Inverse Problem for a Pseudoparabolic Equation with a Non-Self-Adjoint Involutive Second-Order Differential Operator

Abstract

In this paper, we consider a partial differential equation with mixed derivatives of first order in time and second order in the spatial variable. Such equations are usually referred to as one-dimensional pseudoparabolic equations. We prove the existence and uniqueness of a classical solution to problems for a pseudoparabolic equation with a second-order differential operator involving pure involution, under certain requirements imposed on the initial data. The possibility of applying the Fourier method is based on the Riesz basis property of the eigenfunctions of the considered non-self-adjoint second-order differential operator with pure involution. Bessel-type inequalities are established for new systems of functions. The presence of a Bessel inequality for Fourier coefficients facilitates the proof of the uniform convergence of differentiated Fourier series. The solutions are obtained explicitly in the form of a Fourier series. Such representations can be used for performing numerical computations.

1. Introduction

In connection with the needs of the solvability theory for equations of mathematical physics, a large number of works have appeared devoted to the properties of eigenfunctions of ordinary differential operators. Nevertheless, many questions of the spectral theory of differential operators with an involution still remain unresolved. Apparently, the main results of the spectral theory for differential operators with an involution can be found in works [1,2,3,4,5,6,7,8,9]. The results of these studies have been used by various authors for different purposes [10,11,12,13].

Let us write the second-order differential equation with an involution in the following form:

$$\beta u^{″}\left(x\right)+\alpha u^{″}(-x)=\lambda u\left(x\right),-1<x<1,$$

which we shall refer to as an equation with an involution, where $\alpha$ and $\beta$ are certain constants. In the particular case $\beta \equiv 0$, this relation will be called an equation with a pure involution. Sometimes, spectral problems for equations with an involution will be referred to as generalized spectral problems. The main question in the theory of boundary value problems with a spectral parameter is the question of whether the eigenfunctions form a basis.

In the spectral theory of ordinary differential operators, the following fundamental result is well known.

Theorem 1

([14]). The eigenfunctions of a self-adjoint ordinary differential operator form a complete orthonormal system in $L_2\left(-1,1\right)$.

The stated property of self-adjoint operators turns out to be convenient when studying direct and inverse problems for partial differential equations with differential operators, including those with involution.

In works [15,16,17], inverse initial–boundary value problems for the heat-conduction equation of fractional order in time with an involutive perturbation were investigated. Papers [18,19] are devoted to the study of inverse problems for heat-conduction equations with an involutive perturbation.

In recent years, researchers have been increasingly interested in problems described by pseudoparabolic equations [20,21], including those with involutive perturbation [22,23,24,25]. However, inverse problems for pseudoparabolic equations and their fractional analogues have been studied relatively little.

In the present work, a special class of inverse problems generated by pseudoparabolic equations is considered. These equations are characterized by the presence of an involution term and non-self-adjoint boundary conditions.

Compared with existing works:

-

unlike refs. [22,23,24,25] (direct problems of pseudoparabolic equations with involution), we focus on inverse problems for source term determination;

-

differing from [15,16,17,18,19] (heat equations with involution), we address well-posed pseudoparabolic equations with unbounded involutive operators;

-

we establish Bessel-type inequalities for new function systems to ensure uniform convergence of differentiated Fourier series.

In recent decades, mixed non-local problems for a wide class of partial differential equations of parabolic, hyperbolic, pseudoparabolic, and pseudohyperbolic types have attracted significant attention from scientists (see, for example, [26,27,28,29,30,31,32,33,34]). These problems are primarily associated with new mathematical models in engineering, physics, biology, ecology, and other fields.

A number of difficulties arise in the study of problems for partial differential equations with non-self-adjoint differential operators, since their eigenvector systems are non-orthogonal. The concept of orthogonality is closely related to the space $L_2$. In the case of non-orthogonal systems, the concept of biorthogonal systems is introduced (see, for example, [35]).

Two systems of functions $\left\{{\psi }_n\left(x\right)\right\}$ and $\left\{{\mathsf{\Psi }}_n\left(x\right)\right\}$ from $L_2\left(-1,1\right)$ form a biorthogonal system if

$$\left({\psi }_n\left(x\right),{\mathsf{\Psi }}_k\left(x\right)\right)=\underset{-1}{\overset{1}{\int }}{\psi }_n\left(x\right){\mathsf{\Psi }}_k\left(x\right)dx={\delta }_{kn}=\left\{\begin{array}{c}1,n=k,\hfill \\ 0,n\ne k.\hfill \end{array}\right.$$

In this case, the system $\left\{{\mathsf{\Psi }}_n\left(x\right)\right\}$ is called conjugate to the system and $\left\{{\psi }_n\left(x\right)\right\}$.

A system is called minimal if none of the functions in this system belong to the linear span of the remaining functions of the system.

