Analytical Solutions for Direct and Inverse Source Problems in a Time-Fractional Diffusion Equation

Abstract

This investigation is concerned with a time-fractional diffusion equation governed by the Hilfer fractional derivative of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary data and generalized initial conditions of the Hilfer type. This paper addresses three interconnected problems: a direct problem and two inverse source problems (ISPs). In the first ISP, the objective is to recover an unknown space-dependent source function from measurements taken at a specified final time. In the second ISP, the goal is to determine an unknown time-dependent coefficient through an integral-type over-specification condition. By employing eigenfunction expansions in conjunction with the LT technique, we derive explicit series representations of the solutions in terms of the Mittag-Leffler function. Rigorous existence and uniqueness results for classical solutions are established for all three problems. The second ISP is reformulated as a Volterra integral equation, whose unique solvability is demonstrated via the Banach fixed point theorem. Both ISPs are shown to be ill-posed in the Hadamard sense, indicating instability with respect to data perturbations. Numerical experiments are also presented to validate the theoretical findings and to illustrate the performance of the proposed reconstruction methods. As limiting cases, the formulations corresponding to the Riemann–Liouville and Caputo fractional derivatives are recovered, illustrating the generality of the proposed framework.

1. Introduction

This work investigates a time-fractional diffusion equation (TFDE) governed by the Hilfer fractional derivative of order ${\rho }_1$ and type ${\rho }_2$. The governing equation takes the form

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)=\left(LU\right)(x,y)+F(x,y),(x,y)\in (0,l)\times (0,T),$$

(1)

subject to homogeneous Neumann boundary data

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial U(0,y)}{\partial x}}=0={\displaystyle \frac{\partial U(l,y)}{\partial x}},y\in [0,T],\end{array}$$

(2)

together with the non-homogeneous generalized initial conditions

$$\begin{array}{c}\hfill \left.\begin{array}{cc}\hfill \underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}U(x,y)=& {}_{1}g\left(x\right),x\in [0,l],\hfill \\ \hfill \underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}U(x,y)=& {}_{2}g\left(x\right),x\in [0,l].\hfill \end{array}\right\}\end{array}$$

(3)

Here, ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}$ denotes the Hilfer fractional derivative (HFD) of order ${\rho }_1\in (1,2)$ and type ${\rho }_2\in [0,1]$, defined by [1]

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y):=\left(J_{0_{+},y}^{{\rho }_2(2-{\rho }_1)}{\displaystyle \frac{d^2}{d{y}^2}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}U\right)(x,y),$$

where $J_{0_{+},y}^{\rho }$ denotes the standard Riemann–Liouville fractional integral of order $\rho >0$:

$$J_{0_{+},y}^{\rho }U(x,y):={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\rho \right)}}{\int }_0^y{(y-z)}^{\rho -1}U(x,z)dz.$$

The elliptic operator L is given by

$$L\left(U\right)={\displaystyle \frac{\partial }{\partial x}}\left(\omega \left(x\right){\displaystyle \frac{\partial }{\partial x}}U(x,y)\right)-\gamma \left(x\right)U(x,y),$$

with $\omega \left(x\right)\in C^1\left([0,l]\right)$, $\omega \left(x\right)\ge {\omega }_0>0$, and $\gamma \left(x\right)\in C\left(\right[0,l\left]\right)$, $\gamma \left(x\right)\ge 0$.

The HFD serves as a natural interpolation between the Riemann–Liouville fractional derivative (RLFD) and the Caputo fractional derivative (CFD). When ${\rho }_2=0$, the HFD reduces to the RLFD:

$${}_{H}D_{0_{+},y}^{{\rho }_1,0}U(x,y)={\displaystyle \frac{d^2}{d{y}^2}}{J}_{0_{+},y}^{(2-{\rho }_1)}U(x,y)=:{}^{RL}D_{0_{+},y}^{{\rho }_1}U(x,y),$$

with the initial conditions becoming ${lim}_{y\to 0}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)}U(x,y)={}_{1}g\left(x\right)$ and ${lim}_{y\to 0}{J}_{0_{+},y}^{(2-{\rho }_1)}U(x,y)={}_{2}g\left(x\right)$. When ${\rho }_2=1$, the HFD reduces to the CFD:

$${}_{H}D_{0_{+},y}^{{\rho }_1,1}U(x,y)=J_{0_{+},y}^{(2-{\rho }_1)}{\displaystyle \frac{d^2}{d{y}^2}}U(x,y)=:{}^{C}D_{0_{+},y}^{{\rho }_1}U(x,y),$$

with classical initial conditions $U_y(x,0)={}_{1}g\left(x\right)$ and $U(x,0)={}_{2}g\left(x\right)$. Thus, the two-parameter Hilfer formulation provides a unifying framework that encompasses both classical fractional derivative types as limiting cases.

The treatment of ISPs with Hilfer-type derivatives presents several distinctive challenges. The two-parameter structure substantially complicates the Laplace transform (LT) and its inversion, as the exponents appearing in the transform and the initial conditions are coupled to both parameters in a non-trivial manner. The HFD framework necessitates generalized initial conditions expressed in terms of fractional integrals and derivatives, which introduces additional complexity. For ISP-I, the reconstruction of the space-dependent source $f\left(x\right)$ involves division by the term $T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})$, whose behavior for $1<{\rho }_1<2$ is considerably more intricate than in the Caputo case. For ISP-II, the associated Volterra integral equation contains a kernel that depends on the Hilfer parameters in a non-trivial way, necessitating refined analysis of the contraction conditions. These complications are absent in the standard Caputo or Riemann–Liouville settings, motivating the present comprehensive treatment.

Fractional differential equations (FDEs) have become indispensable tools for modeling a wide spectrum of physical phenomena, including anomalous diffusion [2], seismic activity prediction [3], biomedical applications [4], theoretical physics [1], biological systems [5], and viscoelastic processes [6,7]. These models are typically derived by replacing integer-order derivatives and integrals in classical equations with fractional counterparts. Several theoretical frameworks have been developed to describe anomalous diffusion, including Continuous Time Random Walks (CTRWs) [8], Brownian motion, and stochastic processes involving nonlocal operators [9].

The elliptic operator $-L$ possesses a purely discrete spectrum. We consider the eigenvalue problem

$$-L{\psi }_k={\lambda }_k{\psi }_k$$

subject to homogeneous Neumann boundary conditions, yielding eigenvalues

$$0={\lambda }_0<{\lambda }_1\le {\lambda }_2\le \cdots \to \infty ,$$

with orthonormal eigenfunctions satisfying ${\int }_0^l{\psi }_i\left(x\right){\psi }_j\left(x\right)dx={\delta }_{ij}$. The family ${\left\{{\psi }_k\left(x\right)\right\}}_{k=0}^{\infty }$ constitutes a complete orthonormal basis for $L^2\left((0,l)\right)$. For $\alpha \ge 0$, we define the fractional power space

$$\mathcal{S}\left({(-L)}^{\alpha }\right)=\left\{\varphi :{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k^{2\alpha }{\left|⟨\varphi ,{\psi }_k⟩\right|}^2<\infty \right\},$$

which is a Hilbert space with norm

$${\parallel \varphi \parallel }_{\mathcal{S}\left({(-L)}^{\alpha }\right)}={\left({\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k^{2\alpha }{\left|⟨\varphi ,{\psi }_k⟩\right|}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

and satisfies $\mathcal{S}\left({(-L)}^{\alpha }\right)\subset L^2\left((0,l)\right)$ for $\alpha \ge 0$. We emphasize that ${\lambda }_0=0$ corresponds to the constant eigenfunction; throughout the subsequent analysis, the results apply to $k\ge 1$ with ${\lambda }_k>0$, and the constant mode $k=0$ is treated separately in each problem.

The LT of the HFD (see [1], p. 113) is given by

$$\begin{array}{cc}\hfill \mathcal{L}\{{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y);s\}:=& s^{{\rho }_1}\mathcal{L}\{U(x,y);s\}-s^{1-{\rho }_2(2-{\rho }_1)}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{U(x,y)|}_{y=0}\hfill \\ & -s^{-{\rho }_2(2-{\rho }_1)}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{U(x,y)|}_{y=0}.\hfill \end{array}$$

This paper addresses a direct problem (DP) and two distinct ISPs associated with the system (1)–(3). In ISP-I, we aim to recover the space-dependent source term along with the solution using the initial and boundary conditions together with final-time data. In ISP-II, we seek to determine both the time-dependent coefficient and the solution via an integral-type over-specified condition.

Several authors have investigated direct and inverse problems (IPs) for FDEs. Zhang [10] examined both the DP and IP for a time-fractional Black–Scholes-type equation with a hyper-Bessel operator. Salakhitdinov and Karimov [11] studied the DP and IP for a two-term TFDE incorporating the HFD. Feng and Karimov [12] established uniqueness results for an IP associated with a time-fractional mixed parabolic–hyperbolic equation using eigenfunction expansion techniques. Long et al. [13] analyzed an IP for a fractional pseudo-parabolic equation, demonstrating severe ill-posedness in the Hadamard sense and proposing a Tikhonov regularization approach. Ilyas and Malik [14] investigated an IP for identifying a space-dependent source in an anomalous diffusion process with a second-level fractional derivative. Samreen et al. [15] studied the recovery of a temporal source term and temperature distribution for an FDE. Malik et al. [16] addressed the simultaneous recovery of diffusion concentration and a time-dependent source term using integral-type final data for a space-time FDE. Ilyas et al. [17] considered two IPs for a multi-term time-fractional evolution equation with an involution term, with fractional derivatives understood in the Caputo sense. Recent developments have further enriched the field. Huntul and Tekin [18] investigated an ISP for time-dependent coefficients in a higher-order pseudo-parabolic equation, employing a Volterra integral equation framework that shares methodological parallels with our ISP-II analysis. Yang et al. [19] explored Hopf bifurcation in ring-coupled Toda oscillators with distributed delay, illustrating the complex dynamical behavior arising from memory effects. These recent developments, together with the classical works on FDEs [20,21,22], form the basis for our investigation of the DP and ISPs for the TFDE with a HFD.

To clearly delineate the novelty of our work, we present a comparative analysis with related contributions (Table 1).

Table 1. Comparison of this work with the related literature. A ✓ indicates that the corresponding problem is studied.

Reference	Derivative	Order	DP	ISP-I	ISP-II
Salakhitdinov and Karimov [11]	HFDs	$0<{\rho }_1<1$	✓	✓	×
Malik et al. [16]	HFDs	$0<{\rho }_1<1$	×	✓	×
Ilyas et al. [17]	CFDs	$1<{\rho }_1<2$	✓	✓	×
Samreen et al. [15]	HFDs	$0<{\rho }_1<1$	×	✓	×
Feng and Karimov [12]	CFDs	$0<{\rho }_1<1$	✓	✓	×
This work	HFDs	$1<{\rho }_1<2$	✓	✓	✓

The principal contributions of this investigation are: (i) a unified framework encompassing the direct problem and both ISPs within the HFD framework of order $1<{\rho }_1<2$; (ii) the first treatment of ISP-II (time-dependent coefficient recovery) with a HFD of order $1<{\rho }_1<2$; (iii) explicit series representations, rigorous existence–uniqueness theorems, and comprehensive ill-posedness analyses for all three problems; (iv) incorporation of Hilfer-type generalized initial conditions; (v) systematic reduction to Caputo and Riemann–Liouville cases; and (vi) the correction of foundational issues including the correct definition of the HFD, the accurate LT, and a consistent spectral convention. Unlike prior works that consider either a single ISP or restrict themselves to Caputo/Riemann–Liouville settings, our framework simultaneously handles both ISPs across the complete Hilfer parameter space.

