Determination of Initial Data in the Time-Fractional Pseudo-Hyperbolic Equation

Abstract

We examine a time-fractional pseudo-hyperbolic equation involving positive operators. We explore the determination of initial velocity and perturbation. It is demonstrated that these initial inverse problems are ill posed. Additionally, we prove that under certain conditions, the inverse problems exhibit well-posedness properties. Our focus is on developing a theoretical framework for these initial inverse problems associated with time-fractional pseudo-hyperbolic equations, laying the groundwork for future studies on numerical algorithms to solve these problems. This investigation is crucial for understanding the fundamental behavior of the equations under various initial conditions and perturbations. By establishing a rigorous theoretical framework, we pave the way for future research to focus on practical numerical methods and simulations. Our results provide a deeper insight into the mathematical structure of time-fractional pseudo-hyperbolic equations, ensuring that future computational approaches are built on a solid theoretical foundation.

1. Introduction

Consider a separable Hilbert space $\mathcal{H}$ with operators $\mathcal{L}$ and $\mathcal{M}$, which have discrete spectra ${\left\{{\lambda }_{\xi }\right\}}_{\xi \in \mathcal{I}}$ and ${\left\{{\mu }_{\xi }\right\}}_{\xi \in \mathcal{I}}$, respectively, where $\mathcal{I}$ is a countable index set.

In this work, we investigate the conditions under which inverse problems are solvable for the time-fractional pseudo-hyperbolic equation within the time range $T>0$

$${\mathcal{D}}_t^{\alpha }[u\left(t\right)+\mathcal{L}u\left(t\right)]+\mathcal{M}u\left(t\right)=0,\mathrm{for}\mathrm{all}t\in (0,T),$$

(1)

with initial conditions

$$u\left(0\right)=\phi \in \mathcal{H},$$

(2)

$$u_t\left(0\right)=\psi \in \mathcal{H},$$

(3)

and with the final condition

$$u\left(T\right)=\varphi \in \mathcal{H}.$$

(4)

The statement of the inverse problems $\left(\mathbf{IP}\right)$ is given by the following:

IP1: With the initial functions $\varphi$ and $\phi$ specified, find the pair of functions $(u,\psi )$ that satisfy the problem.

IP2: Given the functions $\psi$ and $\phi$, determine the pair of functions $(u,\varphi )$ that meet the criteria.

Here, ${\mathcal{D}}_t^{\alpha }$ is the Caputo derivative (see [1]) with the order $1<\alpha \le 2$ and, in particular, ${\mathcal{D}}_t^{\alpha }:={\partial }_t^2$ for $\alpha =2$. The definition of the Caputo derivative of order $1<\alpha <2$ for any smooth function g with an absolutely continuous derivative $g^{\prime }$ on the interval $[0,T]$ is as follows:

$${\mathcal{D}}_t^{\alpha }\left[g\right]\left(t\right)={\displaystyle \frac{1}{\Gamma (2-\alpha )}}{\int }_0^t{(t-s)}^{1-\alpha }{g}^{″}\left(s\right)ds\mathrm{for}t\in [0,T],$$

where the gamma function is denoted by $\Gamma (·)$.

The main focus of this study is to identify well-posedness conditions and develop a theoretical framework for inverse problems aimed at determining the initial data for time-fractional pseudo-hyperbolic equations. Traditional diffusion equations often fall short in accurately representing macroscopic behavior when random walks exhibit correlations, non-Gaussian statistics, or non-Markovian “memory” effects. To address this, we will explore how extending the concept of a derivative could enhance the classical diffusion framework—a concept rooted in a rich mathematical history [2]. Furthermore, pseudo-hyperbolic equations of the form (1) are widely used to model various physical phenomena, including heat and mass transfer, nerve conduction, and reaction–diffusion processes [3,4,5]. Inverse problems IP1 and IP2 are significant in real-world contexts, such as determining the initial distribution of contaminants based on present density data. It should be highlighted that backward problems for the wave equation with finite-time observation have been studied extensively in [6,7], and research into time-fractional diffusion-wave equations can be found in [8,9,10,11,12,13,14,15]. The foundational work on the well posedness of the backward problem in time for the time-fractional diffusion equation was conducted in [11]. Building on this, various theoretical and numerical studies have explored backward problems for $0<\alpha <1$. In particular, ref. [12] extended the well-posedness results to non-symmetric elliptic operators in time-fractional diffusion equations, generalizing previous results that focused on symmetric operators. In [9,10], numerical approaches were developed for inverse problems aimed at determining the initial conditions for time-fractional wave equations. Refs. [13,14] focused on recovering initial data using final time measurements. In [13], the well posedness of this inverse problem was theoretically demonstrated, while [14] provided a more comprehensive analysis, incorporating both theoretical and numerical aspects. Lastly, ref. [15] addressed the inverse problem of determining two initial conditions for a time-fractional diffusion-wave equation from partial boundary data, exploring the well posedness from both theoretical and numerical viewpoints. However, studies on inverse problems related to finding initial data for time-fractional pseudo-hyperbolic equations, including those involving positive operators, are lacking. Our paper aims to address this issue by investigating these inverse problems for time-fractional pseudo-hyperbolic equations with general positive operators. We explore generalized solutions represented as series expansions, utilizing methods from nonharmonic analysis (see [16,17,18]). We also assess the convergence of these series. Our main objective is to determine the well-posedness criteria and to develop the theoretical foundation for these inverse problems, which will support the development of numerical algorithms.