A biorthogonal expansion of a function $f\left(x\right)$ from the class $L_2\left(-1,1\right)$ with respect to the system $\left\{{\psi }_n\left(x\right)\right\}$ is the series

$$f\left(x\right)\sim {\displaystyle \sum _{n=1}^{\infty }}\left(f\left(x\right),{\mathsf{\Psi }}_n\left(x\right)\right){\psi }_n\left(x\right).$$

Similarly, a biorthogonal series with respect to the system $\left\{{\mathsf{\Psi }}_n\left(x\right)\right\}$ is defined as

$$f\left(x\right)\sim {\displaystyle \sum _{n=1}^{\infty }}\left(f\left(x\right),{\psi }_n\left(x\right)\right){\mathsf{\Psi }}_n\left(x\right).$$

A system $\left\{{\psi }_n\left(x\right)\right\}$ is called a basis in the space $L_2\left(-1,1\right)$ if for every function $f\left(x\right)$ from the class $L_2\left(-1,1\right)$ there exists a unique series ${\displaystyle \sum _{n=1}^{\infty }}{C}_n{\psi }_n\left(x\right)$, which converges to the function $f\left(x\right)$ in the norm of the space $L_2\left(-1,1\right)$.

2. Second-Order Differential Operator with Pure Involution and Non-Self-Adjoint Boundary Conditions

In paper [1], the basis property in the class $L_2\left(-1,1\right)$ of the eigenfunctions of a second-order differential operator with pure involution

$$-u^{″}\left(-x\right)=\lambda u\left(x\right),-1<x<1,$$

(1)

with non-self-adjoint boundary conditions

$$u^{\prime }(-1)=u^{\prime }\left(1\right),u(-1)=0$$

(2)

was established. The problem has two series of eigenvalues:

$${\lambda }_{k1}=-{\left(k\pi \right)}^2,{\lambda }_{k2}={\left(k\pi \right)}^2,k=0,1,2,\dots .$$

(3)

The system of non-orthogonal eigenfunctions of the spectral problem (1), (2) has the form

$$u_0\left(x\right)=x+1,u_{k1}\left(x\right)=sink\pi x,u_{k2}\left(x\right)={(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x.$$

(4)

The function $u_0\left(x\right)$ corresponds to the eigenvalue $\lambda =0$. The remaining eigenfunctions correspond to the non-zero eigenvalues $k=1,2,\dots$.

In a standard way, one writes the eigenfunctions

$$v_0\left(x\right)={\displaystyle \frac{1}{2}},v_{k2}\left(x\right)=cosk\pi x,v_{k1}\left(x\right)={(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x,k=1,2,\dots .$$

(5)

of the conjugate problem

$$-v^{″}(-x)=\overline{\lambda }v\left(x\right),v^{\prime }(-1)=0,v(-1)=v\left(1\right).$$

(6)

The following statement holds.

Theorem 2

([1]). Systems (4) and (5) are complete and minimal biorthogonally conjugate systems. Each of the systems (4) and (5) is a Riesz basis.

Every function from the class $L_2\left(-1,1\right)$ has two expansions with respect to the biorthogonally conjugate systems (4) and (5).

3. A Pseudoparabolic Problem with Pure Involution in the Variable x

We now turn to the study of inverse problems for pseudoparabolic equations. As the previous section shows, a second-order linear differential operator with pure involution is unbounded from above and below. Problems for the heat equation with such an operator may turn out to be ill-posed. However, problems for pseudoparabolic equations with second-order linear differential operators with pure involution turn out to be well posed and solvable.

The spatial operator in Equation (7) is identical to that in (1), Section 2, with the same $L_2\left(-1,1\right)$ domain and non-self-adjoint boundary conditions, justifying direct inheritance of the Riesz basis property.

The result presented in the previous section provides justification for applying the Fourier method to the following problem.

Let us consider the problem for a pseudoparabolic equation with involution

$$u_t\left(x,t\right)-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left[u_{xx}\left(-x,t\right)\right]-u_{xx}\left(-x,t\right)=f\left(x\right),-1<x<1,t>0,$$

(7)

with initial data

$$u\left(x,0\right)=\phi \left(x\right),-1<x<1,$$

(8)

and overdetermination conditions

$$u\left(x,T\right)=\psi \left(x\right),-1<x<1,$$

(9)

as well as with non-self-adjoint boundary conditions

$$u\left(-1,t\right)=0,u_x\left(-1,t\right)=u_x\left(1,t\right),0<t<T.$$

(10)

Note that the initial conditions (8), (9) and boundary conditions (10) are related by compatibility conditions. It will be shown below that the initial conditions completely determine the solution of the equation. Only $\phi$, $\psi$ satisfying these conditions guarantee $f\left(x\right)\in C(-1,1)$ via Bessel inequalities (Lemma 3, Corollary 1).

Inverse problem. The task is to find a pair of functions $\left\{u\left(x,t\right),f\left(x\right)\right\}$ satisfying conditions (7)–(10).

We introduce the definition of a solution to the inverse problem. A pair of functions $\left\{u\left(x,t\right),f\left(x\right)\right\}$ is called a classical solution of the inverse problem; if $u\left(x,t\right)\in C_{x,t}^{2,1}\left(\left(-1,1\right),\left[0,T\right]\right)\cap C_x^1\left(\left[-1,1\right]\right),f\left(x\right)\in C\left(-1,1\right)$, they satisfy conditions (7)–(10).