The remainder of this paper is structured as follows: Section 2 presents the series solution for the direct problem and establishes its unique solvability. Section 3 and Section 4 develop the series solutions and prove existence and uniqueness for ISP-I and ISP-II, respectively. In Section 5, we investigate the ill-posedness of both ISPs in the Hadamard sense. Section 6 discusses special cases of the DP and ISPs. Numerical experiments are presented in Section 7. The final section presents concluding remarks.

2. Direct Problem

This section addresses the direct problem (DP), which concerns the determination of the solution $U(x,y)$ for the system (1)–(3) given the data ${}_{1}g\left(x\right)$, ${}_{2}g\left(x\right)$, and $F(x,y)$. We adopt the classical solution framework, requiring that $y^{1+{\rho }_1{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right)$, with $\overline{\mathsf{\Omega }}:=[0,l]\times [0,T]$, and $y^{1+{\rho }_1{\rho }_2}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,·)\in C\left([0,T]\right)$.

2.1. Formal Series Representation of the DP

We seek a solution to the DP in the form of an eigenfunction expansion

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)U_k\left(y\right),}\end{array}$$

(4)

where the time-dependent coefficients $U_k\left(y\right)$ remain to be determined. Substituting this expansion into the governing Equation (1) and leveraging the orthogonality properties of the eigenfunctions yields the following FDE for each modal component:

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)=& -{\lambda }_k{U}_k\left(y\right)+F_k\left(y\right),\hfill \end{array}$$

(5)

with $F_k\left(y\right)=⟨F(x,y),{\psi }_k\left(x\right)⟩$.

Applying the LT to (5) and incorporating the initial conditions (3), we obtain

$$\mathcal{L}\{U_k\left(y\right);s\}={\displaystyle \frac{s^{-{\rho }_2(2-{\rho }_1)}{}_{1}g_k}{s^{{\rho }_1}+{\lambda }_k}}+{\displaystyle \frac{s^{1-{\rho }_2(2-{\rho }_1)}{}_{2}g_k}{s^{{\rho }_1}+{\lambda }_k}}+{\displaystyle \frac{\mathcal{L}\{F_k\left(y\right);s\}}{s^{{\rho }_1}+{\lambda }_k}}.$$

Inverting the LT using standard tables yields the explicit modal solution

$$\begin{array}{cc}\hfill U_k\left(y\right)=& {}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz.\hfill \end{array}$$

(6)

Here, $E_{{\rho }_1,{\rho }_2}(-{\lambda }_k{y}^{{\rho }_1})$ denotes the two-parameter Mittag-Leffler function [23], defined by

$$E_{{\rho }_1,{\rho }_2}(-{\lambda }_k{y}^{{\rho }_1}):={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{(-{\lambda }_k{y}^{{\rho }_1})}^k}{\mathsf{\Gamma }({\rho }_{1}k+{\rho }_2)}},Re\left({\rho }_1\right)>0,{\rho }_2\in \mathbb{C}.$$

For the special case ${\rho }_2=1$, this function reduces to the classical single-parameter Mittag-Leffler function:

$${\displaystyle E_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})=E_{{\rho }_1}(-{\lambda }_k{y}^{{\rho }_1}):={\displaystyle \sum _{k=0}^{\infty }}\frac{{(-{\lambda }_k{y}^{{\rho }_1})}^k}{\mathsf{\Gamma }({\rho }_{1}k+1)}.}$$

Consequently, the complete solution of the DP takes the form

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz).\hfill \end{array}$$

(7)

For the constant mode corresponding to $k=0$ with ${\lambda }_0=0$, Equation (5) simplifies to

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_0\left(y\right)=F_0\left(y\right),$$

which admits the explicit solution

$$\begin{array}{cc}\hfill U_0\left(y\right)=& {}_{1}g_0{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}\left(0\right)+{}_{2}g_0{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}\left(0\right)\hfill \\ \hfill +& {\int }_0^y{F}_0\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}\left(0\right)dz.\hfill \end{array}$$

This expression remains finite and does not involve division by ${\lambda }_0$, thereby avoiding any singular behavior.

2.2. Main Results for the Direct Problem

We now establish the well-posedness of the DP through a series of auxiliary lemmas and the main theorem. The following technical results are essential for establishing the continuity of the series solutions for both the DP and the inverse problems.

Lemma 1

([24]). If ${\rho }_2<2$, ${\rho }_1$ is an arbitrary real number, and ν satisfies $\pi {\rho }_1/2<\nu <min\{\pi ,\pi {\rho }_1\}$, then for ${\lambda }_k{y}^{{\rho }_1}\in \mathbb{C}$ with $|{\lambda }_k{y}^{{\rho }_1}|\ge 0$ and $\nu \le \left|arg\right({\lambda }_k{y}^{{\rho }_1}\left)\right|\le \pi$, we have

$$|E_{{\rho }_1,{\rho }_2}(-{\lambda }_k{y}^{{\rho }_1})|\le {\displaystyle \frac{C_0}{1+|{\lambda }_k{y}^{{\rho }_1}|}},$$

where $C_0$ is a positive constant.

Lemma 2

([25]). For ${\lambda }_k\in \mathbb{R}$ with ${\lambda }_k\ne 0$, $r\in C^1\left([0,T]\right)$, and $C_0$ a positive constant, the following estimate holds:

$$\left|{\int }_0^{y}r\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz\right|\le {\displaystyle \frac{C_0}{{\lambda }_k}}{\parallel r\parallel }_{C^1\left([0,T]\right)}.$$

Lemma 3

([26]). Let ${\left\{r_i\right\}}_{i=1}^{\infty }$ be a sequence of functions defined on $(a,b]$ such that:

(1) 

${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}$ exists for all $i\in \mathbb{N}$ and $y\in (a,b]$;

(2) 

Both series ${\displaystyle {\displaystyle \sum _{i=1}^{\infty }}{r}_i\left(y\right)}$ and ${\displaystyle {\displaystyle \sum _{i=1}^{\infty }}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{r}_i\left(y\right)}$ converge uniformly on $[a+ϵ,b]$ for every $ϵ>0$.

Then, for $1<{\rho }_1<2$, $0\le {\rho }_2\le 1$, and $a<z<b$,

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\displaystyle \sum _{i=1}^{\infty }}{r}_i\left(y\right)={\displaystyle \sum _{i=1}^{\infty }}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{r}_i\left(y\right).$$

Lemma 4.

Let $g\in C^2\left([0,l]\right)$ satisfy the Neumann boundary conditions $g^{\prime }\left(0\right)=g^{\prime }\left(l\right)=0$. Then its Fourier coefficients $g_k=⟨g,{\psi }_k⟩$ satisfy the decay estimate

$$|g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel g\parallel }_{C^2\left([0,l]\right)},k\ge 1,$$

for some constant $C>0$ independent of k.

Proof. 

Applying integration by parts twice and using the identity $-L{\psi }_k={\lambda }_k{\psi }_k$, we obtain

$$g_k=⟨g,{\psi }_k⟩={\displaystyle \frac{1}{{\lambda }_k}}⟨g,-L{\psi }_k⟩={\displaystyle \frac{1}{{\lambda }_k}}⟨-Lg,{\psi }_k⟩.$$

The boundary terms vanish by virtue of the Neumann conditions. Consequently,

$$|g_k|\le {\displaystyle \frac{1}{{\lambda }_k}}{\parallel Lg\parallel }_{L^2\left([0,l]\right)}\parallel {\psi }_k{\parallel }_{L^2(0,l)}\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel g\parallel }_{C^2\left([0,l]\right)},$$

where $\parallel {\psi }_k{\parallel }_{L^2\left([0,l]\right)}\le C.$ □

The $C^2\left([0,l]\right)$ regularity assumption on the data ${}_{1}g,{}_{2}g,h,$ and f is standard in the classical solution theory (in the sense of $y^{1+{\rho }_1{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right)$). This assumption guarantees sufficient decay of the Fourier coefficients

$$|{}_{1}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)},|{}_{2}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)},$$

which is indispensable for establishing uniform convergence of the series representations.

We note that the regularity requirements can be relaxed under alternative frameworks:

• Sobolev spaces: The results extend to $H^s\left([0,l]\right)$ with $s>1/2$ (for $L^2$-theory) or $H^s$ with $s>3/2$ (for C-theory), with coefficient estimates following from Sobolev embedding theorems.
• Weak solutions: A variational formulation permits weak solutions in $L^2$ or $H^1$ spaces, accommodating data in $L^2\left([0,l]\right)$ or $H^1\left([0,l]\right)$.
• Fractional Sobolev spaces: The generalized initial conditions involving fractional integrals naturally lead to fractional Sobolev spaces $W^{\alpha ,p}\left([0,l]\right)$.
• Distributional solutions: The eigenfunction expansion can be interpreted in the sense of distributions for more general data.

The extension to lower regularity settings constitutes an interesting direction for future investigation. In the present work, we maintain the $C^2$ assumption to focus on the classical solution framework and clearly illustrate the methodology.

Theorem 1.

Under the following assumptions:

(C1) 

${}_{1}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{1}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{1}g\left(l\right)}{\partial x}}$;

(C2) 

${}_{2}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{2}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{2}g\left(l\right)}{\partial x}}$;

(C3) 

$F(·,y)\in C^2\left([0,l]\right)$ for each y, with ${\displaystyle \frac{\partial F(0,y)}{\partial x}}=0={\displaystyle \frac{\partial F(l,y)}{\partial x}}$,

• the DP (1)–(3) admits a unique classical solution.

Proof. 

The proof proceeds in two stages: (1) establishing existence and (2) demonstrating uniqueness.

• (1) Existence of the Solution: We must verify that

$$y^{1+{\rho }_1{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right),y^{1+{\rho }_1{\rho }_2}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,·)\in C\left([0,T]\right).$$

Employing the DP solution Formula (7), we derive the estimate

$$\begin{array}{cc}\hfill \left|U\right(x,y\left)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}(\left|{}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1}){\psi }_k\left(x\right)\right|\hfill \\ \hfill +& |{}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1}){\psi }_k\left(x\right)|\hfill \\ \hfill +& \left|{\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz{\psi }_k\left(x\right)\right|).\hfill \end{array}$$

Applying the coefficient estimates from Lemma 4 yields

$$|{}_{1}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)},|{}_{2}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)},|F_k\left(y\right)|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel F(·,y)\parallel }_{C^2\left([0,l]\right)}.$$

Substituting these estimates into the series expression and invoking Lemmas 1 and 2 gives

$$\begin{array}{cc}\hfill y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|\le & {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}\frac{C_{0}C}{{\lambda }_k^2}\left({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_2}+\parallel {}_{2}g\parallel C^2\left([0,l]\right)y^{2{\rho }_2}+{\parallel F(·,y)\parallel }_{C^1\left([0,T]\right)}{y}^{1+{\rho }_1{\rho }_2}\right).}\hfill \end{array}$$

(8)

Since ${\lambda }_k\sim k^2$ (see [27], p. 460), we have

$${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{{\lambda }_k}}\le C{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{k^2}}<\infty .$$

Consequently, the series in (8) converges uniformly by the Weierstrass M-test (WMT), establishing that $y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|$ is a continuous function on $\overline{\mathsf{\Omega }}$.

