Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Symmetry Group ◽  
Group Method ◽  
Radial Solutions ◽  
Radial Wave ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Wave ◽  
Radial Wave Equation

AbstractIn this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

scholarly journals Some General New Einstein Walker Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Partial Differential Equations ◽  
Symmetry Group ◽  
Lie Symmetry ◽  
Group Method ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Walker Manifold

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

scholarly journals A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables

2019 ◽  
Vol 27 (1) ◽  
pp. 25-34
Author(s):  
Sora Otsuki ◽  
Pauline N. Kawamoto ◽  
Hiroshi Yamazaki
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Derivative ◽  
Separation Of Variables ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Partial Differential Equations ◽  
Dimensional Wave

Summary In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.

A Bremmer series for radial wave equations

1977 ◽  
Vol 55 (24) ◽  
pp. 2150-2157 ◽  
Author(s):  
W. E. Couch ◽  
R. J. Torrence
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Wkb Approximation ◽  
Spherically Symmetric ◽  
Radial Wave ◽  
One Dimensional ◽  
Similar Series ◽  
Leading Term ◽  
The One ◽  
Radial Wave Equation

The Bremmer series solution of the one-dimensional Helmholtz equation with variable velocity is generalized to obtain a similar series for the radial wave equation with a spherically symmetric velocity function. Since the leading term of Bremmer's series is the one-dimensional WKB approximation, we obtain an approximation for the radial wave equation analogous to the WKB approximation.

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

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.

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