An integral-boundary value problem for a partial differential equation of second order

An integral-boundary value problem for a hyperbolic partial differential equation in two independent variables is considered. By introducing additional functional parameters, we investigate the solvability of the problem and develop an algorithm for finding its approximate solutions. The problem is reduced to an equivalent one, consisting of the Goursat problem for a hyperbolic equation with parameters and boundary value problems with an integral condition for ODEs with respect to the parameters entered. We propose an algorithm to find an approximate solution to the original problem, which is based on the algorithm for finding a solution to the equivalent problem. The convergence of the algorithms is proved. A coefficient criterion for the unique solvability of the integral-boundary value problem is established.


Introduction
On the domain Ω = [0, T ] × [0, ω], we consider the integral-boundary value problem for the hyperbolic equation of second order where u(t, x) is an unknown function; the functions A(t, x) , B(t, x) , C(t, x) , and f (t, x) are continuous on Ω ; the functions K(t, x) and P (t) , ψ(t), are continuously differentiable with respect to t on Ω and [0, T ] , respectively; the functions M (t, x) and S(x) , φ(x), are continuously differentiable with respect to x on Ω and [0, ω], respectively; and 0 < a ≤ ω , 0 < b ≤ T . The compatibility condition is given below.
Under the assumption of continuous differentiability of the equation coefficients, the conditions for the unique solvability of that problem have been obtained. In [31], the contractive mapping principle is used to study problem (1)-(3) for P (t) = 0 , S(x) = 0 and K(t, x) = K(x) , M (t, x) = M (t). In [7], the unique solvability conditions for the problem, where P (t) = 0 , S(x) = 0 , and K(t, x) = M (t, x), were established in terms of initial data.
A boundary value problem for hyperbolic equations subject to general integral conditions is one of the rarely studied problems of mathematical physics. This formulation of the problem is considered for the first time.
The aim of this work is to develop an algorithm for finding a solution to problem (1)-(3) and establish conditions for the existence and uniqueness of its classical solution.
In Section 2, a scheme of the method from [4][5][6] is presented, which will be used to investigate the problem.
We introduce new unknown functions as linear combinations of the solutions' values on the characteristics. The problem (1)-(3) is reduced to an equivalent one consisting of the Goursat problem for hyperbolic equations with functional parameters and boundary value problems with integral conditions for ODEs with respect to the parameters entered. An algorithm for finding an approximate solution to the problem under investigation is proposed. The algorithm consists of two parts. First, we solve two boundary value problems with integral conditions for ODEs. Such problems have been intensively studied in recent years [1][2][3] and they occur in numerous areas of applied mathematics. In the second part of the algorithm, we solve the Goursat problem for a hyperbolic equation with parameters. In Section 3, the conditions for the existence of a unique solution to the boundary value problem with integral condition for ODEs are provided. In Section 3, the convergence of the algorithm is proved, and the conditions for the unique solvability of problem (1)-(3) are given in terms of initial data.

The description of the method and the algorithm
We introduce the following notations: where the latter is a new unknown function.

By virtue of conditions
[ Taking into account [ Assumptions on the data of problem (1)-(3) allow us to differentiate (7) and (8) with respect to t and x , respectively. We then obtain We x) ∂t and the following notations: Then equations (13) and (14) can be written in the following forms: Thus, we have a closed system of equations (4)-(6), (15), (12), (16), and (11) for determining unknown

and µ(t).
Relation (15) in conjunction with (12) presents a boundary value problem with integral condition for a differential equation with respect to µ(t) , and the relation (16) in conjunction with (11) presents a boundary value problem with integral condition for a differential equation with respect to λ(x) .
, from the boundary value problem with integral condition (16) and (11).
The constructed algorithm consists of two parts: we solve the boundary value problems with integral condition for the ordinary differential equations (15), (12) and (16), (11) in the first part, and we solve the Goursat problem for hyperbolic equations with functional parameters in the second part.

Boundary value problems with integral conditions for the ordinary differential equations
Consider the boundary value problem with integral condition for the ordinary differential equationṡ [ We also consider the boundary value problem with integral condition for the ordinary differential equation of the following type:λ [ where the functions A 2 (x) and g 2 (x) are continuous on [0, ω], the function K(t, x) is continuous on Ω , and P (0) and ψ(0) are some constants, 0 < a ≤ ω .
Below we give conditions for the unique solvability of boundary value problems with integral condition (17), (18) and (19), (20). Suppose Then the solutions to problems (17), (18) and (19), (20) can be written in the following forms: and respectively. Relations (21) and (22) follow from the representation of the solution to the Cauchy problem for the ordinary differential equations according to the qualitative theory of differential equations.
The following statements are true.

Conditions for convergence of the algorithm and the main result
In Section 2, an algorithm for finding a solution to problem (4)- (8), which is equivalent to problem (1)-(3), is constructed. To formulate the main result, assume that We introduce the following notations: In Section 3, the conditions for unique solvability of boundary value problems with integral condition (17), (18) and (19), (20) are established. For fixed v(t, x) , w(t, x), u(t, x) in each step of the algorithm we solve the boundary value problems with integral condition (15), (12) and (16), (11). For fixedλ(x) ,μ(t), λ(x) , µ(t), we solve the Goursat problem (4), (6).
The following statement provides the conditions for the convergence of the proposed algorithm and the existence of a unique solution to problem (4)-(8). A(t, x), B(t, x), C(t, x) , and f (t, x) be continuous on Ω;
The following inequalities are valid: x) on the m th step of the algorithm and obtain µ (m+1 Suppose ) .
Theorem 4 is proved.