Let us prove the following lemma.

Lemma 1.

(Formal Representation of the Candidate Solution). The formal solution of the problem (7)–(10) has the following form

$$\begin{array}{c}f\left(x\right)={\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[\left({\psi }_{k1}-e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k1}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}sink\pi x\right]\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left[\left({\psi }_{k2}-e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k2}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right],\hfill \end{array}$$

(11)

$$\begin{array}{c}u\left(x,t\right)=\left({\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k1}\right)sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k2}\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

The formulas for calculating the Fourier coefficients ${\phi }_0,{\psi }_0,{\phi }_{kj},{\psi }_{kj}$ will be given below.

Proof. 

Since the function $f\left(x\right)$ is continuous, and the system (4) forms a Riesz basis in $L_2\left(-1,1\right)$, it has a biorthogonal expansion with respect to the system (4). For any function $f\left(x\right)\in L_2\left(-1,1\right)$, this expansion converges to the function $f\left(x\right)$ in the norm of $L_2\left(-1,1\right)$. The function $f\left(x\right)$ also has another convergent biorthogonal expansion with respect to the system (5). Since systems (4) and (5) are equivalent, we may choose either of them. In this work, the biorthogonal expansion with respect to the system (4) is chosen. The Fourier coefficients are calculated using the following formulas:

$$\begin{array}{c}{f}_0={\displaystyle \frac{1}{2}}\underset{-1}{\overset{1}{\int }}f\left(x\right)dx,\hfill \\ f_{k1}=\underset{-1}{\overset{1}{\int }}f\left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx,\hfill \\ f_{k2}=\underset{-1}{\overset{1}{\int }}f\left(x\right)cosk\pi xdx.\hfill \end{array}$$

(12)

We will look for a function $u\left(x,t\right)$ that formally satisfies Equation (7) and conditions (8)–(10) in the form

$$\begin{array}{c}u\left(x,t\right)={\displaystyle \sum _{k=0}^{\infty }}{c}_k\left(t\right)u_k\left(x\right)=c_0\left(t\right)u_0\left(x\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(t\right)u_{k1}\left(x\right)+c_{k2}\left(t\right)u_{k2}\left(x\right)\right]\hfill \\ =c_0\left(t\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(t\right)sink\pi x+c_{k2}\left(t\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right].\hfill \end{array}$$

(13)

The function (13) must satisfy condition (8). Therefore,

$$u\left(x,0\right)=\phi \left(x\right)=c_0\left(0\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(0\right)sink\pi x+c_{k2}\left(0\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right].$$

$$\begin{array}{c}{c}_0\left(0\right)={\phi }_0={\displaystyle \frac{1}{2}}\underset{-1}{\overset{1}{\int }}\phi \left(x\right)dx,\hfill \\ c_{k1}\left(0\right)={\phi }_{k1}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx,\hfill \\ c_{k2}\left(0\right)={\phi }_{k2}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)cosk\pi xdx.\hfill \end{array}$$

(14)

The function (13) must satisfy condition (9). Therefore,

$$\begin{array}{c}u\left(x,T\right)=\psi \left(x\right)=c_0\left(T\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(T\right)sink\pi x+c_{k2}\left(T\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right].\hfill \end{array}$$

(15)

Consequently,

$$\begin{array}{c}{c}_0\left(T\right)={\psi }_0={\displaystyle \frac{1}{2}}\underset{-1}{\overset{1}{\int }}\psi \left(x\right)dx,\hfill \\ c_{k1}\left(T\right)={\psi }_{k1}=\underset{-1}{\overset{1}{\int }}\psi \left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx,\hfill \\ c_{k2}\left(T\right)={\psi }_{k2}=\underset{-1}{\overset{1}{\int }}\psi \left(x\right)cosk\pi xdx.\hfill \end{array}$$

(16)

The formal derivatives of the function (13) are given by

$$u_t\left(x,t\right)=c_0^{\prime }\left(t\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k1}\left(t\right)sink\pi x+{c^{\prime }}_{k2}\left(t\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right],$$

$$u_{xx}\left(x,t\right)={\displaystyle \sum _{k=1}^{\infty }}\left[-c_{k1}\left(t\right)k^2{\pi }^{2}sink\pi x+c_{k2}\left(t\right)k^2{\pi }^2\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)\right],$$

$$u_{xx}\left(-x,t\right)={\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(t\right)k^2{\pi }^{2}sink\pi x-c_{k2}\left(t\right)k^2{\pi }^2\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right],$$

$$u_{xxt}\left(-x,t\right)={\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k1}\left(t\right)k^2{\pi }^{2}sink\pi x-{c^{\prime }}_{k2}\left(t\right)k^2{\pi }^2\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right].$$

Substituting them into Equation (7) gives

$$\begin{array}{c}{c^{\prime }}_0\left(t\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k1}\left(t\right)sink\pi x+{c^{\prime }}_{k2}\left(t\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right]\hfill \\ -{\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k1}\left(t\right)k^2{\pi }^{2}sink\pi x-{c^{\prime }}_{k2}\left(t\right)k^2{\pi }^2\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right]\hfill \\ -{\displaystyle \sum _{k=1}^{\infty }}\left[c_{k1}\left(t\right)k^2{\pi }^{2}sink\pi x-c_{k2}\left(t\right)k^2{\pi }^2\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right]\hfill \\ =f_0\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[f_{k1}\left(t\right)sink\pi x+f_{k2}\left(t\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right].\hfill \end{array}$$