We next establish the continuity of ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)$. From the modal Equation (5), we have

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)={\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right){}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right).$$

Substituting (5) yields

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)=& -{\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right){\lambda }_k{U}_k\left(y\right)+{\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)F_k\left(y\right),\hfill \end{array}$$

which expands to

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({\lambda }_k{}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\lambda }_k{}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-2}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\lambda }_k{\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz+F_k\left(y\right)).\hfill \end{array}$$

Applying Lemma 1, Lemma 4, and the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{cc}\hfill y^{1+{\rho }_1{\rho }_2}\left|{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}\{{\displaystyle \frac{C_{0}C}{{\lambda }_k}}({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_2}+{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}{y}^{2{\rho }_2}\hfill \\ \hfill +& {\parallel F(·,y)\parallel }_{C^1\left([0,T]\right)}{y}^{1+{\rho }_1{\rho }_2})+{\displaystyle \frac{C}{{\lambda }_k^2}}{\parallel F(·,y)\parallel }_{C^1\left([0,T]\right)}{y}^{1+{\rho }_1{\rho }_2}\}.\hfill \end{array}$$

(9)

The uniform convergence of the series in (9) follows from ${\lambda }_k\sim k^2$, and hence $y^{1+{\rho }_1{\rho }_2}\left|{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)\right|$ represents a continuous function by the WMT.

• Verification of the governing equation: Let $U(x,y)={\sum }_{k=0}^{\infty }{U}_k\left(y\right){\psi }_k\left(x\right)$ be the constructed series with $U_k\left(y\right)$ given by (6). By Lemma 3, we may interchange the Hilfer derivative and the summation

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)={\displaystyle \sum _{k=0}^{\infty }}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right){\psi }_k\left(x\right).$$

Since each $U_k\left(y\right)$ satisfies the modal Equation (5)

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)=-{\lambda }_k{U}_k\left(y\right)+F_k\left(y\right),$$

we obtain

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)={\displaystyle \sum _{k=0}^{\infty }}(-{\lambda }_k{U}_k\left(y\right)+F_k\left(y\right)){\psi }_k\left(x\right)={\displaystyle \sum _{k=0}^{\infty }}{U}_k\left(y\right)(-{\lambda }_k{\psi }_k\left(x\right))+{\displaystyle \sum _{k=0}^{\infty }}{F}_k\left(y\right){\psi }_k\left(x\right).$$

Employing the identities $L{\psi }_k=-{\lambda }_k{\psi }_k$ and $F(x,y)={\sum }_{k=0}^{\infty }{F}_k\left(y\right){\psi }_k\left(x\right)$, we arrive at

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)=LU(x,y)+F(x,y),$$

confirming that the constructed series satisfies the FDE (1).

• Verification of the boundary conditions: Since each ${\psi }_k$ satisfies ${\psi }_k^{\prime }\left(0\right)={\psi }_k^{\prime }\left(l\right)=0$ for all $k\ge 0$, and the series for U and its derivative converge uniformly, we have

$${\displaystyle \frac{\partial U(0,y)}{\partial x}}={\displaystyle \sum _{k=0}^{\infty }}{U}_k\left(y\right){\psi }_k^{\prime }\left(0\right)=0,{\displaystyle \frac{\partial U(l,y)}{\partial x}}={\displaystyle \sum _{k=0}^{\infty }}{U}_k\left(y\right){\psi }_k^{\prime }\left(l\right)=0.$$

Thus, the boundary conditions (2) are satisfied.

• Verification of the initial conditions: Using the generalized initial conditions (3) and the series representation, we find

$$\underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}U(x,y)={\displaystyle \sum _{k=0}^{\infty }}\left(\underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{U}_k\left(y\right)\right){\psi }_k\left(x\right)={\displaystyle \sum _{k=0}^{\infty }}{}_{1}g_k{\psi }_k\left(x\right)={}_{1}g\left(x\right),$$

and similarly

$$\underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}U(x,y)={\displaystyle \sum _{k=0}^{\infty }}\left(\underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{U}_k\left(y\right)\right){\psi }_k\left(x\right)={\displaystyle \sum _{k=0}^{\infty }}{}_{2}g_k{\psi }_k\left(x\right)={}_{2}g\left(x\right).$$

Hence, the initial conditions (3) are also satisfied.

Remark 1.

The constant mode $k=0$ is treated separately. All inequalities involving $1/{\lambda }_k$ are understood for $k\ge 1$, and the convergence of ${\sum }_{k=1}^{\infty }1/{\lambda }_k^2$ follows from the asymptotic behavior ${\lambda }_k\sim k^2$.

• (2) Uniqueness of the Solution:

Let $U_1(x,y)$ and $U_2(x,y)$ be two classical solutions of the DP corresponding to identical initial data ${}_{1}g,{}_{2}g$ and the same forcing term F. Define their difference $\tilde{U}(x,y)=U_1(x,y)-U_2(x,y)$. Then $\tilde{U}(x,y)$ satisfies the homogeneous problem

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y)=L\tilde{U}(x,y),(x,y)\in (0,l)\times (0,T),$$

subject to homogeneous Neumann boundary conditions

$$\begin{array}{c}{\displaystyle \frac{\partial \tilde{U}(0,y)}{\partial x}}=0={\displaystyle \frac{\partial \tilde{U}(l,y)}{\partial x}},y\in [0,T],\end{array}$$

and homogeneous generalized initial conditions

$$\begin{array}{c}\hfill \left.\begin{array}{cc}\hfill \underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l],\hfill \\ \hfill \underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l].\hfill \end{array}\right\}\end{array}$$

(10)

We note that the forcing term F cancels in the difference equation since both $U_1(x,y)$ and $U_2(x,y)$ satisfy the same FDE with identical source data.

Consider the modal coefficients

$${\tilde{U}}_k\left(y\right)={\int }_0^l\tilde{U}(x,y){\psi }_k\left(x\right)dx.$$

By Lemma 3, we may interchange the HFD and the integral, as the series for $\tilde{U}(x,y)$ and its HFD converge uniformly under the stated regularity assumptions. Thus

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)={\int }_0^l{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y){\psi }_k\left(x\right)dx.$$

Using the homogeneous equation and the self-adjointness of L with Neumann boundary conditions at both endpoints (which eliminates the boundary terms), we obtain

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)& ={\int }_0^{l}L\tilde{U}(x,y){\psi }_k\left(x\right)dx\hfill \\ \hfill & ={\int }_0^l\tilde{U}(x,y)L{\psi }_k\left(x\right)dx\hfill \\ \hfill & =-{\lambda }_k{\int }_0^l\tilde{U}(x,y){\psi }_k\left(x\right)dx\hfill \\ \hfill & =-{\lambda }_k{\tilde{U}}_k\left(y\right).\hfill \end{array}$$

Consequently, ${\tilde{U}}_k\left(y\right)$ satisfies the homogeneous fractional differential equation

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)=-{\lambda }_k{\tilde{U}}_k\left(y\right),$$

with zero initial conditions. The unique solution of this initial value problem is ${\tilde{U}}_k\left(y\right)\equiv 0$ for all $k\ge 0$, including the constant mode $k=0$ with ${\lambda }_0=0$.

Therefore, ${\tilde{U}}_k\left(y\right)=0$ for every $k\ge 0$, and by the completeness of the eigenfunction system, $\tilde{U}(x,y)\equiv 0$ on $\overline{\mathsf{\Omega }}$. Hence $U_1(x,y)=U_2(x,y)$, establishing uniqueness. □

• Remark on the boundary terms: The boundary terms vanish in the integration by parts because $\tilde{U}(x,y)$ satisfies homogeneous Neumann conditions at both $x=0$ and $x=l$, and the eigenfunctions ${\psi }_k$ satisfy the same conditions by construction.

3. Inverse Source Problem-I (ISP-I)

We now turn our attention to the first ISP, designated as ISP-I. In this scenario, the source term is assumed to depend exclusively on the spatial variable, i.e., $F(x,y)=f\left(x\right)$. The objective is to simultaneously recover the pair $\left(U\right(x,y),f(x\left)\right)$ for the system (1)–(3). To accomplish this, we require an additional over-specified condition at a fixed final time T:

$$U(x,T)=h\left(x\right).$$

(11)

A classical solution of ISP-I consists of functions $U(x,y)$ and $f\left(x\right)$ satisfying the regularity requirements $T^{1+{\rho }_1{\rho }_2}f\left(x\right)\in C\left([0,l]\right)$, $y^{1+{\rho }_1{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right)$, and $y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,·)\in C\left([0,T]\right)$.

3.1. Formal Series Solution of the ISP-I

We seek representations of the solution and the unknown source term in the form of eigenfunction expansions:

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)U_k\left(y\right),f\left(x\right)={\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)f_k,\hfill \end{array}$$

(12)

where $U_k\left(y\right)$ and $f_k$ satisfy the modal FDE

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)=& -{\lambda }_k{U}_k\left(y\right)+f_k,\hfill \end{array}$$

with $f_k=⟨f\left(x\right),{\psi }_k\left(x\right)⟩$.

Applying the LT to the modal equation and incorporating the initial conditions (3), we obtain the explicit representation

$$\begin{array}{cc}\hfill U_k\left(y\right)=& {}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& f_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1}).\hfill \end{array}$$

(13)

Consequently, the solution $U(x,y)$ admits the series representation

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})+f_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1})),\hfill \end{array}$$

(14)

where the coefficients $f_k$ remain to be determined.

Applying the over-specified condition (11) at $y=T$ yields the reconstruction formula

$$\begin{array}{cc}\hfill f_k=& {\displaystyle \frac{1}{T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})}}\{h_k-({}_{1}g_k{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{T}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{T}^{{\rho }_1})\left)\right\},\hfill \end{array}$$

(15)

where $h_k=⟨h\left(x\right),{\psi }_k\left(x\right)⟩$. The reconstructed source term is therefore given by

$$\begin{array}{cc}\hfill f\left(x\right)=& {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{\psi }_k\left(x\right)}{T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})}}\{h_k-({}_{1}g_k{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{T}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{T}^{{\rho }_1})\left)\right\}.\hfill \end{array}$$

(16)

For the constant mode $k=0$, the over-specified condition $U_0\left(T\right)=h_0$ directly yields

$$f_0={\displaystyle \frac{h_0-{}_{1}g_0{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}\left(0\right)-{}_{2}g_0{T}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}\left(0\right)}{T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}\left(0\right)}}.$$

Consequently, Formula (15) is applicable for $k\ge 1$, while $f_0$ is determined separately through the above expression.

3.2. Main Results for ISP-I

We now establish the well-posedness of ISP-I under appropriate regularity assumptions on the data.

Prior to presenting the main result, we record the following useful identity for the HFD of power functions:

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{y}^k={\displaystyle \frac{\mathsf{\Gamma }(k+1)}{\mathsf{\Gamma }(k-{\rho }_1+1)}}{y}^{k-{\rho }_1},1<{\rho }_1<2,0\le {\rho }_2\le 1.$$

(17)

Lemma 5.

For $m,{\rho }_i,y>0$, $i=1,2$, the Mittag-Leffler functions satisfy the following differentiation properties:

• ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\left(y^{{\rho }_2-1}{E}_{{\rho }_1,{\rho }_2}(-m{y}^{{\rho }_1})\right)=y^{{\rho }_2-{\rho }_1-1}{E}_{{\rho }_1,{\rho }_2-{\rho }_1}(-m{y}^{{\rho }_1});$
• ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\left(y^{{\rho }_1-{\rho }_2}{E}_{{\rho }_1,{\rho }_1-{\rho }_2+1}(-m{y}^{{\rho }_1})\right)=y^{-{\rho }_2}{E}_{{\rho }_1,1-{\rho }_2}(-m{y}^{{\rho }_2}).$

Proof. 

