2019
Том 71
№ 11

Khoma N. H.

Articles: 9
Article (Ukrainian)

Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 931-935

On the basis of the exact solution of the linear Dirichlet problem $u_{tt} - u_{xx} = f\left( {x,t} \right)$ , $u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0,{\text{ }}u\left( {x,0} \right) = u\left( {x,2\pi } \right) = 0,$ $0 \leqslant x \leqslant \pi ,{\text{ }}0 \leqslant t \leqslant 2\pi ,$ we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u ttu xx = f(x, t, u, u t).

Brief Communications (Ukrainian)

An exact solution of boundary-value problem

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1142-1143

We establish conditions under which the problem $u_{tt} — u_{xx} = f(x, t),\; u(x, 0) = u(x, \pi) = 0$ possesses the classical solution.

Brief Communications (Ukrainian)

A linear periodic boundary-value problem for a second-order hyperbolic equation

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 281–284

We study the boundary-value problemu tt -u xx =g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of $\frac{\pi }{q} - , \frac{{2\pi }}{{2s - 1}} -$ , and $\frac{{4\pi }}{{2s - 1}}$ -periodic functions (q and s are natural numbers). We obtain the results only for sets of periods $T_1 = (2p - 1)\frac{\pi }{q}, T_2 = (2p - 1)\frac{{2\pi }}{{2s - 1}}$ , and $T_3 = (2p - 1)\frac{{4\pi }}{{2s - 1}}$ which characterize the classes of π-, 2π -, and 4π-periodic functions.

Article (Ukrainian)

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1680–1685

In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator.

Article (Ukrainian)

Linear periodic boundary-value problem for a second-order hyperbolic equation. I

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1537–1544

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u tta 2 u xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ

Article (Russian)

Conditions of solvability of quasilinear periodic boundary-value problems for hyperbolic equations of the second order

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 818–821

On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem $$u_{tt} - u_{xx} = F\left[ {u,u_t } \right], u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0, u\left( {x,t + 2\pi } \right) = u\left( {x,t} \right)$$ .

Brief Communications (Ukrainian)

Smooth solution of one boundary-value problem

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1712–1716

We study the boundary value problem for the quasilinear equation u u − uxx=F[u, ut], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.

Article (Ukrainian)

Generalized periodic solutions of quasilinear equations

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 406-411

We study a boundary-value periodic problem for the quasilinear equationu ff u xx =F[u,u f u x ],u(0,t) =u (π,t),u (x, t + π/q) =u(x, t), 0 ≤xπ,t ∈ ℝ,q ∈ ℕ. We establish conditions under which the theorem on the uniqueness of a smooth solution is true.

Brief Communications (Ukrainian)

Existence of a smooth solution of one boundary-value problem

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1717–1719

We study a periodic boundary-value problem for the quasilinear equationu tt−uxx=F[u, ut], u(0, t)=u(π, t)=0,u(x, t+2π)=u(x, t). We establish conditions that guarantee the validity of the uniqueness theorem.