After grouping terms, we obtain the equality

$$\begin{array}{c}{c^{\prime }}_0\left(t\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k1}\left(t\right)-{c^{\prime }}_{k1}\left(t\right)k^2{\pi }^2-c_{k1}\left(t\right)k^2{\pi }^2\right]sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left[{c^{\prime }}_{k2}\left(t\right)+{c^{\prime }}_{k2}\left(t\right)k^2{\pi }^2+c_{k2}\left(t\right)k^2{\pi }^2\right]\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\hfill \\ =f_0\left(x+1\right)+f_{k1}\left(t\right)sink\pi x+f_{k2}\left(t\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

From this and from (8), we obtain the following first-order differential equations with initial data

$$\left\{\begin{array}{c}{c^{\prime }}_0\left(t\right)=f_0,\hfill \\ c_0\left(0\right)={\phi }_0,\hfill \end{array}\right.$$

$$\left\{\begin{array}{c}{c^{\prime }}_{k1}\left(t\right)-c_{k1}\left(t\right){\displaystyle \frac{k^2{\pi }^2}{1-k^2{\pi }^2}}={\displaystyle \frac{f_{k1}}{1-k^2{\pi }^2}},\hfill \\ c_{k1}\left(0\right)={\phi }_{k1},\hfill \end{array}\right.$$

$$\left\{\begin{array}{c}{c^{\prime }}_{k2}\left(t\right)+c_{k2}\left(t\right){\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}={\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}},\hfill \\ c_{k2}\left(0\right)={\phi }_{k2}.\hfill \end{array}\right.$$

Let us find the solution of the first problem: $c_0\left(t\right)=f_{0}t+{\phi }_0$. Let us find the solution of the second problem. It has the form

$$c_{k1}\left(t\right)=e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k1}-{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}.$$

Let us find the solution of the third problem. It has the form

$$c_{k2}\left(t\right)=e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k2}-{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}.$$

Thus, we have the following relations

$$c_0\left(t\right)=f_{0}t+{\phi }_0,c_{k1}\left(t\right)=e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k1}-{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}},$$

$$c_{k2}\left(t\right)=e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k2}-{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}.$$

(17)

In the obtained relations, the unknowns are the quantities $c_{kj}\left(t\right),f_{kj}$ and $c_0\left(t\right),f_0$. The values of ${\phi }_0,{\phi }_{k1},{\phi }_{k2}$ can be calculated using formulas (14).

Substituting the dependencies found from formulas (17) into (15) and using formulas (16), we then obtain

$$c_0\left(T\right)={\psi }_0=f_{0}T+{\phi }_0,c_{k1}\left(T\right)={\psi }_{k1}=e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}\left({\phi }_{k1}-{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}},$$

$$c_{k2}\left(T\right)={\psi }_{k2}=e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}\left({\phi }_{k2}-{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}.$$

From the obtained equalities, we determine the values of the unknown quantities.

$$\begin{array}{c}\hfill f_0={\displaystyle \frac{{\psi }_0-{\phi }_0}{T}},f_{k1}=\left({\psi }_{k1}-e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k1}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}},\\ \hfill f_{k2}=\left({\psi }_{k2}-e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k2}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}.\end{array}$$

(18)

Now, using Formula (17), the unknowns $c_{kj}\left(t\right),c_0\left(t\right)$ can be easily determined.

$$\begin{array}{c}{c}_0\left(t\right)={\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0,\hfill \\ c_{k1}\left(t\right)=e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k1}-{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k1}}{1+k^2{\pi }^2}}\hfill \\ =e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}{f}_{k1}=e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\left({\psi }_{k1}-e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k1}\right)\hfill \\ =\left(e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}-{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{e}^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}\right){\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k1},\hfill \end{array}$$

$$\begin{array}{c}{c}_{k2}\left(t\right)=e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}\left({\phi }_{k2}-{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}\right)+{\displaystyle \frac{f_{k2}}{1+k^2{\pi }^2}}=e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}{f}_{k2}\hfill \\ =e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\left({\psi }_{k2}-e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k2}\right).\hfill \end{array}$$

Thus, the formal solution of the problem (7)–(10) can be written as a series

$$\begin{array}{c}u\left(x,t\right)=\left({\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left(e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}{f}_{k1}\right)sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left(e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}{f}_{k2}\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

