A Bremmer series for radial wave equations

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.

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0323 ◽

2018 ◽

Vol 73 (2) ◽

pp. 161-170 ◽

Cited By ~ 2

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.

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Dirac Systems with Discrete Spectra

Canadian Journal of Mathematics ◽

10.4153/cjm-1987-006-0 ◽

1987 ◽

Vol 39 (1) ◽

pp. 100-122 ◽

Cited By ~ 10

Author(s):

D. B. Hinton ◽

J. K. Shaw

Keyword(s):

Separation Of Variables ◽

Relativistic Quantum ◽

Three Dimensional ◽

Relativistic Quantum Mechanics ◽

Radial Wave ◽

One Dimensional ◽

The One ◽

Image Position ◽

Discrete Spectra ◽

Radial Wave Equation

In this paper we consider the one dimensional Dirac system1.1where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.

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

Journal of Applied Physics ◽

10.1063/5.0050895 ◽

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

Doklady of the National Academy of Sciences of Belarus ◽

10.29235/1561-8323-2020-64-6-657-662 ◽

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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