The recent progress in numerical methods for solving the fractional wave equation is detailed in references [19,20].

The structure of this paper is as follows. In Section 2, we present our main results and proofs of them. The last section is the conclusion.

2. Main Results and Methods

In this part, we present our main results and their proofs.

Before describing our main results, let us provide the preliminary theorem that is necessary for our investigation.

Theorem 1

([21] Theorem 1.6). Consider $\alpha <2$ and any real number β, with the constraint $\pi \alpha /2<\mu <min\{\pi ,\pi \alpha \}$. For the Mittag-Leffler function given by

$$E_{\alpha ,\beta }\left(z\right)=\sum _{j=0}^{\infty }{\displaystyle \frac{z^j}{\Gamma (\alpha j+\beta )}},$$

(5)

there exists a positive constant $C_0$ such that

$$|E_{\alpha ,\beta }\left(z\right)|\le {\displaystyle \frac{C_0}{1+\left|z\right|}},$$

(6)

for all $\mu \le \left|arg\right(z\left)\right|\le \pi$ and $\left|z\right|\ge 0.$

2.1. Inverse Initial Velocity Problem

In this subsection, we study IP1. First, we note that the Mittag-Leffler function $E_{\alpha ,2}$ has no zeros for $0<\alpha \le {\displaystyle \frac{4}{3}}$ [22] and a finite number of zeros for ${\displaystyle \frac{4}{3}}<\alpha <2$ [23]. For more details on roots of the function $E_{\alpha ,2}$, the reader is referred to [24,25,26]. Let us denote by $\Theta :={\left\{{\theta }_{\eta }\right\}}_{\eta \in \mathcal{B}}$ the set of all negative roots of the function $E_{\alpha ,2}$, where $\mathcal{B}$ is some finite set. Also, let

$$\Lambda :={\left\{{\left(-{\displaystyle \frac{{\theta }_{\eta }}{{\lambda }_{\xi }}}\right)}^{{\displaystyle \frac{1}{\alpha }}}\right\}}_{\xi \in \mathcal{I},\eta \in \mathcal{B}}.$$

Then, we say that the set $\mathbb{T}:={\mathbb{R}}_{-}\backslash \Lambda$ is admissible. Indeed, the set $\mathbb{T}$ is countable because $\Theta$ and $\Lambda$ are countable sets (see [27]).

Throughout this section, we define ${\mathcal{H}}_{\mathcal{L},\mathcal{M}}^{l,m}$ as

$${\mathcal{H}}_{\mathcal{L},\mathcal{M}}^{l,m}:=\{u\in \mathcal{H}:{\mathcal{L}}^l{\mathcal{M}}^{m}u\in \mathcal{H}\},$$

for any $l,m\in \mathbb{R}$. In view of this, we can define ${\mathcal{H}}_{\mathcal{L}}^l,{\mathcal{H}}_{\mathcal{M}}^m$ correspondingly

$${\mathcal{H}}_{\mathcal{L}}^l:={\mathcal{H}}^l:=\{u\in \mathcal{H}:{\mathcal{L}}^{l}u\in \mathcal{H}\},$$

$${\mathcal{H}}_{\mathcal{M}}^m:=\{u\in \mathcal{H}:{\mathcal{M}}^{m}u\in \mathcal{H}\},$$

for any $l,m\in \mathbb{R}$.