(19)

and the series (11), where the quantities $f_0,f_{kj};{\phi }_0,{\phi }_{kj};{\psi }_0,{\psi }_{kj}$ are determined by formulas (18), (14), and (16), respectively. Let us substitute them

$$\begin{array}{c}u\left(x,t\right)=\left({\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0\right)\left(x+1\right)\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left(e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}\left({\psi }_{k1}-e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k1}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\right)sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left(e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1+k^2{\pi }^2}}\left({\psi }_{k2}-e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k2}\right){\displaystyle \frac{1+k^2{\pi }^2}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right),\hfill \end{array}$$

$$\begin{array}{c}u\left(x,t\right)=\left({\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left(e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\left({\psi }_{k1}-e^{{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k1}\right)\right)sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left(e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}t}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}\left({\psi }_{k2}-e^{-{\displaystyle \frac{k^2{\pi }^2}{1+k^2{\pi }^2}}T}{\phi }_{k2}\right)\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

Thus, the formal solution of the problem (7)–(10) in its final form can be written as a series

$$\begin{array}{c}u\left(x,t\right)=\left({\displaystyle \frac{{\psi }_0-{\phi }_0}{T}}t+{\phi }_0\right)\left(x+1\right)+{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k1}\right)sink\pi x\hfill \\ +{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k2}\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

The termwise differentiation and substitution into Equation (7) are formal operations, with rigor justified in Theorem 3 via series uniform convergence.

The lemma is proved. □

In what follows, we will need the following lemmas.

Lemma 2

([36]). Let $f\left(x\right)\in L_2(0,1)$ and $a_k=\underset{0}{\overset{1}{\int }}f\left(x\right)e^{-\lambda kx}dx,b_k=\underset{0}{\overset{1}{\int }}f\left(x\right)e^{\lambda k(x-1)}dx$, where λ is an arbitrary complex number with a positive real part $\left(\lambda =\alpha +i\beta ,\alpha >0\right)$. Then, the series converge $\sum {\left|a_k\right|}^2,\sum {\left|b_k\right|}^2$.

Lemma 3.

If an arbitrary function $\phi \left(x\right)\in C^3\left[-1,1\right]$ and for each fixed $j=0,1,2$ satisfy the conditions ${\phi }^{\left(j-1\right)}\left(-1\right)={\phi }^{\left(j\right)}\left(1\right)$, then for the Fourier coefficients of each function $\phi \left(x\right),{\phi }^{\prime }\left(x\right),{\phi }^{″}\left(x\right),{\phi }^{‴}\left(x\right)$ with respect to the correspondingly iterated integrated systems (5), the Bessel inequalities hold.

$$\begin{array}{c}{\displaystyle \sum _{k=1}^{\infty }}{\left|\left({\phi }^{\prime },{(-1)}^k{\displaystyle \frac{e^{k\pi x}{e}^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)\right|}^2<\infty ,\hfill \end{array}$$

(20)

$$\begin{array}{c}{\displaystyle \sum _{k=1}^{\infty }}{|({\phi }^{″}\left(x\right),{(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-sink\pi x)|}^2<\infty ,\hfill \end{array}$$

(21)

$$\begin{array}{c}{\displaystyle \sum _{k=1}^{\infty }}{\left|\left({\phi }^{‴},{(-1)}^k{\displaystyle \frac{e^{k\pi x}{e}^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right|}^2<\infty \hfill \end{array}.$$

(22)

Corollary 1.

If an arbitrary function $\phi \left(x\right)\in C^3\left[-1,1\right]$ and for each $j=0,1,2$ satisfy the conditions ${\phi }^{\left(j-1\right)}\left(-1\right)={\phi }^{\left(j\right)}\left(1\right)=0,$, then for the Fourier coefficients of each function $\phi \left(x\right),{\phi }^{\prime }\left(x\right),{\phi }^{‴}\left(x\right),{\phi }^{‴}\left(x\right)$ with respect to the correspondingly iterated integrated systems (5), the Bessel inequalities (20)–(22) hold.

Proof. 

The proofs of Lemma 3 and Corollary 1 are similar.

Let us prove Lemma 3. To prove the lemma, it is sufficient to integrate by parts the integrals three times.

$$\begin{array}{c}{\phi }_{k1}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx,\hfill \\ {\phi }_{k2}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)cosk\pi xdx.\hfill \end{array}$$

We do not write the coefficient ${\phi }_0$, since a single number does not affect the convergence of the series.

After integrating the first integral by parts three times, we obtain three equalities for the functions $\phi \left(x\right),{\phi }^{\prime }\left(x\right),{\phi }^{″}\left(x\right),{\phi }^{‴}\left(x\right)$

$$\begin{array}{c}{\phi }_{k1}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx\hfill \\ ={\left.{\displaystyle \frac{\phi \left(x\right)}{k\pi }}\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)\right|}_{-1}^1\hfill \\ -{\displaystyle \frac{1}{k\pi }}\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)dx;\hfill \\ -{\displaystyle \frac{1}{k\pi }}\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)dx\hfill \\ ={\left.-{\displaystyle \frac{{\phi }^{\prime }\left(x\right)}{k^2{\pi }^2}}\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-sink\pi x\right)\right|}_{-1}^1\hfill \\ +{\displaystyle \frac{1}{k^2{\pi }^2}}\underset{-1}{\overset{1}{\int }}{\phi }^{″}\left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-sink\pi x\right)dx;\hfill \\ +{\displaystyle \frac{1}{k^2{\pi }^2}}\underset{-1}{\overset{1}{\int }}{\phi }^{″}\left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-sink\pi x\right)dx\hfill \\ ={\left.{\displaystyle \frac{{\phi }^{\prime \prime }\left(x\right)}{k^3{\pi }^3}}\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)\right|}_{-1}^1\hfill \\ -{\displaystyle \frac{1}{k^3{\pi }^3}}\underset{-1}{\overset{1}{\int }}{\phi }^{‴}\left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right)dx.\hfill \end{array}$$

(23)

All three boundary terms vanish by the conditions of the lemma.

