Direct and inverse problems for integro-differential equations of hyperbolic type represent a rapidly developing area in modern mathematical physics. Problems of this nature frequently occur across various areas of applied science, including electrodynamics, acoustics, quantum scattering theory, geophysics, astronomy, and related disciplines. These phenomena give rise to integro-differential equations that govern the propagation of elastic and electromagnetic waves in media with memory, where the present state depends not only on the current parameters but also on the entire history of the system. Mathematically, this introduces convolution-type integral terms into the right-hand sides of the classical wave equations, accounting for delay or aftereffect phenomena.

One of the pressing problems in modern mathematical physics is the investigation of the existence and uniqueness of solutions to initial value problems for linear Volterra convolution-type integro-differential equations in Banach spaces. The specificity of such objects lies in their dual nature. Integro-differential equations are functional equations in which the unknown function and its derivatives appear both under the integral sign and outside of it. Consequently, unlike integral equations, they also contain derivatives of the unknown function. Similar to integral equations, integro-differential equations are classified as Fredholm or Volterra. In contemporary literature, such technical and natural systems are referred to as systems with aftereffect, heredity, or dynamic memory. Volterra proposed the mathematical description of the governing laws for such systems in a series of articles [1] and a review [2] based on integral and integro-differential equations, which were subsequently named after him and remain relevant to this day.

A series of profound results on the well-posedness of Volterra integro-differential equations of first and second order that are not resolved with respect to the highest-order derivative, along with findings on the spectrum of corresponding operator-valued functions, were established by Kopachevsky and his students [3], [4]. It should be noted that in these works, solvability was examined in spaces of continuously differentiable functions over a finite time interval.

References [5] and [6] present a comprehensive study of different approaches to formulating direct and inverse problems for second-order partial differential equations.

This study addresses the problem of finding a solution to a nonhomogeneous integro-differential equation of viscoelasticity, in which the principal part contains a general hyperbolic operator, taking into account the conditions that define the direct (initial-boundary) problem. Similar issues for homogeneous integro-differential equations were previously examined in [7, 8].

In the domain, we consider the integro-differential equation.

with initial conditions

boundary conditions

where and are given functions .

In geophysics, integro-differential equations arise when studying the properties of a medium using seismic waves. Essentially, under smoothness assumptions, the system of equations for the non-elastic Boltzmann model (one of the most general for a linear non-elastic medium) reduces to this equation.

Definition 1 ([9]). A function that satisfies all the equations and conditions of the mixed Problems (1)–(3) is called its classical solution.

We introduce a new function , defining it by the relation:

It can readily be verified [10] that the formula is expressed via the function:

where

For simplicity, we assume that . Consequently, as follows , for the new functions and Eqs. (1)-(3) acquire the form:

where we have introduced the notation

We will seek the solution to Problems (5)-(7) in the form of a Fourier series

If we substitute Expression Eq. (8) into the given equation, the resulting relation can be rewritten in the form of a spectral problem. In particular, the eigenvalues of this operator are expressed by the functions .

In contrast, the eigenfunctions associated with these eigenvalues take the form of Fourier

coefficients, calculated using the formula, which are numerical parameters that determine the amplitude and phase of the harmonic components of a function.

Let be the Fourier coefficients.

of the functions and

Remark. In this case, the existence of Fourier series expansions for the functions is assumed. By expressing Eqs. (5) and (6) in terms of the variable using Eq. (8), we obtain the following mathematical problem, which constitutes a system of equations with specific conditions, the solution of which allows us to investigate the behavior of the original function.

With initial conditions

The solutions to Problems (11) and (12) can be expressed as corresponding integral equations, providing a convenient framework for further investigation. This formulation not only allows for the determination of the existence and uniqueness of the solution but also facilitates the application of analytical methods for its construction. In particular, expressing the problem in integral form is well-suited to the Fourier method, which decomposes functions into series and reduces the original integro-differential equation to a system of algebraic equations for the expansion coefficients. This approach simplifies the study of the solution’s properties and allows both analytical and numerical examination of the system’s behavior.

For each fixed , Eq. (13) is a Volterra integral equation of the second kind with respect to the function . From the theory of integral equations, it follows that the solution to this equation is unique and can be obtained by the method of successive approximations.

Lemma. When they are sufficiently large, the following inequalities hold:

with the same data an, bn and fn(t), where S1, S2 − are positive constants depending only on T;

Proof. From Eq. (13) it follows that

The estimate Eq. (14) is obtained directly from Eq. (16) by applying the well-known Grönwall inequality. To derive Eq. (15), we perform a formal double differentiation of series Eq. (13) with respect to the corresponding variable and repeat a similar procedure. This method allows careful control over the behavior of the derivatives and ensures the correctness of the resulting expressions when taking limits. Moreover, the use of formal differentiation of the series allows one to reveal the internal structure of the solution and determine the influence of each term on the overall dynamics of the function. This approach also provides a foundation for subsequent convergence analysis. By formally differentiating series Eq. (8) with respect to the variables x and t, we obtain the corresponding series expansions:

For the functional expansion Eq. (8), a majorant series is selected.

By applying the estimates Eqs. (14) and (15) to series Eqs. (17) and (18), the following inequalities can be established:

where h1, h2 and h3 denote positive constants that depend exclusively on T. Eqs. (19)–(21) imply that, for any (x, t) ∈ Ω, the series Eqs. (8), (17), (18) are dominated by the series

where h4 − is a constant that depends only on T.

The main result of this work is the following theorem.

4 3 3

Theorem 1. Let ( ) ∈ [0, ], ( ) ∈ [0, ], ( ) ∈ [0, ] и ( ) ∈ [0, ], and additionally let the following boundary conditions hold:

a(0) = a(p) = a (0) = a (p) = 0.

b(0) = b(p) = b (0) = b (p) = 0.

f(0) = f(p) = f (0) = f (p) = 0.

Then there exists a unique solution to Eqs. (5)-(7).

Proof: By performing a fourfold integration by parts on Eq. (9), while accounting for all conditions specified in the theorem, we obtain a form that incorporates the boundary effects and facilitates the derivation of the necessary estimates. This approach transfers the differential operators onto auxiliary functions, simplifying the subsequent analysis and yielding a final expression suitable for further investigation and for establishing the theorem’s claims.

From this, we obtain the following representation for |an|:

Performing a similar procedure for |bn|, ∥ fn(t) ∥, we have

From these formulas, using Bessel's inequality, we obtain

From these relations, the convergence of the series Eq. (22) follows. Consequently, the series Eqs. (8), (18), (19) converge uniformly. Thus, the function u(x, t) defined by Eq. (8) is a solution to Problems (5)-(7). The uniqueness of this solution is established by the usual method. The function w(x, t) is related to the function u(x, t) by a relation. Consequently, the function is a solution to Problems (1)-(3).

The theorem is proven.

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