scholarly journals The classical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2021 ◽  
Vol 57 (1) ◽  
pp. 23-32
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Wave Equation ◽  
Classical Solution ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom of the boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at a point. The smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved, and the conditions under which a piecewise-smooth solution exists are established. The problem with conjugate conditions is considered

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

scholarly journals Classical solution of the mixed problem for a one-dimensional wave equation with second-order derivatives at boundary conditions

2020 ◽  
Vol 55 (4) ◽  
pp. 406-412
Author(s):  
V. I. Korzyuk ◽  
S. N. Naumavets ◽  
V. A. Sevastyuk
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Analytical Form ◽  
Second Order ◽  
Compatibility Conditions ◽  
One Dimensional ◽  
Dimensional Wave

This paper considers the mixed problem for a one-dimensional wave equation with second-order derivatives at boundary conditions. Using the method of characteristics, a classical solution to this problem is found in analytical form. Its uniqueness is proved under the relevant compatibility conditions.

scholarly journals The classical solution of one problem of an absolutely inelastic impact on a long elastic semi-infinite bar

2021 ◽  
Vol 57 (4) ◽  
pp. 417-427
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Classical Solution ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Bottom Boundary ◽  
One Dimensional ◽  
Inelastic Impact ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom boundary, the Cauchy conditions are specified, meanwhile, the second of them has a discontinuity of the first kind at one point. The smooth boundary condition, which has the first and the second order derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved and the conditions are established under which a piecewise-smooth solution exists. The problem with matcing conditions is considered.

scholarly journals Solutions of problems for the wave equation with conditions on the characteristics

2021 ◽  
Vol 57 (2) ◽  
pp. 148-155
Author(s):  
V. I. Korzyuk ◽  
O. A. Kovnatskaya
Keyword(s):  
Wave Equation ◽  
Analytical Solution ◽  
Classical Solution ◽  
Method Of Characteristics ◽  
Matching Conditions ◽  
One Dimensional ◽  
The Method Of Characteristics ◽  
The One ◽  
The Given ◽  
Dimensional Wave

In this paper we obtain a classical solution of the one-dimensional wave equation with conditions on the characteristics for different areas this problem is considered in. The analytical solution is constructed by the method of characteristics. In addition, the uniqueness of the obtained solution is proved. The necessity and sufficiency of the matching conditions for given functions of the problem are proved. When these conditions are satisfied and the given functions are smooth enough, the classical solution of the considered problem exists.

scholarly journals Mixed problem for a one-dimensional wave equation with conjugation conditions and second-order derivatives in boundary conditions

2020 ◽  
Vol 56 (3) ◽  
pp. 287-297
Author(s):  
V. I. Korzyuk ◽  
S. N. Naumavets ◽  
V. P. Serikov
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Mixed Problem ◽  
Analytical Form ◽  
Second Order ◽  
Conjugation Conditions ◽  
One Dimensional ◽  
Characteristics Method ◽  
Half Strip ◽  
Dimensional Wave

In this paper, we consider the boundary problem for the half-strip on the plane for the case of two independent variables. This mixed problem is solved for a one-dimensional wave equation with Cauchy conditions on the basis of the half-strip and boundary conditions for lateral parts of the area border containing second-order derivatives. Moreover, the conjugation conditions are specified for the required function and its derivatives for the case when the homogeneous matching conditions are not satisfied. A classical solution to this problem is found in an analytical form by the characteristics method. This solution is approved to be unique if the relevant conditions are fulfilled.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave
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