For IP1, now we present the main theorem of this section.

Theorem 2.

Assume that $\phi \in \mathcal{H},\varphi \in {\mathcal{H}}^1$, and $T>0$.

(i) 

Let $1<\alpha \le {\displaystyle \frac{4}{3}}$;

(ii) 

Let ${\displaystyle \frac{4}{3}}<\alpha <2$. Suppose that T is a sufficiently large number or from the admissible set $\mathbb{T}$.

A unique solution $\left(u\right(t),\psi )$ to the problem IP1 is available, where u is in $C^{\alpha }([0,T];\mathcal{H})\cap C([0,T];{\mathcal{H}}^1)$ and ψ belongs to $\mathcal{H}$. The form of this solution is

$$\begin{array}{cc}\hfill u\left(t\right)=& \sum _{\xi \in \mathcal{I}}[\left(E_{\alpha ,1}(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha })-{\displaystyle \frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}}{E}_{\alpha ,1}(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{T}^{\alpha })\right){\phi }_{\xi }\hfill \\ \hfill & +{\displaystyle \frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}}{\varphi }_{\xi }]e_{\xi },\hfill \end{array}$$

for all $t\in [0,T]$, and

$$\psi =u_t\left(t\right){|}_{t=0}=\sum _{\xi \in \mathcal{I}}{\displaystyle \frac{{\varphi }_{\xi }-{\phi }_{\xi }{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{T{E}_{\alpha ,2}(-{\lambda }_{\xi }{T}^{\alpha })}}{e}_{\xi }.$$

where ${\phi }_{\xi }={(\phi ,e_{\xi })}_{\mathcal{H}}$ and ${\varphi }_{\xi }={(\varphi ,e_{\xi })}_{\mathcal{H}}$.

Remark 1.

We assume that T is either sufficiently large or belongs to the admissible set $\mathbb{T}$ as per assumption (ii). If neither condition holds, we cannot ensure that

$$E_{\alpha ,2}(-{\lambda }_{\xi }{T}^{\alpha })\ne 0,$$

which is essential for proving the well posedness of IP1. A similar assumption will be used to establish the well posedness of IP2.

Existence. We seek the solution in the following form:

$$u\left(t\right)=\sum _{\xi \in \mathcal{I}}{u}_{\xi }\left(t\right)e_{\xi }.$$

(7)

Substituting (7) into Equation (1) and the conditions (2) and (4), we obtain the following:

$${\mathcal{D}}_t^{\alpha }{u}_{\xi }\left(t\right)+{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{u}_{\xi }\left(t\right)=0,$$

(8)

$$u_{\xi }\left(0\right)={\phi }_{\xi },$$

(9)

$$u_{\xi }\left(T\right)={\varphi }_{\xi }.$$

(10)

According to [28], the solution of (8) can be expressed by the following formula:

$$u_{\xi }\left(t\right)=A_{\xi }{E}_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right)+B_{\xi }t{E}_{\alpha ,2}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right),$$

(11)

where $A_{\xi }$ and $B_{\xi }$ are unknown coefficients. We determine these coefficients using the conditions (9) and (10). Substituting (11) into (9) and (10), we obtain

$$A_{\xi }={\phi }_{\xi },B_{\xi }={\displaystyle \frac{{\varphi }_{\xi }-{\phi }_{\xi }{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}},$$

provided that for all $\xi \in \mathcal{I}$, the following condition holds:

$$E_{\alpha ,2}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{T}^{\alpha }\right)\ne 0.$$

(12)

This condition is guaranteed by assumption (i) or (ii).

Substituting the values of $A_{\xi }$ and $B_{\xi }$ into the Formula (11), we finally obtain

$$\begin{array}{cc}\hfill u_{\xi }\left(t\right)=& \left(E_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right)-{\displaystyle \frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}}{E}_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{T}^{\alpha }\right)\right){\phi }_{\xi }\hfill \\ \hfill & +{\displaystyle \frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}}{\varphi }_{\xi },\hfill \end{array}$$

(13)

for all $\xi \in \mathcal{I}$.