Let us consider the Fourier coefficients ${\phi }^{\prime }\left(x\right)$ and estimate them. From the expression

$$\begin{array}{c}\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)dx,\hfill \end{array}$$

we obtain the inequality

$$\begin{array}{c}{\left|\left({\phi }^{\prime },{(-1)}^k{\displaystyle \frac{e^{k\pi x}{e}^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)\right|}^2\le 2{\left|\left({\phi }^{\prime },{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}\right)\right|}^2+2{\left|\left({\phi }^{\prime },cosk\pi x\right)\right|}^2.\hfill \end{array}$$

Here, the second term on the right-hand side is the Fourier coefficient of the function ${\phi }^{\prime }\left(x\right)$ with respect to an orthonormal system, for which the Bessel inequalities hold. Therefore, let us consider the first term and transform it.

$$\begin{array}{c}\left|\left({\phi }^{\prime },{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}\right)\right|=\left|\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right){\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}dx\right|\hfill \\ =\left|{\displaystyle \frac{1}{e^{k\pi }-e^{-k\pi }}}\left[\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{k\pi x}dx-\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{-k\pi x}dx\right]\right|\hfill \\ =\left|{\displaystyle \frac{1}{e^{k\pi }-e^{-k\pi }}}\left[\underset{-1}{\overset{0}{\int }}{\phi }^{\prime }\left(x\right)e^{k\pi x}dx+\underset{0}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{k\pi x}dx-\underset{-1}{\overset{0}{\int }}{\phi }^{\prime }\left(x\right)e^{-k\pi x}dx-\underset{0}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{-k\pi x}dx\right]\right|\hfill \\ \le {\displaystyle \frac{2}{e^{k\pi }}}\left[\underset{0}{\overset{1}{\int }}{\phi }^{\prime }(-x)e^{-k\pi x}dx+\underset{0}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{k\pi x}dx-\underset{0}{\overset{1}{\int }}{\phi }^{\prime }(-x)e^{k\pi x}dx-\underset{0}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)e^{-k\pi x}dx\right]\hfill \\ =2\left|\left\{\underset{0}{\overset{1}{\int }}[{\phi }^{\prime }(-x)-{\phi }^{\prime }\left(x\right)]e^{-k\pi (x+1)}dx+\underset{0}{\overset{1}{\int }}[{\phi }^{\prime }\left(x\right)-{\phi }^{\prime }(-x)]e^{k\pi (x-1)}dx\right\}\right|.\hfill \end{array}$$

Based on Lemma 2, it follows that the Bessel inequality (20) holds. In the same way, the inequalities (21) and (22) are proved. Now, let us turn to the expression

$$\begin{array}{c}{\phi }_{k2}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)cosk\pi xdx.\hfill \end{array}$$

As in the previous case, we integrate by parts three times.

$$\begin{array}{c}{\phi }_{k2}=\underset{-1}{\overset{1}{\int }}\phi \left(x\right)cosk\pi xdx={\left.{\displaystyle \frac{1}{k\pi }}\phi \left(x\right)sink\pi x\right|}_{-1}^1-{\displaystyle \frac{1}{k\pi }}\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)sink\pi xdx\hfill \\ ={\displaystyle \frac{1}{k^2{\pi }^2}}{\left.{\phi }^{\prime }\left(x\right)cosk\pi x\right|}_{-1}^1-{\displaystyle \frac{1}{k^2{\pi }^2}}\underset{-1}{\overset{1}{\int }}{\phi }^{″}\left(x\right)cosk\pi xdx\hfill \\ ={\left.-{\displaystyle \frac{1}{k^3{\pi }^3}}{\phi }^{″}\left(x\right)sink\pi x\right|}_{-1}^1+{\displaystyle \frac{1}{k^3{\pi }^3}}\underset{-1}{\overset{1}{\int }}{\phi }^{‴}\left(x\right)sink\pi xdx.\hfill \end{array}$$

Here, the boundary terms vanish for any function $\phi \left(x\right)\in C^3\left[-1,1\right]$. For the Fourier coefficients of the functions $\phi \left(x\right),{\phi }^{\prime }\left(x\right),{\phi }^{″}\left(x\right),{\phi }^{‴}\left(x\right)$ with respect to an orthonormal system of trigonometric functions, the Bessel inequalities always hold. The lemma is proved. □

Let us now prove the following theorem.

Theorem 3.

If the initial function $\phi \left(x\right),\psi \left(x\right)\in C^3\left[-1,1\right]$ satisfies the conditions ${\phi }^{\left(j\right)}\left(-1\right)={\phi }^{\left(j\right)}\left(1\right)=0,{\psi }^{\left(j\right)}\left(-1\right)={\psi }^{\left(j\right)}\left(1\right)=0,j=0,1,2$, then the inverse problem (7)–(10) has a unique solution in the form of the Fourier series (11).