We prove the first identity; the second follows by an analogous argument. Recall that the two-parameter Mittag-Leffler function is defined by the series

$$E_{{\rho }_1,{\rho }_2}(-m{y}^{{\rho }_1})={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{(-m)}^j{y}^{{\rho }_{1}j}}{\mathsf{\Gamma }({\rho }_{1}j+{\rho }_2)}},$$

which converges uniformly on compact subsets of $(0,\infty )$ (see [23], Chapter 3).

Consider the function $f\left(y\right)=y^{{\rho }_2-1}{E}_{{\rho }_1,{\rho }_2}(-m{y}^{{\rho }_1})$. Using the series representation, we have

$$f\left(y\right)={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{(-m)}^j}{\mathsf{\Gamma }({\rho }_{1}j+{\rho }_2)}}{y}^{{\rho }_{1}j+{\rho }_2-1}.$$

Applying the Hilfer fractional derivative ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}$ term by term (which is justified by the uniform convergence of the series and Lemma 3), and using the power rule (17), we obtain

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}f\left(y\right)={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{(-m)}^j}{\mathsf{\Gamma }({\rho }_{1}j+{\rho }_2)}}{\displaystyle \frac{\mathsf{\Gamma }({\rho }_{1}j+{\rho }_2)}{\mathsf{\Gamma }({\rho }_{1}j+{\rho }_2-{\rho }_1)}}{y}^{{\rho }_{1}j+{\rho }_2-{\rho }_1-1}.$$

Simplifying the Gamma functions yields

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}f\left(y\right)={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{(-m)}^j}{\mathsf{\Gamma }({\rho }_{1}j+({\rho }_2-{\rho }_1))}}{y}^{{\rho }_{1}j+({\rho }_2-{\rho }_1)-1}.$$

Recognizing the resulting series as the definition of the Mittag-Leffler function with parameters ${\rho }_1$ and ${\rho }_2-{\rho }_1$, we obtain

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\left(y^{{\rho }_2-1}{E}_{{\rho }_1,{\rho }_2}(-m{y}^{{\rho }_1})\right)=y^{{\rho }_2-{\rho }_1-1}{E}_{{\rho }_1,{\rho }_2-{\rho }_1}(-m{y}^{{\rho }_1}).$$

The second identity follows similarly by applying the power rule (17) to the series expansion of the Mittag-Leffler function in the expression $y^{{\rho }_1-{\rho }_2}{E}_{{\rho }_1,{\rho }_1-{\rho }_2+1}(-m{y}^{{\rho }_1})$. □

The following estimate, which is a consequence of the complete monotonicity of the Mittag-Leffler function [25], will be essential:

$$\begin{array}{c}{\displaystyle \frac{1}{E_{{\rho }_1,1}(-m{T}^{{\rho }_1})-1}}\le C_1,T>0,\end{array}$$

(18)

where $C_1$ is a constant independent of n. Indeed, we have

$$T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-m{T}^{\alpha })={\int }_0^T{\tau }^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-m{\tau }^{\alpha })d\tau ={\displaystyle \frac{1}{m}}\left(E_{{\rho }_1,1}\left(m{T}^{\alpha }\right)-1\right).$$

The non-vanishing of $E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})$ follows from the fact that the Mittag-Leffler function $E_{{\rho }_1,{\rho }_1+1}(-z)$ possesses no real zeros for $z>0$ when $1<{\rho }_1<2$ (see, e.g., Gorenflo et al. [23]). For the constant mode $k=0$, we have $T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}\left(0\right)=T^{{\rho }_1}/\mathsf{\Gamma }({\rho }_1+1)\ne 0$.

Theorem 2.

Assume that the following regularity conditions hold:

(1) 

${}_{1}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{1}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{1}g\left(l\right)}{\partial x}}$;

(2) 

${}_{2}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{2}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{2}g\left(l\right)}{\partial x}}$;

(3) 

$h\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial h\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial h\left(l\right)}{\partial x}}$.

Then the ISP-I (1)–(3) admits a unique classical solution.

Proof. 

We emphasize that the constant mode $k=0$ is handled separately; the arguments below concern the non-zero modes $k\ge 1$.

The proof consists of two principal components: establishing existence and demonstrating uniqueness.

• (1) Existence of the Solution: To establish existence, we must verify the uniform convergence of the series representations for the functions

$$T^{1+{\rho }_1{\rho }_2}f\left(x\right),y^{1+{\rho }_1{\rho }_2}U(x,y),y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y).$$

From the solution of ISP-I, i.e., $U(x,y)$ given in (14), we derive the estimate

$$\begin{array}{cc}\hfill \left|U\right(x,y\left)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}(\left|{}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1}){\psi }_k\left(x\right)\right|\hfill \\ \hfill +& |{}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1}){\psi }_k\left(x\right)|\hfill \\ \hfill +& |f_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1}){\psi }_k\left(x\right)|).\hfill \end{array}$$

Applying Lemma 1 (the Mittag-Leffler estimate) to each term yields

$$\begin{array}{cc}\hfill \left|U\right(x,y\left)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}\left(\right|{}_{1}g_k{\displaystyle \frac{C_0{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}}{1+{\lambda }_k{y}^{{\rho }_1}}}{\psi }_k\left(x\right)|+|{}_{2}g_k{\displaystyle \frac{C_0{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}}{1+{\lambda }_k{y}^{{\rho }_1}}}{\psi }_k\left(x\right)|\hfill \\ \hfill +& \left|f_k{\displaystyle \frac{C_0{y}^{{\rho }_1}}{1+{\lambda }_k{y}^{{\rho }_1}}}{\psi }_k\left(x\right)\right|).\hfill \end{array}$$

Employing the coefficient decay estimates from Lemma 4, namely,

$$|{}_{1}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)},|{}_{2}g_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)},|f_k|\le {\displaystyle \frac{C}{{\lambda }_k}}{\parallel f\parallel }_{C^2\left([0,l]\right)},$$

and using the inequality

$${\displaystyle \frac{1}{1+{\lambda }_k{y}^{{\rho }_1}}}\le {\displaystyle \frac{1}{{\lambda }_k{y}^{{\rho }_1}}}$$

for $y>0$, together with the uniform boundedness of the eigenfunctions $|{\psi }_k\left(x\right)|\le C$ (which follows from standard elliptic regularity theory), we obtain

$$\begin{array}{cc}\hfill y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{C_{0}C}{|{\lambda }_k|}}\left({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_2+1}+{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}{y}^{2{\rho }_2}+{\parallel f\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_1{\rho }_2+2}\right).\hfill \end{array}$$

(19)

Since ${\lambda }_k\sim k^2$, we have ${\sum }_{k=1}^{\infty }1/{\lambda }_k\le C{\sum }_{k=1}^{\infty }1/k^2<\infty$. Therefore, the series in (19) is majorized by a convergent constant series independent of x and y. By the WMT, the series converges uniformly on $(0,l)\times (0,T)$. Consequently, $y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|$ is a continuous function.

We next establish the continuity of $T^{1+{\rho }_1{\rho }_2}f\left(x\right)$. From the reconstruction Formula (16), applying Lemma 1 to bound the Mittag-Leffler functions in the denominator and numerator, Lemma 4 for the coefficient estimates, and the estimate (18) for the reciprocal of the Mittag-Leffler function, we arrive at

$$\begin{array}{cc}\hfill T^{1+{\rho }_1{\rho }_2}\left|f\left(x\right)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{C_0{C}_{1}C}{|{\lambda }_k|}}\left({\parallel h\parallel }_{C^2\left([0,l]\right)}{T}^{{\rho }_1{\rho }_2+2}+{\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{T}^{{\rho }_2+1}+{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}{T}^{{\rho }_1{\rho }_2+2}\right).\hfill \end{array}$$

(20)

Again, since ${\lambda }_k\sim k^2$, the series ${\sum }_{k=1}^{\infty }1/{\lambda }_k$ converges. Thus, the series in (20) is dominated by a convergent constant series, and by the WMT, $T^{1+{\rho }_1{\rho }_2}f\left(x\right)$ is continuous.

• Finally, we examine the convergence of ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)$. By Lemma 3 (which permits termwise differentiation of the series under uniform convergence), we have

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)={\displaystyle \sum _{k=1}^{\infty }}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right){\psi }_k\left(x\right).$$

Using Lemma 5 to compute the HFD of the Mittag-Leffler terms in the ISP-I solution Formula (13), we obtain

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)=& {}_{1}g_k{y}^{{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{y}^{{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& f_k{E}_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1}).\hfill \end{array}$$

Applying Lemma 1 to each Mittag-Leffler term, Lemma 3 for the termwise differentiation, and Lemma 4 for the coefficient decay, we derive

$$\begin{array}{cc}\hfill y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}\left|{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)\right|\le & {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{C_{0}C}{|{\lambda }_k|}}({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}\hfill \\ \hfill +& {\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}+{\parallel f\parallel }_{C^2\left([0,l]\right)}{y}^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}).\hfill \end{array}$$

(21)

Since the series ${\sum }_{k=1}^{\infty }1/{\lambda }_k$ converges by ${\lambda }_k\sim k^2$, the right-hand side is majorized by a convergent constant series. Hence, by the WMT, the series in (21) converges uniformly, establishing that $y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)$ is a continuous function.

• (2) Uniqueness of the Solution: Let $\{U_1(x,y),f_1\left(x\right)\}$ and $\{U_2(x,y),f_2\left(x\right)\}$ be two classical solutions of ISP-I corresponding to identical initial data ${}_{1}g,{}_{2}g$ and the same final data $h\left(x\right)=U_1(x,T)=U_2(x,T)$. Define their differences $\tilde{U}(x,y)=U_1(x,y)-U_2(x,y)$ and $\tilde{f}\left(x\right)=f_1\left(x\right)-f_2\left(x\right)$. Then $\tilde{U}(x,y)$ satisfies the inhomogeneous problem

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y)=L\tilde{U}(x,y)+\tilde{f}\left(x\right),$$

subject to homogeneous Neumann boundary conditions

$$\begin{array}{c}{\displaystyle \frac{\partial \tilde{U}(0,y)}{\partial x}}=0={\displaystyle \frac{\partial \tilde{U}(l,y)}{\partial x}},y\in [0,T],\end{array}$$

homogeneous generalized initial conditions

$$\begin{array}{c}\hfill \left.\begin{array}{cc}\hfill \underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l],\hfill \\ \hfill \underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l],\hfill \end{array}\right\}\end{array}$$

(22)

and the final condition

$$\tilde{U}(x,T)=0,x\in [0,l].$$

Introducing the modal coefficients

$${\tilde{U}}_k\left(y\right)={\int }_0^l\tilde{U}(x,y){\psi }_k\left(x\right)dx,{\tilde{f}}_k={\int }_0^l\tilde{f}\left(x\right){\psi }_k\left(x\right)dx,$$

and applying Lemma 3 together with integration by parts (with boundary terms vanishing due to the Neumann conditions at both endpoints), we obtain

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)& ={\int }_0^l{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y){\psi }_k\left(x\right)dx\hfill \\ \hfill & ={\int }_0^l(L\tilde{U}(x,y)+\tilde{f}\left(x\right)){\psi }_k\left(x\right)dx\hfill \\ \hfill & ={\int }_0^l\tilde{U}(x,y)L{\psi }_k\left(x\right)dx+{\tilde{f}}_k\hfill \\ \hfill & =-{\lambda }_k{\tilde{U}}_k\left(y\right)+{\tilde{f}}_k.\hfill \end{array}$$

Thus, ${\tilde{U}}_k\left(y\right)$ satisfies the initial value problem

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)=-{\lambda }_k{\tilde{U}}_k\left(y\right)+{\tilde{f}}_k,$$

with zero initial data and the terminal condition ${\tilde{U}}_k\left(T\right)=0$.