Substituting (13) into (7), we have

$$\begin{array}{cc}\hfill u\left(t\right)=& \sum _{\xi \in \mathcal{I}}[\left(E_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right)-{\displaystyle \frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}}{E}_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{T}^{\alpha }\right)\right){\phi }_{\xi }\hfill \\ \hfill & +{\displaystyle \frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}}{\varphi }_{\xi }]e_{\xi }.\hfill \end{array}$$

(14)

Next, we need to find the function $\psi$ corresponding to condition (3). To achieve this, we calculate the derivative of (14) with respect to t. First, we compute the derivatives of (5), obtaining

$${\displaystyle \frac{d}{dt}}{E}_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right)=-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right),$$

(15)

$${\displaystyle \frac{d}{dt}}\left(t{E}_{\alpha ,2}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right)\right)=E_{\alpha ,1}\left(-{\displaystyle \frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{t}^{\alpha }\right).$$

(16)

Using Formulas (15) and (16), we find the following:

$$\begin{array}{c}\hfill u_t\left(t\right)={\displaystyle \sum _{\xi \in \mathcal{I}}}\left[\frac{{\varphi }_{\xi }-{\phi }_{\xi }{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)-{\phi }_{\xi }\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)\right]e_{\xi }.\end{array}$$

Finally, we obtain

$$\psi =u_t\left(t\right){|}_{t=0}=\sum _{\xi \in \mathcal{I}}\frac{{\varphi }_{\xi }-{\phi }_{\xi }{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-{\lambda }_{\xi }{T}^{\alpha }\right)}{e}_{\xi }.$$

(17)

Convergence. We proceed to show that the series defined in (14) converges, as well as its derivatives ${\mathcal{D}}_t^{\alpha }u$, $\mathcal{L}u$, ${\mathcal{D}}_t^{\alpha }\mathcal{L}u$, $\mathcal{M}u$, and (17).

Let us estimate the function

$$K({\lambda }_{\xi },{\mu }_{\xi },t)=\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}.$$

It has been established in [23] that for large positive y, the following asymptotic holds:

$$E_{\alpha ,2}(-y)=\frac{y^{-1}}{\Gamma (2-\alpha )}+O\left(y^{-2}\right),y\to \infty .$$

(18)

Since estimate (18) implies

$$\underset{y\to \infty }{lim}y{E}_{\alpha ,2}(-y)=\frac{1}{\Gamma (2-\alpha )}>0,$$

and in light of (12), we derive

$$\left|\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right|>C_1,$$

where $C_1$ is a constant. Thus,

$$\left|K({\lambda }_{\xi },{\mu }_{\xi },t)\right|\stackrel{\left(6\right)}{\le }\frac{C_0{T}^{\alpha -1}t\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}}{C_1\left(1+\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}\le C_2{t}^{1-\alpha },$$

(19)

where $C_2=\frac{C_0}{C_1}{T}^{\alpha -1}$.

Now, let us estimate

$$\begin{array}{cc}\hfill {\parallel u\left(t\right)\parallel }_{\mathcal{H}}^2& \le C{\displaystyle \sum _{\xi \in \mathcal{I}}}{\left|{\phi }_{\xi }\right|}^2{\left|E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)\right|}^2\hfill \\ & +C{\displaystyle \sum _{\xi \in \mathcal{I}}}{\left|{\phi }_{\xi }\right|}^2\frac{{\left|t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)\right|}^2}{{\left|T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right|}^2}{\left|E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right|}^2\hfill \\ & +C{\displaystyle \sum _{\xi \in \mathcal{I}}}{\left|{\varphi }_{\xi }\right|}^2\frac{{\left|t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)\right|}^2}{{\left|T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right|}^2}\hfill \\ & \stackrel{\left(6\right),\left(19\right)}{\le }\frac{C}{t^{2\alpha }}(1+t^2){\displaystyle \sum _{\xi \in \mathcal{I}}}\left(\frac{|{\phi }_{\xi }{|}^2}{{\mu }_{\xi }^2}+\frac{|{(\mathcal{L}\phi ,e_{\xi })}_{\mathcal{H}}{|}^2}{{\mu }_{\xi }^2}\right)+\frac{C}{t^{2\alpha }}{\parallel \varphi \parallel }_{\mathcal{H}}^2\hfill \end{array}$$

for any $0<t\le T$. In the case that $\frac{4}{3}<\alpha <2$, this estimate holds when T is sufficiently large, or when T belongs to an admissible set $\mathbb{T}$.

