On a Very Weak Solution of the Wave Equation for a Hamiltonian in a Singular Electromagnetic Field

2018 ◽  
Vol 103 (5-6) ◽  
pp. 856-858 ◽  
Author(s):  
M. V. Ruzhansky ◽  
N. E. Tokmagambetov
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Weak Solution ◽  
Very Weak Solution

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 701-716 ◽  
Author(s):  
REMIGIO RUSSO ◽  
ALFONSINA TARTAGLIONE
Keyword(s):  
Boundary Layer ◽  
Weak Solution ◽  
Lipschitz Domain ◽  
Stokes System ◽  
Navier Stokes ◽  
Robin Problem ◽  
Layer Potentials ◽  
Very Weak Solution ◽  
Stokes Systems ◽  
Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

Singularities of electron kernel functions in an external electromagnetic field

1954 ◽  
Vol 222 (1148) ◽  
pp. 93-108 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Perturbation Theory ◽  
Wave Equation ◽  
Vacuum Polarization ◽  
Proper Time ◽  
External Electromagnetic Field ◽  
Kernel Functions ◽  
Second Order ◽  
Significant Part ◽  
Second Order Wave Equation

The electron kernel functions are derived from solutions of the second-order wave equation, using the proper-time parametrization. Iterated kernel functions are introduced and a gauge-independent perturbation theory is developed. The separation of singular parts proceeds in terms of the iterated kernel functions valid in the absence of an electromagnetic field, and the singular expressions which have to be compensated in order to determine the physically significant part of the vacuum polarization are obtained in a more transparent form than those given originally by Heisenberg.

scholarly journals Some applications of the Riesz potential to the theory of the electromagnetic field and the meson field

1946 ◽  
Vol 188 (1012) ◽  
pp. 18-31 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Differential Equations ◽  
Riesz Potential ◽  
Analytical Continuation ◽  
Neutral Meson ◽  
Meson Field ◽  
Hyperbolic Differential Equations ◽  
Proper Energy

This paper contains some applications of the method of Marcel Riesz in the solution of normal hyperbolic differential equations, in particular the wave equation, where the known difficulties, due to the occurrence of divergent integrals, are avoided by a process of analytical continuation. In the theory of the electromagnetic field the method yields simple deductions of classical results, but also the results recently obtained by Dirac regarding the proper energy and proper momentum of an electron are obtained without any addition of new assumptions. The corresponding problem in Bhabha’s analogous theory for the neutral meson field are also studied.

scholarly journals Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1283
Author(s):  
Karel Van Bockstal
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Differential Operator ◽  
Unique Weak Solution ◽  
Fractional Wave Equation ◽  
Equation Of Time ◽  
Initial Boundary Value ◽  
Distributed Order

We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

2007 ◽  
Vol 137 (2) ◽  
pp. 349-371 ◽  
Author(s):  
Shuguan Ji ◽  
Yong Li
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Time Periodic Solutions ◽  
Time Periodic ◽  
Dimensional Wave

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

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