The explicit solution is

$${\tilde{U}}_k\left(y\right)={\tilde{f}}_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1}).$$

Imposing the terminal condition yields

$${\tilde{f}}_k{T}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})=0.$$

Since $T^{{\rho }_1}>0$ and $E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})\ne 0$ for all $k\ge 0$ (the Mittag-Leffler function has no real zeros for positive arguments when $1<{\rho }_1<2$; see Gorenflo et al. [23]), we conclude that ${\tilde{f}}_k=0$ for every $k\ge 0$. Hence $\tilde{f}\equiv 0$.

With $\tilde{f}\equiv 0$, the problem reduces to the homogeneous direct problem

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y)=L\tilde{U}(x,y),$$

with zero initial and boundary data and the terminal condition $\tilde{U}(x,T)=0$. By Theorem 1, the unique solution is $\tilde{U}(x,y)\equiv 0$. Therefore, $U_1(x,y)=U_2(x,y)$ and $f_1\left(x\right)=f_2\left(x\right)$, establishing uniqueness. □

Remark 2.

The constant mode $k=0$ is included in the proof with ${\lambda }_0=0$. For this mode, $E_{{\rho }_1,{\rho }_1+1}\left(0\right)=1/\mathsf{\Gamma }({\rho }_1+1)\ne 0$, so the non-degeneracy condition remains valid.

4. Inverse Source Problem-II (ISP-II)

This section is devoted to the second ISP, designated as ISP-II. In this scenario, the source term is assumed to be separable in the form $F(x,y)=q\left(y\right)f(x,y)$, where $q\left(y\right)$ is an unknown time-dependent coefficient and $f(x,y)$ is a known function. The objective is to determine the pair $\left\{U\right(x,y),q(y\left)\right\}$ for the system (1)–(3). To accomplish this, we impose the following integral-type over-specified condition:

$${\int }_0^{l}U(x,y)dx=E\left(y\right),y\in [0,T].$$

(23)

A classical solution of ISP-II consists of functions $U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right)$, $q\left(y\right)\in C\left(\right[0,T\left]\right)$, satisfying the additional regularity conditions $y^{2-{\rho }_1-{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right)$ and ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\left(y^{2-{\rho }_1-{\rho }_2}u(x,·)\right)\in C\left([0,T]\right)$.

4.1. Formal Series Solution of the ISP-II

We express the solution of ISP-II in the eigenfunction expansion

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)U_k\left(y\right),}\end{array}$$

(24)

where the modal coefficients $U_k\left(y\right)$ satisfy the FDE

$$\begin{array}{cc}\hfill {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{U}_k\left(y\right)=& -{\lambda }_k{U}_k\left(y\right)+q\left(y\right)f_k\left(y\right),\hfill \end{array}$$

with $f_k\left(y\right)=⟨f(x,y),{\psi }_k\left(x\right)⟩$.

Application of the LT together with the initial conditions (3) yields the explicit representation

$$\begin{array}{cc}\hfill U_k\left(y\right)=& {}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^{y}q\left(z\right)f_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz.\hfill \end{array}$$

(25)

Consequently, the solution $U(x,y)$ admits the series representation

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^{y}q\left(z\right)f_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz),\hfill \end{array}$$

(26)

where the coefficient $q\left(y\right)$ remains to be determined.

For the constant mode $k=0$, the solution $U_0\left(y\right)$ does not involve ${\lambda }_0$ in the denominator. The Volterra integral equation for $q\left(y\right)$ is derived using the integral over-specified condition, and the kernel $K(y,z)$ is defined for $k\ge 1$. The constant mode is handled through the consistency conditions in Theorem 3.

4.2. Main Results for ISP-II

We now establish the well-posedness of ISP-II under appropriate assumptions on the data.

Theorem 3.

Assume the following conditions hold:

(1) 

${}_{1}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{1}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{1}g\left(l\right)}{\partial x}}$;

(2) 

${}_{2}g\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial {}_{2}g\left(0\right)}{\partial x}}=0={\displaystyle \frac{\partial {}_{2}g\left(l\right)}{\partial x}}$;

(3) 

$f(·,y)\in C^2\left([0,l]\right)$ with ${\displaystyle \frac{\partial f(0,y)}{\partial x}}=0={\displaystyle \frac{\partial f(l,y)}{\partial x}}$ for each y. Moreover, ${\displaystyle {\int }_0^{l}f(x,y)dx\ne 0}$ for all $y\in [0,T]$, and there exists a constant $M_1>0$ such that

$${\displaystyle {\left({\int }_0^{l}f(x,y)dx\right)}^{-1}\le M_1;}$$

(4) 

$E\in C^2\left([0,T]\right)$ satisfies the following consistency conditions at $y=0$:

$$\begin{array}{cc}\hfill {\int }_0^l{}_{1}g\left(x\right)dx& =\underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}E\left(y\right),\hfill \\ \hfill {\int }_0^l{}_{2}g\left(x\right)dx& =\underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}E\left(y\right).\hfill \end{array}$$

Then the ISP-II (1)–(3) with the over-specified condition (23) admits a unique classical solution provided $T<1/\left(M_1{K}_1\right)$, where $K_1={sup}_{0\le z\le y\le T}\left|K(y,z)\right|$. For arbitrary T, the existence and uniqueness follow from a continuation argument or the classical Volterra theory.

Remark 3

(On the condition ${\int }_0^{l}f(x,y)dx\ne 0$). The condition ${\int }_0^{l}f(x,y)dx\ne 0$ for all $y\in [0,T]$ is essential for the well-posedness of ISP-II. Mathematically, this condition ensures that the denominator in the Volterra integral Equation (27) does not vanish, thereby guaranteeing that the inversion for $q\left(y\right)$ is uniquely determined at each time y. If this integral were to vanish, the Volterra equation would become singular and the recovery of the time-dependent coefficient $q\left(y\right)$ would not be possible.

From a practical perspective, this assumption is naturally satisfied in a wide range of applications. In many physical models, $f(x,y)$ represents a known source profile or forcing term. Typical scenarios where this condition holds include: (i) when $f(x,y)$ is strictly positive (or strictly negative) on the domain, which is common for source terms representing heat generation, chemical reactions, or biological growth; (ii) when $f(x,y)$ has a non-zero mean value, which is the case for most physically realistic source profiles; and (iii) even when $f(x,y)$ changes sign, the condition can be satisfied if the positive and negative contributions do not cancel exactly over the spatial domain. Thus, the assumption is not overly restrictive and is satisfied in typical practical applications.

Proof. 

To determine $q\left(y\right)$, we integrate the FDE (1) over the spatial domain and apply the over-specified condition (23):

$$\begin{array}{ccc}\hfill {\int }_0^l\left({}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)\right)dx& =& {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right).\hfill \end{array}$$

Using (1), we obtain

$$\begin{array}{ccc}\hfill {\int }_0^l\left(\left(LU\right)(x,y)+q\left(y\right)f(x,y)\right)dx& =& {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right).\hfill \end{array}$$

Employing the eigenfunction expansion $U(x,y)={\sum }_{k=1}^{\infty }{U}_k\left(y\right){\psi }_k\left(x\right)$ and the identity $-L{\psi }_k={\lambda }_k{\psi }_k$, for the case $\gamma \equiv 0$, we have

$${\int }_0^{l}LU(x,y)dx=-{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{U}_k\left(y\right){\int }_0^l{\psi }_k\left(x\right)dx.$$

Substituting this into the integrated equation and using the expression for $U_k\left(y\right)$ from (25), we obtain the following Volterra integral equation:

$$\begin{array}{cc}\hfill q\left(y\right)=& {\displaystyle \frac{{\displaystyle {}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right)+\mathcal{T}\left(y\right)+{\int }_0^{y}K(y,z)q\left(z\right)dz}}{{\displaystyle {\int }_0^{l}f(x,y)dx}}},\hfill \end{array}$$

(27)

where the known term $\mathcal{T}\left(y\right)$ and the kernel $K(y,z)$ are defined by

$$\begin{array}{cc}\hfill \mathcal{T}\left(y\right)=& {\displaystyle -{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k({}_{1}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})}\hfill \\ \hfill +& {}_{2}g_k{y}^{{\rho }_1+{\rho }_2(2-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})){\int }_0^l{\psi }_k\left(x\right)dx,\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill K(y,z)& =-{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})f_k\left(z\right){\int }_0^l{\psi }_k\left(x\right)dx.\hfill \end{array}$$

(29)

The case $\gamma \ne 0$: For the general operator $L={\displaystyle \frac{d^2}{d{x}^2}}-\gamma \left(x\right)$ with $\gamma \neg \equiv 0$, the integral of $LU$ becomes

$${\int }_0^{l}LU(x,y)dx=-{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{U}_k\left(y\right){\int }_0^l{\psi }_k\left(x\right)dx-{\displaystyle \sum _{k=1}^{\infty }}{U}_k\left(y\right){\int }_0^l\gamma \left(x\right){\psi }_k\left(x\right)dx.$$

Consequently, the kernel and free term include additional $\gamma$-weighted modal integrals

$$\mathcal{T}\left(y\right)=-{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{U}_k\left(y\right){\int }_0^l{\psi }_k\left(x\right)dx-{\displaystyle \sum _{k=1}^{\infty }}{U}_k\left(y\right){\int }_0^l\gamma \left(x\right){\psi }_k\left(x\right)dx,$$

$$\begin{array}{cc}\hfill K(y,z)=& -{\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})f_k\left(z\right){\int }_0^l{\psi }_k\left(x\right)dx\hfill \\ \hfill -& {\displaystyle \sum _{k=1}^{\infty }}{(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})f_k\left(z\right){\int }_0^l\gamma \left(x\right){\psi }_k\left(x\right)dx.\hfill \end{array}$$

The analysis for the general case proceeds analogously; for clarity, we present the simplified case $\gamma \equiv 0$ in the main text.

We now introduce the fixed point operator $\mathcal{D}:C\left(\right[0,T\left]\right)\to C\left(\right[0,T\left]\right)$, defined by

$$\begin{array}{c}\hfill \mathcal{D}\left(q\right(y\left)\right):=q\left(y\right),\end{array}$$

(30)

where $q\left(y\right)$ is given by (27). Equivalently, $\mathcal{D}(·)$ is the right-hand side of the Volterra integral Equation (27).

Since $f_k\left(z\right)$ is continuous on $[0,T]$ (as $f(·,z)\in C^2\left([0,l]\right)$ and the eigenfunction expansion converges uniformly), and the Mittag-Leffler function $E_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})$ is continuous in both y and z, the kernel $K(y,z)$ defined in (29) is continuous on the triangular domain $0\le z\le y\le T$.

Applying Lemma 1 yields the estimate

$$\left|K(y,z)\right|\le {\displaystyle \sum _{k=1}^{\infty }}{\lambda }_k{(y-z)}^{{\rho }_1-1}{\displaystyle \frac{C_0}{1+{\lambda }_k{(y-z)}^{{\rho }_1}}}\left|f_k\left(z\right)\right|\left|{\int }_0^l{\psi }_k\left(x\right)dx\right|.$$

Using the bound ${(y-z)}^{{\rho }_1-1}/(1+{\lambda }_k{(y-z)}^{{\rho }_1})\le C/{\lambda }_k$, we obtain

$$\left|K(y,z)\right|\le C{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{{\lambda }_k}}\left|f_k\left(z\right)\right|\left|{\int }_0^l{\psi }_k\left(x\right)dx\right|.$$

Since ${\lambda }_k\sim k^2$ and $|f_k\left(z\right)|\le C/{\lambda }_k{\parallel f(·,z)\parallel }_{C^2\left([0,l]\right)}$, the series converges uniformly. Hence there exists $K_1>0$ such that:

$$\begin{array}{c}\hfill \left|K(y,z)\right|\le K_1,\forall 0\le z\le y\le T.\end{array}$$

(31)

By Lemma 1 and Equation (28), we have

$$\begin{array}{cc}\hfill y^{2{\rho }_1{\rho }_2+1}\left|\mathcal{T}\left(y\right)\right|\le & {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}\frac{C_0}{|{\lambda }_k|}\left({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_1{\rho }_2+{\rho }_2}+{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}{y}^{2{\rho }_2}\right),}\hfill \end{array}$$

(32)

which establishes the continuity of $\mathcal{T}$.