Finally, from the last estimate we obtain

$${\parallel u\parallel }_{C\left(\right[0,T],\mathcal{H})}\le C{\displaystyle \sum _{\xi \in \mathcal{I}}}\left(\frac{|{\phi }_{\xi }{|}^2}{{\mu }_{\xi }^2}+\frac{|{(\mathcal{L}\phi ,e_{\xi })}_{\mathcal{H}}{|}^2}{{\mu }_{\xi }^2}\right)+C{\parallel \varphi \parallel }_{\mathcal{H}}^2.$$

Let us calculate

$$\begin{array}{cc}\hfill \mathcal{L}u\left(t\right)=& {\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)-\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right){\phi }_{\xi }\hfill \\ & +\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{\varphi }_{\xi }]\mathcal{L}{e}_{\xi }\hfill \\ \hfill =& {\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)-\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right){\lambda }_{\xi }{\phi }_{\xi }\hfill \\ & +\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{\lambda }_{\xi }{\varphi }_{\xi }]e_{\xi }.\hfill \end{array}$$

Similarly, we calculate

$$\begin{array}{cc}\hfill \mathcal{M}u\left(t\right)=& {\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)-\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{E}_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\right){\mu }_{\xi }{\phi }_{\xi }\hfill \\ & +\frac{t{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }\right)}{T{E}_{\alpha ,2}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)}{\mu }_{\xi }{\varphi }_{\xi }]e_{\xi }.\hfill \end{array}$$

Next, for ${\mathcal{D}}_t^{\alpha }u\left(t\right)$, we have

$$\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }u\left(t\right)=& {\displaystyle \sum _{\xi \in \mathcal{I}}}{\mathcal{D}}_t^{\alpha }{u}_{\xi }\left(t\right)e_{\xi }=-{\displaystyle \sum _{\xi \in \mathcal{I}}}\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{u}_{\xi }\left(t\right)e_{\xi }\hfill \\ \hfill & =-{\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })-\frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })\right)\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{\phi }_{\xi }\hfill \\ \hfill & +\frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{\varphi }_{\xi }]e_{\xi },\hfill \end{array}$$

we also have

$$\begin{array}{cc}\hfill {\mathcal{D}}_t^{\alpha }\mathcal{L}u\left(t\right)=& -{\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })-\frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })\right)\frac{{\lambda }_{\xi }{\mu }_{\xi }}{1+{\lambda }_{\xi }}{\phi }_{\xi }\hfill \\ \hfill & +\frac{t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}\frac{{\lambda }_{\xi }{\mu }_{\xi }}{1+{\lambda }_{\xi }}{\varphi }_{\xi }]e_{\xi },\hfill \end{array}$$

Similarly to the way the estimate got to ${\parallel u\parallel }_{C\left(\right[0,T];\mathcal{H})}$, it can be demonstrated that

$${\parallel \mathcal{M}u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2=\parallel {\mathcal{D}}_t^{\alpha }{u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2\le {C\parallel \phi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

$$\parallel {\mathcal{D}}_t^{\alpha }{\mathcal{L}u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2\le {C\parallel \phi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

and

$${\parallel \psi \parallel }_{\mathcal{H}}^2\le {C\parallel \phi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

ending the proof.

Uniqueness. Assume that $(u_1,{\psi }_1)$ and $(u_2,{\psi }_2)$ are solutions to the problem described by (1) and (4). Let $u=u_1-u_2$ and $\psi ={\psi }_1-{\psi }_2$. Consequently, $u\left(t\right)$ and $\psi$ satisfy the following equation:

$${\mathcal{D}}_t^{\alpha }[u+\mathcal{L}u]+\mathcal{M}u=0,$$

(20)

with the conditions

$$u\left(0\right)=0,$$

(21)

$$u_t\left(0\right)=\psi ,$$

(22)

and

$$u\left(T\right)=0.$$

(23)

We also have the expressions

$$u_{\xi }\left(t\right)={(u\left(t\right),e_{\xi })}_{\mathcal{H}},\xi \in \mathcal{I},$$

(24)

and

$${\psi }_{\xi }={(\psi ,e_{\xi })}_{\mathcal{H}},\xi \in \mathcal{I}.$$

(25)

Applying the operator ${\mathcal{D}}_t^{\alpha }$ to (24), we obtain

$${\mathcal{D}}_t^{\alpha }{u}_{\xi }\left(t\right)={({\mathcal{D}}_t^{\alpha }u\left(t\right),e_{\xi })}_{\mathcal{H}},\xi \in \mathcal{I}.$$

(26)

