# An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

## Abstract

**:**

## 1. Introduction

## 2. Assumptions on the Memory Kernel

- (P1)
- The class $\mathcal{CMF}$ is closed under point-wise multiplication.
- (P2)
- If $\phi \in \mathcal{CMF}$ and $\psi \in \mathcal{BF}$ then the composite function $\phi \left(\psi \right)\in \mathcal{CMF}$.
- (P3)
- $\phi \in \mathcal{CBF}$ if and only if $1/\phi \in \mathcal{SF}$.
- (P4)
- If $\phi \in \mathcal{BF}$ then it admits a continuous extension to ${\overline{\mathbb{C}}}_{+}$, which is holomorphic in ${\mathbb{C}}_{+}$ and satisfies $\Re \phi \left(s\right)>0$ for all $\Re s>0$.

## 3. Two Examples of Generalized Subdiffusion Equations

#### 3.1. Two Time-Scale Diffusion Model

#### 3.2. Fractional Jeffreys-Type Heat Conduction Equation

## 4. Generalized Relaxation Equation

**Proposition**

**1.**

## 5. Biorthonormal Pair of Riesz Bases

**Proposition**

**2.**

**Proof.**

## 6. Formal Spectral Expansions for the Solution

## 7. Uniqueness of Solution and Stability Estimates in Sobolev Spaces

**Proposition**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 8. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Bazhlekova, E.
An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions. *Fractal Fract.* **2021**, *5*, 63.
https://doi.org/10.3390/fractalfract5030063

**AMA Style**

Bazhlekova E.
An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions. *Fractal and Fractional*. 2021; 5(3):63.
https://doi.org/10.3390/fractalfract5030063

**Chicago/Turabian Style**

Bazhlekova, Emilia.
2021. "An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions" *Fractal and Fractional* 5, no. 3: 63.
https://doi.org/10.3390/fractalfract5030063