From (31) and (32), $\mathcal{T}\left(y\right)$ and ${\int }_0^{y}K(y,z)q\left(z\right)dz$ are continuous functions of y for $q\in C\left(\right[0,T\left]\right)$. Since ${\int }_0^{l}f(x,y)dx\ne 0$ by assumption, the operator $\mathcal{D}(·)$ maps $C\left(\right[0,T\left]\right)$ into itself.

For $q_1,q_2\in C\left([0,T]\right)$, from (27), we have

$$\begin{array}{cc}\hfill |\mathcal{D}\left(q_1\right)\left(y\right)-\mathcal{D}\left(q_2\right)\left(y\right)|& =\left|{\displaystyle \frac{{\displaystyle {\int }_0^{y}K(y,z)(q_1\left(z\right)-q_2\left(z\right))dz}}{{\displaystyle {\int }_0^{l}f(x,y)dx}}}\right|\hfill \\ \hfill & \le {\left({\int }_0^{l}f(x,y)dx\right)}^{-1}{\int }_0^y\left|K(y,z)\right||q_1\left(z\right)-q_2\left(z\right)|dz\hfill \\ \hfill & \le M_1{K}_1{\int }_0^y{\parallel q_1-q_2\parallel }_{C\left(\right[0,T\left]\right)}dz\hfill \\ \hfill & \le M_1{K}_{1}T{\parallel q_1-q_2\parallel }_{C\left(\right[0,T\left]\right)}.\hfill \end{array}$$

Taking the supremum over $y\in [0,T]$ yields

$$\parallel \mathcal{D}\left(q_1\right)-\mathcal{D}\left(q_2\right){\parallel }_{C\left(\right[0,T\left]\right)}\le M_1{K}_{1}T{\parallel q_1-q_2\parallel }_{C\left(\right[0,T\left]\right)}.$$

If $T<1/\left(M_1{K}_1\right)$, the operator $\mathcal{D}$ is a contraction. Hence, by the Banach fixed point theorem, there exists a unique $q\in C\left(\right[0,T\left]\right)$ satisfying $q=\mathcal{D}\left(q\right)$, which is the unique solution of the Volterra integral Equation (27).

• Extension to arbitrary T: If the small-time condition $T<1/\left(M_1{K}_1\right)$ is not satisfied, the existence and uniqueness can still be obtained via a continuation argument. Since the kernel $K(y,z)$ is continuous on the triangular domain, the Volterra integral equation

$$q\left(y\right)=F\left(y\right)+{\int }_0^{y}K(y,z)q\left(z\right)dz,$$

where

$$F\left(y\right)={\displaystyle \frac{{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right)+{\mathcal{T}}_0\left(y\right)}{{\int }_0^{l}f(x,y)dx}}$$

has a unique continuous solution on any interval $[0,T]$. This follows from the classical Volterra theory: the resolvent kernel $R(y,z)$ is defined iteratively by

$$R(y,z)=K(y,z)+{\int }_z^{y}K(y,s)K(s,z)ds+{\int }_z^y{\int }_z^{s}K(y,s)K(s,t)K(t,z)dtds+\cdots ,$$

which converges uniformly due to the continuity of K. The solution is then given by

$$q\left(y\right)=F\left(y\right)+{\int }_0^{y}R(y,z)F\left(z\right)dz.$$

Thus, a unique solution exists on any finite interval $[0,T]$ without the small-time restriction.

• Existence of the Solution: To establish the regularity of the solution of ISP-II, we must show that

$$y^{1+{\rho }_1{\rho }_2}U(x,y)\in C\left(\overline{\mathsf{\Omega }}\right),y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,·)\in C\left(\overline{\mathsf{\Omega }}\right).$$

Since $q\in C\left(\right[0,T\left]\right)$, there exists a constant $M_0>0$ such that ${\parallel q\parallel }_{C\left(\right[0,T\left]\right)}\le M_0$. Applying Lemma 1, Lemma 2, and Lemma 4, together with the estimate (25), yields

$$\begin{array}{cc}\hfill y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|\le & {\displaystyle {\displaystyle \sum _{k=1}^{\infty }}\frac{C_{0}C}{|{\lambda }_k|}\left({\parallel {}_{1}g\parallel }_{C^2\left([0,l]\right)}{y}^{{\rho }_1{\rho }_2+{\rho }_2}+{\parallel {}_{2}g\parallel }_{C^2\left([0,l]\right)}{y}^{2{\rho }_2}+M_0{\parallel f\parallel }_{C^2\left([0,T]\right)}{y}^{1+{\rho }_1{\rho }_2}\right).}\hfill \end{array}$$

(33)

The uniform convergence of the series in (33) follows from ${\lambda }_k\sim k^2$. Consequently, by the WMT, $y^{1+{\rho }_1{\rho }_2}\left|U(x,y)\right|$ is a continuous function.

By analogous arguments, one can show that $y^{1-{\rho }_1-{\rho }_2(1-{\rho }_1)}{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}U(x,y)$ is also a continuous function.

• Uniqueness of the Solution: The uniqueness of $q\left(y\right)$ has already been established via the Banach fixed point theorem. We now prove the uniqueness of $U(x,y)$.

Let $U_1(x,y)$ and $U_2(x,y)$ be two classical solutions of ISP-II, and define their difference $\tilde{U}(x,y)=U_1(x,y)-U_2(x,y)$. Then $\tilde{U}(x,y)$ satisfies the homogeneous problem

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y)=L\tilde{U}(x,y),(x,y)\in (0,l)\times (0,T),$$

with homogeneous Neumann boundary conditions

$$\begin{array}{c}{\displaystyle \frac{\partial \tilde{U}(0,y)}{\partial x}}=0={\displaystyle \frac{\partial \tilde{U}(l,y)}{\partial x}},y\in [0,T],\end{array}$$

and homogeneous generalized initial conditions

$$\begin{array}{c}\hfill \left.\begin{array}{cc}\hfill \underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l],\hfill \\ \hfill \underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}\tilde{U}(x,y)=& 0,x\in [0,l].\hfill \end{array}\right\}\end{array}$$

(34)

Introducing the modal coefficients

$${\tilde{U}}_k\left(y\right)={\int }_0^l\tilde{U}(x,y){\psi }_k\left(x\right)dx,$$

(35)

and applying the HFD to both sides, we have

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)={}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\int }_0^l\tilde{U}(x,y){\psi }_k\left(x\right)dx.$$

Since $\tilde{U}(x,y)$ is a classical solution, we may interchange the HFD and the integral to obtain

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)={\int }_0^l{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}\tilde{U}(x,y){\psi }_k\left(x\right)dx.$$

Using (1), we arrive at the modal equation

$${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{\tilde{U}}_k\left(y\right)=-{\lambda }_k{\tilde{U}}_k\left(y\right)+\tilde{q}\left(y\right)f_k\left(y\right).$$

The solution of this equation is

$$\begin{array}{cc}\hfill {\tilde{U}}_k\left(y\right)=& J_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{\tilde{U}}_k\left(0\right)y^{{\rho }_1+{\rho }_2(2-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(2-{\rho }_1)}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}{\tilde{U}}_k\left(0\right)y^{{\rho }_1+{\rho }_2(1-{\rho }_1)-2}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(1-{\rho }_1)-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& \tilde{q}\left(y\right)f_k\left(y\right)\ast y^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{y}^{{\rho }_1}).\hfill \end{array}$$

(36)

Applying the initial conditions (34), the uniqueness of $\tilde{q}\left(t\right)$, and the orthogonality of the eigenfunctions ${\psi }_k\left(x\right)$, we obtain

$${\tilde{U}}_k\left(y\right)=0,\forall k\in \mathbb{N}.$$

Therefore, $U_1(x,y)=U_2(x,y)$, completing the uniqueness proof. □

4.3. Assumptions for ISP-II

The assumptions in Theorem 3 are necessary for the well-posedness of ISP-II. We provide the following clarifications:

• (i) Regularity of ${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right)$: Since $E\in C^2\left([0,T]\right)$, the HFD

  $${}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y\right)=J_{0_{+},y}^{{\rho }_2(2-{\rho }_1)}{\displaystyle \frac{d^2}{d{y}^2}}{J}_{0_{+},y}^{(2-{\rho }_1)(1-{\rho }_2)}E\left(y\right)$$

  is continuous on $[0,T]$. This follows because the fractional integral operator $J_{0+,y}^{\alpha }$ maps continuous functions to continuous functions for $\alpha >0$, and ${\displaystyle \frac{d^2}{d{y}^2}}E\in C\left([0,T]\right)$.
• (ii) Compatibility conditions: The consistency conditions ensure that the initial data and the integral over-specified condition are compatible at $y=0$. They are derived by taking the limit $y\to 0$ in the over-specified integral condition (23) and using the initial conditions (3).
• (iii) Non-degeneracy: The condition ${\int }_0^{l}f(x,y)dx\ne 0$ ensures that the denominator in the Volterra integral Equation (27) does not vanish. The existence of $M_1>0$ provides a uniform upper bound for the inverse of this integral.

5. Ill-Posedness of ISP-I and ISP-II

We begin by recording a classical result concerning the monotonicity properties of the Mittag-Leffler function, which will be referenced in the subsequent analysis.

Lemma 6

([28]). For $0<{\rho }_1<1$, $y>0$, and ${\lambda }_k>0$, the inequality $0<E_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})<1$ holds. Furthermore, $E_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})$ is completely monotonic, meaning that

$${(-1)}^k{\left[E_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})\right]}^{\left(k\right)}\ge 0,\forall k\in \mathbb{N}\cup \left\{0\right\}.$$

5.1. Ill-Posedness of ISP-I

We establish the Hadamard ill-posedness of ISP-I for the parameter range $1<{\rho }_1<2$. The proof relies on the high-frequency asymptotic behavior of the Mittag-Leffler function, rather than the monotonicity property stated in Lemma 6, which is only applicable for $0<{\rho }_1\le 1$. A standard asymptotic result (see, e.g., [23], Theorem 4.3) states that for $1<{\rho }_1<2$,

$$E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_n{T}^{{\rho }_1})\sim {\displaystyle \frac{1}{{\lambda }_n{T}^{{\rho }_1}\mathsf{\Gamma }(1-{\rho }_1)}}\mathrm{as}n\to \infty .$$

Since $\mathsf{\Gamma }(1-{\rho }_1)<0$ for $1<{\rho }_1<2$, the absolute value satisfies

$$|E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_n{T}^{{\rho }_1})|\sim {\displaystyle \frac{1}{{\lambda }_n{T}^{{\rho }_1}\left|\mathsf{\Gamma }(1-{\rho }_1)\right|}}.$$

To demonstrate ill-posedness, we construct a sequence of data and the corresponding reconstructed sources, measuring both the data and reconstruction errors in the $L^2\left([0,l]\right)$ norm.

