This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data. The new results obtained here are sharp in low dimension. 1. Introduction 1.1. The results. We consider in this paper second order, nonlinear hyperbolic equations of the form (1.1) gij (u) ∂i ∂j u = q ij (u) ∂i u ∂j u
on R × Rn , with Cauchy data prescribed at time 0, (1.2) u(0, x) = u0 (x) , ∂0 u(0, x) = u1 (x) .
In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Rie2 1 mannian metric g. That is, for X = M × Rt , P = Dt − ∆M , and u ∈ Hloc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with...
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term
ROTHE TIME-DISCRETIZATION METHOD APPLIED TO A QUASILINEAR WAVE EQUATION SUBJECT TO INTEGRAL CONDITIONS
ABDELFATAH BOUZIANI AND NABIL MERAZGA Received 27 January 2004 and in revised form 12 February 2004
This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method. 1.
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Global Existence, Uniqueness, and Asymptotic Behavior of Solution for p-Laplacian Type Wave Equation
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping
Ancient Hebrew and Christian tradition relates that the universe was created in six days,
following which there was a day of rest. What the old chronicles never recorded was that, on
the eighth day, the Creator must have dropped back into the lab to do some tidying up. Only
then, coming across his rough notes for Maxwell’s and the acoustic wave equations, did the
thought occur that the creation of light and sound could be logically extrapolated to x-rays
Partial differential equations (PDEs) are very important in modelling as their solutions
unlock the secrets to a range of important phenomena in engineering and
physics. The PDE known as the wave equation models sound waves, light waves
and water waves. It arises in fields such as acoustics, electromagnetics and fluid
The first is Faraday’s law of induction, the second is Amp`ere’s law as amended by
Maxwell to include the displacement current ∂D/∂t, the third and fourth are Gauss’ laws
for the electric and magnetic fields.
The displacement current term ∂D/∂t in Amp`ere’s law is essential in predicting the
existence of propagating electromagnetic waves. Its role in establishing charge conservation
is discussed in Sec. 1.7.
Eqs. (1.1.1) are in SI units.
This book is concerned with Ultra-Low-Frequency (ULF)-electromagnetic
waves observed on the Earth and in Space. These are so-called geomagnetic
variations or pulsations. Alfv´en’s discovery related to the influence of the
strong magnetic field on the conducting fluids (magnetohydrodynamics) led
to development of the concept that the ULF-waves are magnetospheric magnetohydrodynamic
MHD-waves at their propagation gather information about the magnetosphere,
ionosphere, and the ground. There are two applied aspects based on
using the ULF electromagnetic oscillations.
The subject of acoustic waves might easily be considered a mature one, quite
specialized, with narrow and circumscribed fields of interest and of application. The
present book is an evidence of the opposite: it witnesses how the concept of acoustic
wave, a collective displacement of matter which perturbs an equilibrium
configuration, is a pervasive concept, which emerges in very different fields. This type
of phenomena can be analyzed from different points of view, it can be exploited in
different ways, and is the object of active investigations.
We studied theoretically and experimentally the transformation, attenuation, and setup due to shoaling and breaking of internal waves in a two-layer fluid system on a uniform slope. An image processing technique was used to illustrate 2D instantaneous displacements of density interface. These results were compared with the calculated values by using the method of characteristics, the simple shoaling model with energy dissipation, and the momentum balance equation based on a radiation stress concept.
Document "The Mathematical Theory of Maxwell’s Equations" give you the knowledge: The Variational Expansion into Wave Functions, Scattering From a Perfect Conductor, Approach to the Cavity Problem, Boundary Integral Equation Methods for Lipschitz Domains,...
Wave-Equation Description of Nonlinear Optical Interactions
Các phương trình sóng cho truyền thông quang học phi tuyến
Chúng ta đã thấy trong chương cuối cùng như thế nào phi tuyến trong phản ứng của một hệ thống vật chất vào một lĩnh vực laser cường độ cao có thể gây ra sự phân cực của môi trường để phát triển các thành phần tần số mới không có mặt trong lĩnh vực bức xạ nhiệt.
The wireless era was started by two European scientists, James Clerk Maxwell and Heinrich Rudolf Hertz. In 1864, Maxwell presented Maxwell's equations by unifying the works of Lorentz, Faraday, Ampere, and Gauss. He predicted the propagation of electromagnetic waves in free space at the speed of light. He postulated that light was an electromagnetic phenomenon of a particular wavelength and predicted that radiation would occur at other wavelengths as well. His theory was not well accepted until 20 years later, after Hertz validated the electromagnetic wave (wireless) propagation.