Using Formulas (24)–(26) in (20)–(23), and considering the self-adjoint nature of the operators $\mathcal{L}$ and $\mathcal{M}$, we arrive at the following problem:

$${\mathcal{D}}_t^{\alpha }{u}_{\xi }\left(t\right)+\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{u}_{\xi }\left(t\right)=0,$$

(27)

with the conditions

$$u_{\xi }\left(0\right)=0,$$

(28)

$$u_{\xi }^{\prime }\left(0\right)={\psi }_{\xi },$$

(29)

and

$$u_{\xi }\left(T\right)=0.$$

(30)

The problem characterized by (27)–(30) has the solution $(u_{\xi },{\psi }_{\xi })=(0,0)$ for each $\xi \in \mathcal{I}$. Using the results from (24) and (25), we deduce that

$${(u,e_{\xi })}_{\mathcal{H}}=0\mathrm{and}{(\psi ,e_{\xi })}_{\mathcal{H}}=0\mathrm{for}\mathrm{every}\xi \in \mathcal{I}.$$

Due to the basis property of ${\left\{e_{\xi }\right\}}_{\xi \in \mathcal{I}}$ in $\mathcal{H}$, it follows that $u\left(t\right)=0$ for all $t\in [0,T]$ and $\psi =0$. This implies uniqueness.

2.2. Inverse Initial Perturbation Problem

In this subsection, we study IP2. First, we note that the Mittag-Leffler function $E_{\alpha ,1}$ has a finite number of real-valued roots for $1<\alpha <2$ [23]. For more details on roots of the function $E_{\alpha ,1}$, the reader is referred to [22,24,25,26] and references therein.

Let us denote by $K:={\left\{{\kappa }_{\eta }\right\}}_{\eta \in \mathcal{B}}$ the set of all negative zeros of the function $E_{\alpha ,1}$, where $\mathcal{B}$ is some finite set. Also, let

$$\Delta :={\left\{{\left(-\frac{{\kappa }_{\eta }}{{\lambda }_{\xi }}\right)}^{\frac{1}{\alpha }}\right\}}_{\xi \in \mathcal{I},\eta \in \mathcal{B}}.$$

Then, we say that the set

$$\mathbb{X}:={\mathbb{R}}_{-}\backslash \Delta$$

is admissible. Indeed, the set $\mathbb{X}$ is countable because K and $\Delta$ are so (see [27]).

Theorem 3.

Suppose that $T>0$ is a sufficiently large number or from the admissible set $\mathbb{X}$. Let us assume that $\psi \in \mathcal{H},\varphi \in {\mathcal{H}}^1$. A distinct solution $\left(u\right(t),\phi )$ exists for IP2 such that u is in $C^{\alpha }([0,T];\mathcal{H})\cap C([0,T];{\mathcal{H}}^1)$ and φ is an element of $\mathcal{H}$. This solution can be expressed as follows

$$\begin{array}{cc}\hfill u\left(t\right)=& {\displaystyle \sum _{\xi \in \mathcal{I}}}[\left(t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })-\frac{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })\right){\psi }_{\xi }\hfill \\ \hfill & +\frac{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{\varphi }_{\xi }]e_{\xi },\hfill \end{array}$$

for all $t\in [0,T]$, and

$$\phi ={\displaystyle \sum _{\xi \in \mathcal{I}}}\frac{{\varphi }_{\xi }-{\psi }_{\xi }T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{e}_{\xi },$$

where ${\psi }_{\xi }={(\psi ,e_{\xi })}_{\mathcal{H}}$ and ${\varphi }_{\xi }={(\varphi ,e_{\xi })}_{\mathcal{H}}$.

Existence. We seek the solution as in the previous section, i.e., in the form of (7). Substituting (7) into Equation (1) and conditions (3) and (4), we obtain the ordinary differential Equation (8) with the following conditions:

$$u_{\xi }^{\prime }\left(0\right)={\psi }_{\xi },$$

(31)

and (10).

As we know from section IP1, the solution of (8) is given by (11). In (11), $A_{\xi }$ and $B_{\xi }$ are unknown coefficients, which we determine by using conditions (10) and (31). Before using (31), let us compute the derivative of (11) with respect to t:

$$u_{\xi }^{\prime }\left(t\right)=-A_{\xi }\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha -1}{E}_{\alpha ,\alpha }(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })+B_{\xi }{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha }).$$

(32)

Here, we use the Formulas (15) and (16). Now, substituting (32) into (31), we obtain

$$B_{\xi }={\psi }_{\xi }.$$

(33)

Substituting (11) into (10) and using (33), we find

$$A_{\xi }=\frac{{\varphi }_{\xi }-{\psi }_{\xi }T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })},$$

(34)

provided that

$$E_{\alpha ,1}\left(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha }\right)\ne 0.$$

This can be guaranteed by assumption (i) or (ii).

