Separation of variables in the wave equation. Sets of the type (1.1) and Schr�dinger equation

1991 ◽  
Vol 34 (2) ◽  
pp. 122-126
Author(s):  
V. G. Bagrov ◽  
B. F. Samsonov ◽  
A. V. Shapovalov
Keyword(s):  
Wave Equation ◽  
Separation Of Variables ◽  
Dinger Equation

A Perturbation Method for Low-Frequency Fluid-Structure Interaction Problems

1973 ◽  
Vol 40 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Y. K. Lou
Keyword(s):  
Wave Equation ◽  
Perturbation Method ◽  
Electromagnetic Scattering ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Fluid Structure Interaction ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Fluid Structure ◽  
Structure Interaction

Perturbation methods have been used for electromagnetic scattering and diffraction problems in recent years. A similar method suitable for low-frequency fluid-structure interaction problems is presented. The essence of the method lies in the fact that approximate solutions for fluid-structure interaction problems can be obtained from a set of Poisson’s equations, rather than from the reduced wave equation. The method is particularly useful for those problems where the Poisson’s equation may be solved by the method of separation of variables while the reduced wave equation cannot. As an illustrative example, the vibrations of a submerged spherical shell is studied using the perturbation method and the accuracy of the method is demonstrated.

The generalized radially symmetric wave equation

1956 ◽  
Vol 236 (1205) ◽  
pp. 265-277 ◽  
Keyword(s):  
Wave Equation ◽  
Gas Dynamics ◽  
Initial Conditions ◽  
Separation Of Variables ◽  
Cauchy Data ◽  
Uniqueness Result ◽  
Contour Integral ◽  
Quarter Plane ◽  
Singular Initial Value Problem ◽  
Integral Solutions

In this paper a detailed study is made of solutions of the differential equation ∂ 2 ϕ /∂ R 2 + k/R ∂ ϕ /∂ R – ∂ 2 ϕ /∂ T 2 = 0 in the quarter plane R ≽ 0, T ≽ 0. The boundary value problem considered is that of finding a solution which satisfies Cauchy data on T = 0. The contour integral solutions developed for an equation occurring in gas dynamics, shown to be equivalent to that considered here, are the main aid in the investigation. The solution is obtained first for values of T ≼ R but is continued into the whole quarter plane. This continuation follows from a fundamental uniqueness result that knowledge of ϕ on the characteristic R = T specifies its value in the domain R ≼ T . A point emphasized is that the continuation is not in general the analytic continuation from the domain T ≼ R , even for analytic initial data. Different interpretations of the solutions found are examined. When k is a positive integer the equation is that satisfied by radially symmetric solutions of the wave equation in k + 1 space dimensions, and this leads to the solution of the full wave equation for given initial conditions on T = 0. The Huygens principle is clearly illustrated. For general positive values of k the discussion clarifies a problem in gas dynamics in the study of which the original contour integral solutions were first devised. The general solution is also compared with a solution by separation of variables, and some conclusions are drawn regarding certain infinite integrals involving Bessel functions. In the final section negative values of k are considered. The contour integral representation solves in a concise form the singular initial value problem of finding a solution which takes prescribed values on R = 0, thus generalizing a result well known for positive value of k .

scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

Sign in / Sign up

Export Citation Format

Share Document