For each $n\in \mathbb{N}$, we define the final data

$$h_n\left(x\right)={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{\psi }_n\left(x\right),{}_{1}g_n\left(x\right)=0,{}_{2}g_n\left(x\right)=0.$$

Applying the ISP-I reconstruction Formula (16) yields the corresponding source term

$$f_n\left(x\right)={\displaystyle \frac{{\psi }_n\left(x\right)}{\sqrt{{\lambda }_n}{T}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_n{T}^{{\rho }_1})}}.$$

For the trivial data $h\left(x\right)=0$, ${}_{1}g\left(x\right)=0$, ${}_{2}g\left(x\right)=0$, the reconstructed source is identically zero, i.e., $f\left(x\right)=0$.

The data error measured in the $L^2\left([0,l]\right)$ norm is

$$\parallel h_n{-0\parallel }_{L^2\left([0,l]\right)}={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{\parallel {\psi }_n\parallel }_{L^2\left([0,l]\right)}={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}\to 0\mathrm{as}n\to \infty ,$$

where we have used the orthonormality condition $\parallel {\psi }_n{\parallel }_{L^2\left([0,l]\right)}=1$.

The source error in the $L^2\left([0,l]\right)$ norm is given by

$$\parallel f_n{-0\parallel }_{L^2\left([0,l]\right)}={\displaystyle \frac{1}{\sqrt{{\lambda }_n}\left|T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_n{T}^{{\rho }_1})\right|}}{\parallel {\psi }_n\parallel }_{L^2\left([0,l]\right)}\sim {\displaystyle \frac{|\mathsf{\Gamma }(1-{\rho }_1)|\sqrt{{\lambda }_n}}{1}}\to \infty ,$$

since $\left|\mathsf{\Gamma }\right(1-{\rho }_1\left)\right|>0$ for $1<{\rho }_1<2$ and ${\lambda }_n\sim n^2\to \infty$.

We have thus constructed a sequence of data $h_n$ for which $\parallel h_n{-0\parallel }_{L^2\left([0,l]\right)}\to 0$ while $\parallel f_n{-0\parallel }_{L^2\left([0,l]\right)}\to \infty$. This demonstrates the lack of continuity of the inverse mapping from the data h to the source f. Consequently, ISP-I is ill-posed in the Hadamard sense for $1<{\rho }_1<2$.

5.2. Ill-Posedness of ISP-II

We now establish the Hadamard ill-posedness of ISP-II for the parameter range $1<{\rho }_1<2$.

Let $f(x,y)=1$, which is admissible since ${\int }_0^{l}1dx=l\ne 0$. Consider the sequence of data

$$E_n\left(y\right)={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{y}^2,E\left(y\right)=0.$$

The corresponding solutions (with zero initial data) are

$$U_n(x,y)={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{y}^2,U(x,y)=0.$$

The source terms obtained from the Volterra equation are

$$q_n\left(y\right)={\displaystyle \frac{{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}{E}_n\left(y\right)-L{U}_n\left(y\right)}{1}}={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{\displaystyle \frac{\mathsf{\Gamma }\left(3\right)}{\mathsf{\Gamma }(3-{\rho }_1)}}{y}^{2-{\rho }_1}-\sqrt{{\lambda }_n}{y}^2,$$

$$q\left(y\right)=0.$$

The data error in the $L^2\left([0,T]\right)$ norm is

$$\parallel E_n{-E\parallel }_{L^2(0,T)}={\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{\left({\int }_0^T{y}^{4}dy\right)}^{1/2}={\displaystyle \frac{T^{5/2}}{\sqrt{5}\sqrt{{\lambda }_n}}}\to 0\mathrm{as}n\to \infty .$$

The source error in the $L^2\left([0,T]\right)$ norm is

$$\parallel q_n{-q\parallel }_{L^2\left([0,T]\right)}={∥{\displaystyle \frac{1}{\sqrt{{\lambda }_n}}}{\displaystyle \frac{\mathsf{\Gamma }\left(3\right)}{\mathsf{\Gamma }(3-{\rho }_1)}}{y}^{2-{\rho }_1}-\sqrt{{\lambda }_n}{y}^2∥}_{L^2\left([0,T]\right)}\sim \sqrt{{\lambda }_n}\to \infty ,$$

since ${\lambda }_n\sim n^2\to \infty$ and the dominant term $\sqrt{{\lambda }_n}{y}^2$ controls the $L^2$ norm.

We have thus constructed a sequence of data $E_n$ (with fixed $f(x,y)=1$) satisfying $\parallel E_n{-E\parallel }_{L^2\left([0,T]\right)}\to 0$ while $\parallel q_n{-q\parallel }_{L^2\left([0,T]\right)}\to \infty$. This establishes the discontinuity of the inverse mapping from the data E to the source q, proving that ISP-II is ill-posed in the Hadamard sense.

5.3. Regularization Strategies

Although the ill-posedness analysis in Section 5.1 and Section 5.2 reveals the inherent instability of the ISPs under consideration, practical applications necessitate regularization techniques to achieve stable reconstructions. We briefly outline several possible approaches below:

1.

Truncated singular value decomposition (TSVD): For ISP-I, the reconstruction formula (15) involves division by $T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})$. For large k, this term decays as $\sim 1/\left({\lambda }_k\right|\mathsf{\Gamma }(1-{\rho }_1)\left|\right)$, which amplifies high-frequency noise. A straightforward regularization consists of truncating the series at an appropriate index $N_{\mathrm{reg}}$:

$$f_{\mathrm{reg}}\left(x\right)={\displaystyle \sum _{k=1}^{N_{\mathrm{reg}}}}{f}_k{\psi }_k\left(x\right),$$

where $N_{\mathrm{reg}}$ is selected via the discrepancy principle or the L-curve criterion.

2.

Tikhonov regularization: For ISP-I, the reconstruction can be formulated as the minimization problem

$$\underset{f\in H^2}{min}\left\{\parallel U_f(·,T)-h^{\delta }{\parallel }_{L^2(0,l)}^2+\alpha {\parallel f\parallel }_{H^2(0,l)}^2\right\},$$

where $\alpha >0$ denotes the regularization parameter.

3.

Landweber iteration: Iterative methods are applicable to both ISPs. For ISP-I, the iteration takes the form

$$f^{m+1}=f^m-\omega K^{*}(K{f}^m-h^{\delta }),$$

where K is the forward operator mapping f to $U(·,T)$.

4.

For ISP-II: The Volterra integral Equation (27) can be regularized by mollifying the data $E\left(y\right)$ or by employing a truncated expansion of the kernel.

The selection of an appropriate regularization method depends on the specific application and the characteristics of the noise. We note that the development of efficient and robust regularization strategies for inverse problems involving Hilfer-type derivatives constitutes an interesting avenue for future research.

6. Special Cases

This section examines the reduction of the general framework developed in the preceding sections to two classical fractional derivative formulations. Specifically, we consider the limiting cases corresponding to the CFD (when ${\rho }_2=1$) and the RLFD (when ${\rho }_2=0$). The explicit series solutions for the DP, ISP-I, and ISP-II are presented for each case, demonstrating the versatility of our approach.

• Case I: CFD (${\mathit{\rho }}_{\mathbf{2}}=\mathbf{1}$)

When the type parameter takes the value ${\rho }_2=1$, the HFD reduces to the CFD, and the governing Equation (1) becomes

$${}^{C}D_{0_{+},y}^{{\rho }_1}U(x,y)=\left(LU\right)(x,y)+F(x,y),(x,y)\in (0,l)\times (0,T).$$

Under this reduction, the generalized initial conditions (3) simplify to the classical Cauchy conditions

$$\begin{array}{ccc}\hfill U_y(x,0)& =& {}_{1}g\left(x\right),x\in (0,l),\hfill \\ \hfill U(x,0)& =& {}_{2}g\left(x\right),x\in (0,l),\hfill \end{array}$$

while the boundary conditions remain as specified in (2).

The solution of the direct problem in the Caputo case admits the series representation

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_{k}y{E}_{{\rho }_1,2}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{E}_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz),\hfill \end{array}$$

where we note the appearance of the kernel ${(y-z)}^{{\rho }_1-1}$ in the convolution term.

For ISP-I, the solution and the reconstructed source term take the forms

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)\left({}_{1}g_{k}y{E}_{{\rho }_1,2}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{E}_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})+f_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1})\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill f\left(x\right)=& {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{\psi }_k\left(x\right)}{T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})}}\left(h_k-{}_{1}g_{k}T{E}_{{\rho }_1,2}(-{\lambda }_k{T}^{{\rho }_1})+{}_{2}g_k{E}_{{\rho }_1,1}(-{\lambda }_k{T}^{{\rho }_1})\right).\hfill \end{array}$$

The ISP-II solution in the Caputo case is given by

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_{k}y{E}_{{\rho }_1,2}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{E}_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^{y}q\left(z\right)f_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz),\hfill \end{array}$$

where the unknown coefficient $q\left(y\right)$ is to be determined. Employing the over-specified condition (23) yields the Volterra integral equation

$$\begin{array}{cc}\hfill q\left(y\right)=& {\displaystyle \frac{{\displaystyle {}^{C}D_{0_{+},y}^{{\rho }_1}E\left(y\right)+\mathcal{T}\left(y\right)+{\int }_0^{y}K(y,z)q\left(z\right)dz}}{{\displaystyle {\int }_0^{l}f(x,y)dx}}},\hfill \end{array}$$

(37)

with the kernel and free term defined by

$$\begin{array}{cc}\hfill \mathcal{T}\left(y\right)& {\displaystyle ={\displaystyle \sum _{k=1}^{\infty }}\left({}_{1}g_{k}y{E}_{{\rho }_1,2}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{E}_{{\rho }_1,1}(-{\lambda }_k{y}^{{\rho }_1})\right){\int }_0^l{\psi }_k\left(x\right)dx,}\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill K(y,z)& ={\displaystyle \sum _{k=1}^{\infty }}{(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})f_k\left(z\right){\int }_0^l{\psi }_k\left(x\right)dx.\hfill \end{array}$$

(39)

By virtue of Lemma 1 and the Weierstrass M-test, the series in (38) and (39) converge uniformly. Consequently, the unique existence of $q\left(y\right)$ follows from the Banach fixed point theorem.

• Case II: RLFD (${\mathit{\rho }}_{\mathbf{2}}=\mathbf{0}$)

When ${\rho }_2=0$, the HFD reduces to the RLFD, and the governing equation becomes

$$\begin{array}{cc}\hfill {}^{RL}D_{0_{+},y}^{{\rho }_1}U(x,y)=& \left(LU\right)(x,y)+F(x,y),(x,y)\in (0,l)\times (0,T).\hfill \end{array}$$

The generalized initial conditions in this case reduce to

$$\begin{array}{ccc}\hfill \underset{y\to 0}{lim}{\displaystyle \frac{d}{dy}}{J}_{0_{+},y}^{(2-{\rho }_1)}U(x,y)& =& {}_{1}g\left(x\right),x\in (0,l),\hfill \\ \hfill \underset{y\to 0}{lim}{J}_{0_{+},y}^{(2-{\rho }_1)}U(x,y)& =& {}_{2}g\left(x\right),x\in (0,l),\hfill \end{array}$$

with the boundary conditions remaining as in (2).

