where $0 < \alpha, \beta < 1$.
• Linear Operator: $A$ is an unbounded, positive, self-adjoint operator defined on a domain $D(A) \subset H$. It possesses a complete orthonormal system of eigenfunctions $\{v_k\}$ with corresponding eigenvalues $\lambda_k$ such that $0 < \lambda_1 \le \lambda_2 \le \dots \to +\infty$.
• Initial Conditions:
  • $u(+0) = \phi$
  • $D^\alpha_t u(+0) = \psi$
  • where $\phi, \psi \in H$.
• Source Term: $f(t)$ is a continuous function taking values in $H$.

The goal is to establish the existence and uniqueness of the solution and to provide an explicit representation of the solution using eigenfunction expansions.

2. Methodology

The authors employ a combination of spectral theory and special functions (specifically Mittag-Leffler functions) to solve the problem.

A. Spectral Decomposition

Since $A$ is a self-adjoint operator with a discrete spectrum, the solution $u(t)$ is expanded in terms of the eigenfunctions $\{v_k\}$:
$$u(t) = \sum_{k=1}^{\infty} T_k(t) v_k$$
Substituting this into the original equation reduces the partial differential equation (in the Hilbert space sense) to a sequence of scalar ordinary fractional differential equations for the Fourier coefficients $T_k(t)$:
$$D^\beta_t (D^\alpha_t T_k(t)) + \lambda_k D^\beta_t T_k(t) = f_k(t)$$
where $f_k(t)$, $\phi_k$, and $\psi_k$ are the Fourier coefficients of $f(t)$, $\phi$, and $\psi$, respectively.

B. Analysis of the Scalar Problem

The authors solve the scalar Cauchy problem using the method of successive approximations (Picard iteration). They derive the explicit solution for the scalar case involving the two-parameter Mittag-Leffler function $E_{\alpha, \mu}(z)$:
$$E_{\alpha, \mu}(z) = \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + \mu)}$$
The scalar solution is found to be:
$$T_k(t) = \phi_k E_{\alpha,1}(-\lambda_k t^\alpha) + (\psi_k + \lambda_k \phi_k)t^\alpha E_{\alpha, \alpha+1}(-\lambda_k t^\alpha) + \int_0^t (t-\eta)^{\alpha+\beta-1} E_{\alpha, \alpha+\beta}(-\lambda_k(t-\eta)^\alpha) f_k(\eta) d\eta$$

C. Convergence and Regularity Analysis

To prove that the infinite series constructed from the scalar solutions converges to a valid solution in the Hilbert space, the authors utilize:

1. Asymptotic Estimates: They apply known bounds for the Mittag-Leffler function, specifically $|E_{\alpha, \mu}(-t)| \le \frac{C}{1+t}$.
2. Fractional Calculus Lemmas: They use properties of Caputo derivatives acting on Mittag-Leffler functions (e.g., $D^\gamma_t (t^{\beta-1}E_{\alpha, \beta}(\lambda t^\alpha))$).
3. Norm Estimates: They define weighted norms based on the operator $A^\epsilon$ (where $0 < \epsilon < 1$) to control the growth of the eigenvalues $\lambda_k$.
4. Inequalities: The generalized Minkowski inequality and Parseval's identity are used to bound the series terms and prove uniform convergence.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

The paper establishes the following result:
If the initial data satisfies $\phi \in D(A)$ and $\psi \in H$, and the source term satisfies $f(t) \in C([0, T]; D(A^\epsilon))$ for some fixed $0 < \epsilon < 1$, then the Cauchy problem (1.1) has a unique solution.

The solution is explicitly given by the series:
$$u(t) = \sum_{k=1}^{\infty} \left[ \phi_k E_{\alpha,1}(-\lambda_k t^\alpha) + (\psi_k + \lambda_k \phi_k)t^\alpha E_{\alpha, \alpha+1}(-\lambda_k t^\alpha) + \int_0^t (t-\eta)^{\alpha+\beta-1} E_{\alpha, \alpha+\beta}(-\lambda_k(t-\eta)^\alpha) f_k(\eta) d\eta \right] v_k$$

Regularity of the Solution

The authors rigorously prove that the constructed series satisfies the definition of a solution (Definition 1.1), specifically verifying:

• $u(t) \in AC([0, T]; H)$ (Absolutely continuous).
• $D^\alpha_t u(t) \in C([0, T]; H)$.
• $D^\beta_t (Au(t)) \in C((0, T]; H)$.
• $D^\beta_t (D^\alpha_t u(t)) \in C((0, T]; H)$.

The proof relies on showing that the series for $Au(t)$ and its fractional derivatives converge uniformly in the Hilbert space norm under the specified conditions on $\phi$ and $f$.

4. Significance and Context

• Generalization of Classical Models: The equation generalizes the classical Langevin equation (used for Brownian motion) and the fractional telegraph equation.
  • If $\beta = 0$, it reduces to a fractional sub-diffusion equation.
  • If $\alpha = \beta$, it relates to fractional telegraph equations.
• Handling Two Fractional Orders: Unlike many previous studies that focus on a single fractional order or specific kernels, this work addresses a "Langevin-type" equation with two distinct fractional orders ($\alpha$ and $\beta$) acting on both the time derivative and the operator term.
• Operator-Theoretic Framework: By working in a general separable Hilbert space with an unbounded self-adjoint operator $A$, the results are applicable to a wide range of physical systems modeled by partial differential equations (e.g., heat conduction, wave propagation) where $A$ represents a spatial differential operator (like the Laplacian).
• Explicit Representation: The paper provides a constructive method (eigenfunction expansion) to find the solution, which is crucial for numerical implementation and further analysis of stability or inverse problems.

Conclusion

This paper successfully extends the theory of fractional differential equations to a complex Langevin-type structure involving two Caputo derivatives and an unbounded operator. By leveraging spectral decomposition and the properties of Mittag-Leffler functions, the authors provide a rigorous proof of existence, uniqueness, and an explicit formula for the solution, filling a gap in the literature regarding multi-order fractional Langevin equations in infinite-dimensional spaces.