Proof. 

First, let us show that the function $u\left(x,t\right)$ in the form of a Fourier series satisfies conditions (8)–(10). The fulfillment of the boundary conditions (10) can be checked directly, since the eigenfunctions (4) satisfy them. The series in (19) consists of continuous functions. If this series converges uniformly, then its sum is a continuous function and takes values at the points $t=0$ and $t=T$. From (19), we obtain the inequality

$$\begin{array}{c}\left|u\left(x,t\right)\right|\le 2\left|{\psi }_0\right|+3{\displaystyle \sum _{k=1}^{\infty }}\left(\left|{\phi }_{k1}\right|+\left|{\phi }_{k2}\right|+\left|{\psi }_{k1}\right|+\left|{\psi }_{k2}\right|\right).\hfill \end{array}$$

(24)

Using the differentiability of the functions $\phi \left(x\right),\psi \left(x\right)$, for example, we can write from (23) the estimate

$$\begin{array}{c}\left|{\phi }_{k1}\right|=\left|\underset{-1}{\overset{1}{\int }}\phi \left(x\right)\left({(-1)}^k{\displaystyle \frac{e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}+sink\pi x\right)dx\right|=\left|{\displaystyle \frac{1}{k\pi }}\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)dx\right|\hfill \\ \le {\displaystyle \frac{1}{2k^2{\pi }^2}}+{\displaystyle \frac{1}{2}}{\left|\underset{-1}{\overset{1}{\int }}{\phi }^{\prime }\left(x\right)\left({(-1)}^k{\displaystyle \frac{-e^{-k\pi x}+e^{k\pi x}}{e^{k\pi }-e^{-k\pi }}}-cosk\pi x\right)dx\right|}^2.\hfill \end{array}$$

Similar estimates are obtained for $\left|{\phi }_{k2}\right|,\left|{\psi }_{k1}\right|,\left|{\psi }_{k2}\right|$. By Lemmas 3 and 4, due to the Bessel inequalities, we obtain the uniform convergence of the second series in (11) with respect to the variables $x,t$. This ensures the fulfillment of conditions (8) and (9).

We proceed to prove that the series in (11) satisfy Equation (7). For this purpose, we first establish the uniform convergence of the series for the functions $u_t\left(x,t\right),u_{xx}\left(-x,t\right)$ and $u_{xxt}\left(-x,t\right)$. For the function $u_t\left(x,t\right)$, an estimate of the form (24) holds, and the series $u_t\left(x,t\right)$ converges uniformly with respect to the variables x and t. When estimating the series for the function $u_{xx}\left(-x,t\right)$, we use Equation (1) and obtain

$$\begin{array}{c}{u}_{xx}\left(-x,t\right)=k^2{\pi }^2{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k1}+{\displaystyle \frac{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k1}\right)sink\pi x\hfill \\ -k^2{\pi }^2{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\phi }_{k2}+{\displaystyle \frac{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}t}}{1-e^{-\frac{k^2{\pi }^2}{1+k^2{\pi }^2}T}}}{\psi }_{k2}\right)\left({(-1)}^k{\displaystyle \frac{e^{k\pi x}-e^{-k\pi x}}{e^{k\pi }-e^{-k\pi }}}+cosk\pi x\right).\hfill \end{array}$$

From this, we obtain the estimate

$$\begin{array}{c}\left|u_{xx}\left(-x,t\right)\right|\le 3k^2{\pi }^2{\displaystyle \sum _{k=1}^{\infty }}\left(\left|{\phi }_{k1}\right|+\left|{\phi }_{k2}\right|+\left|{\psi }_{k1}\right|+\left|{\psi }_{k2}\right|\right).\hfill \end{array}$$

After integrating the expressions for ${\phi }_{k1},{\phi }_{k2},{\psi }_{k1},{\psi }_{k2}$ three times, by Lemma 3 and Corollary 1, we obtain that each term in the expression ${\displaystyle \sum _{k=1}^{\infty }}\left(\left|{\phi }_{k1}\right|+\left|{\phi }_{k2}\right|+\left|{\psi }_{k1}\right|+\left|{\psi }_{k2}\right|\right)$ is estimated by a Bessel inequality. Thus, the uniform convergence for $u_{xx}\left(-x,t\right)$ is proved. The uniform convergence for $u_{xxt}\left(-x,t\right)$ is proved in exactly the same way. Using the same method, the uniform convergence for $f\left(x\right)$ is also established.

It remains to prove the uniqueness of the solution. Suppose the contrary. Let there exist two solutions $u_1\left(x,t\right),u_2\left(x,t\right)$ of the problem (5)–(7). Then, for $u\left(x,t\right)=u_1\left(x,t\right)-u_2\left(x,t\right)$, we have Equation (5), condition (7), and $u\left(x,0\right)=0$.

