Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations

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

“…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).…”

“…A triple of functions ( u(t, x), µ(t), λ(x)) satisfying the hyperbolic equation (4), the conditions on characteristics (5) and (6), and the functional relations (7) and (8) for µ(0) = λ(0) is called a solution to problem…”

“…Obviously, the functions u * (t, x), v * (t, x), and w * (t, x) are continuous on Ω. Solving the problems on the (m + 1) th step of the algorithm and passing to the limit as m → ∞ , we obtain that the functions u * (t, x), λ * (x), µ * (t) together with their derivatives satisfy the Goursat problem (4)- (6) and boundary value problems with integral condition (15), (12) and (16), (11).…”

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

“…The criteria for the unique solvability of some classes of linear boundary value problems for hyperbolic equations with variable coefficients were obtained relatively recently [15][16][17][18][19][20][21]. In [15], a nonlocal boundary value 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. It is established that the well-posedness of a nonlocal boundary value problem with an integral condition for systems of hyperbolic equations is equivalent to the well-posedness of a family of two-point boundary value problems for a system of ordinary differential equations.…”

“…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.…”