On the well‐posedness of periodic problems for the system of hyperbolic equations with finite time delay

Abstract

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well‐posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.

An existence solution to an identification parameter problem for higher-order partial differential equations

“…The following theorem provides conditions for the feasibility and convergence of the constructed algorithm, as well as conditions for the existence of a unique solution to problem with parameter (6)-- (10). The functions functions The proof of Theorem 1 is similar to the proof of Theorem 1 in [35].…”

“…The criteria for the unique solvability of some classes of linear nonlocal problems for hyperbolic equations with variable coefficients were obtained relatively recently [34][35][36]. In [34], a nonlocal problem with an integral condition for systems of hyperbolic equations by introducing new unknown functions is reduced to a problem consisting of a family of boundary value problems with an integral condition for systems of ordinary differential equations and functional relations.…”

“…To solve the problem under consideration, we use the method of introducing additional functional parameters [34][35][36] and reduce the original problem to an equivalent problem consisting of a nonlocal problem with parameter for a second-order hyperbolic equation with functional parameters and integral relations. We establish sufficient conditions for the unique solvability of the considered problem in the terms of unique solvability of nonlocal problem with parameter for a second-order hyperbolic equation.…”

On the solvability of a semi-periodic boundary value problem for the nonlinear Goursat equation

“…Here, the problem of finding a solution to a semi-periodic boundary value problem for hyperbolic equations ( 5)-( 8) reduced to a family of periodic boundary value problems for a system of ordinary differential equations ( 9), (10) and functional relationships (11), (12). The problems ( 5)-( 8) and ( 9)-( 12) are equivalent in the sense that if a pair of functions (u * (x, t), z * (x, t)) are the solution to the problem ( 5)-( 8), then three (w * (x, t) = ∂u * (x,t) ∂x , u * (x, t), z * (x, t)) is the solution of the problem ( 9)-( 12) and vice versa, if three (w * (x, t), u * (x, t), z * (x, t))-is the solution of the problem( 9)-( 12), then a pair of functions (u * (x, t), z * (x, t)) is the solution of the problem ( 5)- (8).…”

An Initial-Boundary Value Problem for a Higher-Order Partial Differential Equation

“…We will investigate the questions of the existence and uniqueness of classical solutions to the initialboundary value problem for a higher-order partial differential equation (1) -(3) and the construction of its approximate solutions. For these purposes, we apply the method of introducing additional functional parameters proposed in [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] for solving various nonlocal problems for systems of hyperbolic equations with mixed derivatives. The considered problem is reduced to a nonlocal problem for second-order hyperbolic equations, including additional functions, and integral relations.…”