Substituting (33) and (34) into (11), we have for all $\xi \in \mathcal{I}$:

$$u_{\xi }\left(t\right)=\left(t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })-\frac{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })\right){\psi }_{\xi }+\frac{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{\varphi }_{\xi }.$$

(35)

Finally, we obtain

$${\phi }_{\xi }=\frac{{\varphi }_{\xi }-{\psi }_{\xi }T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}$$

(36)

for all $\xi \in \mathcal{I}$.

Thus, we obtain the solution

$$u\left(t\right)={\displaystyle \sum _{\xi \in \mathcal{I}}}\left[\left(t{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })-\frac{T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E}_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })\right){\psi }_{\xi }+\frac{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{t}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{\varphi }_{\xi }\right]e_{\xi }.$$

Taking into account the initial condition (2), we find

$$\phi ={\displaystyle \sum _{\xi \in \mathcal{I}}}\frac{{\varphi }_{\xi }-{\psi }_{\xi }T{E}_{\alpha ,2}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{E_{\alpha ,1}(-\frac{{\mu }_{\xi }}{1+{\lambda }_{\xi }}{T}^{\alpha })}{e}_{\xi }.$$

(37)

Convergence. For sufficiently large $T>0$ or for T within the admissible set $\mathbb{X}$, the final series holds true, as the Mittag-Leffler function $E_{\alpha ,1}$ has a limited number of real zeros when $1<\alpha <2$ [27]. The convergence of the series $u\left(t\right)$, $\mathcal{L}u\left(t\right)$, ${\mathcal{D}}_t^{\alpha }u\left(t\right)$, ${\mathcal{D}}_t^{\alpha }\mathcal{L}u\left(t\right)$, and $\phi$ can be proven in a manner similar to Theorem 2. Specifically, it can be demonstrated that

$${\parallel u\parallel }_{C\left(\right[0,T];\mathcal{H})}\le {C\parallel \psi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{\mathcal{H}}^2,$$

$${\parallel \mathcal{M}u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2=\parallel {\mathcal{D}}_t^{\alpha }{u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2\le {C\parallel \psi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

$$\parallel {\mathcal{D}}_t^{\alpha }{\mathcal{L}u\parallel }_{C\left(\right(0,T];\mathcal{H})}^2\le {C\parallel \psi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

and

$${\parallel \phi \parallel }_{\mathcal{H}}^2\le {C\parallel \psi \parallel }_{\mathcal{H}}^2+C{\parallel \varphi \parallel }_{{\mathcal{H}}^1}^2,$$

thus concluding the proof.

Uniqueness. Now, we proceed to demonstrate the uniqueness of the solution for IP2. We set $\psi =\varphi \equiv 0$. From Equations (35)–(37), we derive

$${(u\left(t\right),e_{\xi })}_{\mathcal{H}}=0,{(\phi ,e_{\xi })}_{\mathcal{H}}=0,$$

for all $\xi \in \mathcal{I}$. Given the completeness of the basis ${e_{\xi }}_{\xi \in \mathcal{I}}$ in $\mathcal{H}$, we conclude $u\left(t\right)\equiv 0$ and $\phi \equiv 0$, thus proving the uniqueness.

3. Conclusions

This research explores a time-fractional pseudo-hyperbolic equation involving positive operators. We investigated how to determine the initial velocity and perturbation, revealing that these initial inverse problems are ill posed. Additionally, we proved that, under certain conditions, these problems can be well posed. The theoretical aspects of these initial inverse problems for time-fractional pseudo-hyperbolic equations are developed here, providing a foundation for future work on numerical algorithms to solve these problems.

Several important questions in this paper require additional investigation. First, we aim to achieve analogous results for the equation model (1) with a non-trivial right-hand side of the form $f(t,u(t\left)\right)$. Second, we will study the ill-posedness behavior of problems IP1 and IP2 when T is outside the admissible sets. Thirdly, we consider the problems IP1 and IP2 involving the operator $\mathcal{L}$, which exhibits a continuous spectrum.