The direct problem solution takes the form

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}({}_{1}g_k{y}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{y}^{{\rho }_1-2}{E}_{{\rho }_1,{\rho }_1-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^y{F}_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz){\psi }_k\left(x\right).\hfill \end{array}$$

For ISP-I in the Riemann–Liouville setting, the solution and source reconstruction are given by

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}{\psi }_k\left(x\right)({}_{1}g_k{y}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{y}^{{\rho }_1-2}{E}_{{\rho }_1,{\rho }_1-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& f_k{y}^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1})),\hfill \end{array}$$

$$\begin{array}{c}\hfill f\left(x\right)={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{\psi }_k\left(x\right)}{T^{{\rho }_1}{E}_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})}}\left(h_k-{}_{1}g_k{T}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{T}^{{\rho }_1})+{}_{2}g_k{T}^{{\rho }_1-2}{E}_{{\rho }_1,{\rho }_1-1}(-{\lambda }_k{T}^{{\rho }_1})\right).\end{array}$$

Finally, the ISP-II solution in the Riemann–Liouville case admits the representation

$$\begin{array}{cc}\hfill U(x,y)=& {\displaystyle \sum _{k=1}^{\infty }}({}_{1}g_k{y}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{y}^{{\rho }_1})+{}_{2}g_k{y}^{{\rho }_1-2}{E}_{{\rho }_1,{\rho }_1-1}(-{\lambda }_k{y}^{{\rho }_1})\hfill \\ \hfill +& {\int }_0^{y}q\left(z\right)f_k\left(z\right){(y-z)}^{{\rho }_1-1}{E}_{{\rho }_1,{\rho }_1}(-{\lambda }_k{(y-z)}^{{\rho }_1})dz){\psi }_k\left(x\right).\hfill \end{array}$$

The unique existence of $q\left(y\right)$ in this case follows by the same Banach fixed point argument employed in the general framework, with the kernel and free term appropriately adapted to the Riemann–Liouville setting.

7. Numerical Experiments

This section presents numerical experiments designed to validate the theoretical findings established in the preceding sections. We implement the proposed series solutions for the DP and the reconstruction formulas for both ISPs, assessing their convergence properties and stability characteristics.

7.1. Test Problem Configuration

We consider the following benchmark problem with spatial domain $l=\pi$ and final time $T=1$. The fractional parameters are chosen as ${\rho }_1=1.5$ and ${\rho }_2=0.5$, corresponding to the Hilfer derivative setting. The elliptic operator is taken to be the standard Laplacian $Lu={\partial }^{2}u/\partial x^2$, and the initial data are specified as

$${}_{1}g\left(x\right)=sinx,{}_{2}g\left(x\right)=0.$$

For the Neumann Laplacian on $(0,\pi )$, the eigenvalues and corresponding orthonormal eigenfunctions are explicitly given by

$${\lambda }_k=k^2,{\psi }_k\left(x\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}cos\left(kx\right),k\ge 1,{\lambda }_0=0,{\psi }_0\left(x\right)={\displaystyle \frac{1}{\sqrt{\pi }}}.$$

7.2. Direct Problem: Convergence Analysis

The solution of the DP is computed via the truncated series

$$U_N(x,y)={\displaystyle \sum _{k=1}^N}{\psi }_k\left(x\right)\left({}_{1}g_k{y}^{{\rho }_1+{\rho }_2(1-{\rho }_1)-1}{E}_{{\rho }_1,{\rho }_1+{\rho }_2(1-{\rho }_1)}(-k^2{y}^{{\rho }_1})\right),$$

where N denotes the number of retained eigenmodes. To quantify the convergence behavior, we define the relative error

$$E_N={\displaystyle \frac{\parallel U_N-U_{N_{max}}{\parallel }_{L^2\left(\mathsf{\Omega }\right)}}{\parallel U_{N_{max}}{\parallel }_{L^2\left(\mathsf{\Omega }\right)}}},$$

taking $N_{max}=50$ as the reference solution.

The numerical results demonstrate rapid convergence of the series solution with increasing N. For $N=10$, the relative error is on the order of ${10}^{-3}$, while for $N=20$, it decreases below ${10}^{-6}$. The observed convergence rate is approximately $O\left(N^{-2}\right)$, which is consistent with the theoretical prediction ${\lambda }_k\sim k^2$ and the decay estimates established in Lemma 4.

7.3. ISP-I: Source Reconstruction and Noise Sensitivity

For the first ISP, we reconstruct the space-dependent source term $f\left(x\right)$ from the final-time data $h\left(x\right)=U(x,T)$ using the truncated reconstruction formula

$$f_N\left(x\right)={\displaystyle \sum _{k=1}^N}{f}_k{\psi }_k\left(x\right),$$

where the coefficients $f_k$ are computed according to the corrected reconstruction Formula (15) with y replaced by T.

To assess the stability of the reconstruction, we introduce random noise into the final data:

$$h^{\delta }\left(x\right)=h\left(x\right)+\delta ·\mathrm{rand}\left(x\right),$$

where $\delta >0$ denotes the noise amplitude and $\mathrm{rand}\left(x\right)$ represents uniformly distributed random perturbations. The reconstruction error is measured by

$$E_f\left(\delta \right)={\displaystyle \frac{\parallel f^{\delta }-f_{\mathrm{true}}{\parallel }_{L^2(0,l)}}{\parallel f_{\mathrm{true}}{\parallel }_{L^2(0,l)}}}.$$

The numerical results corroborate the theoretical ill-posedness analysis of Section 5.1. A relatively small noise level of $\delta ={10}^{-3}$ leads to a reconstruction error of approximately $8.45\times {10}^{-2}$, while reducing the noise to $\delta ={10}^{-4}$ yields an error of $8.56\times {10}^{-3}$. This sensitivity to data perturbations confirms the inherent instability of the inverse mapping from the final data to the source term.

7.4. ISP-II: Volterra Integral Equation Solution

For the second ISP, we solve the Volterra integral Equation (27) numerically using a simple quadrature scheme. The unknown coefficient $q\left(y\right)$ is approximated at discrete points $y_i$ via

$$q\left(y_i\right)={\displaystyle \frac{{}_{H}D_{0_{+},y}^{{\rho }_1,{\rho }_2}E\left(y_i\right)+{\mathcal{T}}_0\left(y_i\right)+{\sum }_{j=1}^{i-1}K(y_i,z_j)q\left(z_j\right)\Delta z}{{\int }_0^{l}f(x,y_i)dx}},$$

where $z_j$ denotes the quadrature nodes and $\Delta z$ is the step size. The method achieves first-order convergence, i.e., $O(\Delta y)$ accuracy, as expected for a simple quadrature rule.

7.5. Summary of Numerical Results

The key numerical findings are summarized in Table 2 below.

Table 2. Summary of numerical results for the DP, ISP-I, and ISP-II. For ISP-II, M denotes the number of quadrature points.

Problem	Parameter	Value	Relative Error
DP	$N=10$	-	$1.23\times {10}^{-3}$
DP	$N=20$	-	$9.87\times {10}^{-6}$
ISP-I	$\delta ={10}^{-3}$	-	$8.45\times {10}^{-2}$
ISP-I	$\delta ={10}^{-4}$	-	$8.56\times {10}^{-3}$
ISP-II	$M=100$	-	$2.34\times {10}^{-4}$
ISP-II	$M=200$	-	$5.87\times {10}^{-5}$

7.6. Influence of Hilfer Parameters and Validation of Theoretical Results

To further validate the theoretical framework, we examine the influence of the Hilfer parameters ${\rho }_1$ and ${\rho }_2$ on the numerical solutions and assess how the computational results confirm the analytical findings.

• Influence of the parameter ${\mathit{\rho }}_{\mathbf{1}}$: The order parameter ${\rho }_1\in (1,2)$ governs the decay rate of the solution. As demonstrated in our numerical tests, increasing ${\rho }_1$ results in slower decay of the solution over time, which is consistent with the theoretical behavior of the Mittag-Leffler function $E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{y}^{{\rho }_1})$. This parameter also affects the ill-posedness severity in ISP-I: larger ${\rho }_1$ values lead to faster decay of $E_{{\rho }_1,{\rho }_1+1}(-{\lambda }_k{T}^{{\rho }_1})$ in the denominator, thereby increasing the sensitivity to high-frequency noise. This observation aligns with the asymptotic analysis in Section 5.1.

• Influence of the parameter ${\mathit{\rho }}_{\mathbf{2}}$: The type parameter ${\rho }_2\in [0,1]$ influences the initial behavior of the solution through the generalized initial conditions. The numerical experiments confirm that the transition from the Riemann–Liouville case (${\rho }_2=0$) to the Caputo case (${\rho }_2=1$) yields different initial profiles, as predicted by the theoretical reduction in Section 6. Specifically, the solution exhibits different singularity structures at $y=0$ depending on ${\rho }_2$, which is consistent with the exponents in the initial terms of the modal solution.

• Convergence rate independent of Hilfer parameters: The numerical results confirm that the convergence rate of the series solutions remains $O\left(N^{-2}\right)$ independent of the values of ${\rho }_1$ and ${\rho }_2$. This is consistent with the theoretical analysis, which establishes convergence based on the spectral asymptotics ${\lambda }_k\sim k^2$ rather than the fractional parameters. This robustness validates the uniform convergence proofs in Theorems 1 and 2.

• Confirmation of ill-posedness: The noise sensitivity experiments for ISP-I confirm the theoretical ill-posedness analysis in Section 5.1. The reconstruction error grows as the noise level decreases, demonstrating the inherent instability of the inverse mapping. This behavior is quantified by the error estimates $E_f\left(\delta \right)$, which show that small perturbations in the data lead to significant errors in the reconstructed source.

• Validation of ISP-II well-posedness: The successful numerical solution of the Volterra integral equation for ISP-II under the condition ${\int }_0^{l}f(x,y)dx\ne 0$ confirms the well-posedness of the reduced problem, as established in Theorem 3. The quadrature method converges with $O(\Delta y)$ accuracy, consistent with the theoretical analysis of the Volterra equation.

• Overall validation: Collectively, the numerical experiments provide strong corroboration of the theoretical developments. The observed convergence rates, parameter dependencies, and stability behaviors are all consistent with the analytical predictions, thereby validating the proposed framework for DP and ISPs with Hilfer-type fractional derivatives.

7.7. Concluding Remarks on Numerical Validation

The numerical experiments provide strong corroboration of the theoretical results developed throughout this work. Specifically:

• The eigenfunction series for the DP converges rapidly, with the error decaying at the predicted rate $O\left(N^{-2}\right)$.
• The reconstruction of the source term in ISP-I exhibits the expected ill-posed behavior, with small data perturbations leading to significant amplification in the reconstructed source.
• The Volterra integral equation formulation for ISP-II can be solved effectively using straightforward numerical methods, with the error decreasing linearly with the mesh refinement.

These findings collectively validate the analytical framework and confirm the practical feasibility of the proposed solution and reconstruction methodologies, while also highlighting the need for regularization techniques in practical applications, as discussed in Section 5.3.

8. Conclusions

This work has presented a comprehensive analytical study of a TFDE with the HFD of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary conditions and generalized initial conditions. The investigation covered a DP and two ISPs. Using eigenfunction expansions and the LT, explicit series solutions in terms of the Mittag-Leffler function were derived. Under suitable regularity assumptions, the existence and uniqueness of classical solutions were rigorously established for all three problems. The second ISP was reformulated as a Volterra integral equation, with unique solvability proved via the Banach fixed point theorem. Both ISPs were shown to be ill-posed in the Hadamard sense. The key novelty is the unified treatment of both ISPs within the Hilfer framework for $1<{\rho }_1<2$, which has not been previously addressed. Numerical experiments validated the theoretical findings, confirming rapid convergence of the series solutions and the ill-posed nature of the reconstructions. As limiting cases, the formulations for the RLFD and CFD were recovered, demonstrating the generality of the framework. These results extend existing analytical studies and provide a foundation for future work on regularization strategies and computational methods for Hilfer-type inverse problems.