Let us consider the Fourier coefficients $\underset{-1}{\overset{1}{\int }}u\left(x,t\right)sink\pi xdx,\underset{-1}{\overset{1}{\int }}u\left(x,t\right)cos\left(k-{\displaystyle \frac{1}{2}}\right)k\pi xdx$. We differentiate them with respect to the variable t

$$\begin{array}{c}\underset{-1}{\overset{1}{\int }}{u}_t\left(x,t\right)sink\pi xdx=\underset{-1}{\overset{1}{\int }}\left\{{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left[u_{xx}\left(-x,t\right)\right]+u_{xx}\left(-x,t\right)\right\}sink\pi xdx\hfill \\ =-\lambda \underset{-1}{\overset{1}{\int }}\left\{{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left[u\left(x,t\right)\right]+u\left(x,t\right)\right\}sink\pi xdx.\hfill \end{array}$$

We obtain a first-order differential equation.

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }{u}_{k1}\left(t\right)}{\mathsf{\partial }t}}=-\lambda {\displaystyle \frac{\mathsf{\partial }{u}_{k1}\left(t\right)}{\mathsf{\partial }t}}-\lambda u_{k1}\left(t\right),\hfill \end{array}$$

(25)

with the condition

$$\begin{array}{c}{u}_{k1}\left(0\right)=0.\hfill \end{array}$$

(26)

The second coefficient,

$$\begin{array}{c}{u}_{k2}\left(t\right)=\underset{-1}{\overset{1}{\int }}u\left(x,t\right)cos\left(k-{\displaystyle \frac{1}{2}}\right)k\pi xdx,\hfill \end{array}$$

we differentiate with respect to the variable t

$$\begin{array}{c}\underset{-1}{\overset{1}{\int }}{u}_t\left(x,t\right)cos\left(k-{\displaystyle \frac{1}{2}}\right)k\pi xdx=\underset{-1}{\overset{1}{\int }}\left\{{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left[u_{xx}\left(-x,t\right)\right]+u_{xx}\left(-x,t\right)\right\}cos\left(k-{\displaystyle \frac{1}{2}}\right)k\pi xdx\hfill \\ =-\lambda \underset{-1}{\overset{1}{\int }}\left\{{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left[u\left(x,t\right)\right]+u\left(-x,t\right)\right\}cos\left(k-{\displaystyle \frac{1}{2}}\right)k\pi xdx.\hfill \end{array}$$

We obtain the equation

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }{u}_{k2}\left(t\right)}{\mathsf{\partial }t}}=-\lambda {\displaystyle \frac{\mathsf{\partial }{u}_{k2}\left(t\right)}{\mathsf{\partial }t}}-\lambda u_{k2}\left(t\right),\hfill \end{array}$$

(27)

with the condition

$$\begin{array}{c}{u}_{k2}\left(0\right)=0.\hfill \end{array}$$

(28)

Let us write the solutions of Equations (25) and (27).

$$u_{k1}\left(t\right)=A_{k1}{e}^{-{\displaystyle \frac{{\lambda }_{k1}}{{\lambda }_{k1}+1}}t},u_{k2}\left(t\right)=A_{k2}{e}^{-{\displaystyle \frac{{\lambda }_{k2}}{{\lambda }_{k2}+1}}t}.$$

Using the conditions (26) and (28), we obtain

$$\begin{array}{c}{u}_{k1}\left(t\right)=\underset{-1}{\overset{1}{\int }}u\left(x,t\right)sink\pi xdx=0,u_{k2}\left(t\right)=\underset{-1}{\overset{1}{\int }}u\left(x,t\right)cos\left(k-{\displaystyle \frac{1}{2}}\right)\pi xdx=0.\hfill \end{array}$$

Due to the completeness of system (4) in the class $L_2\left(-1,1\right)$, we then obtain $u\left(x,t\right)\equiv 0$, or $u_1\left(x,t\right)=u_2\left(x,t\right)$. The theorem is proved.

Without proof, let us formulate one more theorem.Under different conditions on the functions $\phi \left(x\right),\psi \left(x\right)$, the existence and uniqueness theorem for the solution also holds. □

Theorem 4.

If the $\phi \left(x\right),\psi \left(x\right)\in C^3\left[-1,1\right]$ initial function $\phi \left(x\right),\psi \left(x\right)$ satisfies the conditions $\phi \left(x\right),\psi \left(x\right)\in C^3\left[-1,1\right]$, and ${\phi }^{\left(j\right)}\left(-1\right)={\phi }^{\left(j\right)}\left(1\right),{\psi }^{\left(j\right)}\left(-1\right)={\psi }^{\left(j\right)}\left(1\right),j=0,1,2$, then the inverse problem (7)–(10) has a unique solution in the form of the Fourier series (11).

4. Conclusions

This paper investigates non-self-adjoint problems for pseudoparabolic equations with second-order differential operators involving a pure involution. Second-order differential operators with a pure involution do not possess the property of semiboundedness. Moreover, spectral problems with non-self-adjoint boundary conditions are not always amenable to analysis. This paper studies new systems obtained in the course of the research. The developed approaches can also be applied to other types of non-self-adjoint boundary conditions and demonstrate the possibility of extending the research to problems with operators having variable coefficients.