Wave Equations

Analytical solutions of the wave equation with bi-fractional-order and frictional memory kernel of Mittag-Leffler type are obtained via Caputo-Fabrizio fractional derivative in the Liouville-Caputo sense. Through the method of separation of variables and Laplace transform method we derive closed-form solutions and establish fundamental solutions. Special cases with homogeneous Dirichlet boundary conditions and nonhomogeneous initial conditions, as well as for the external force are considered. Numerical simulations of the special solutions were done and novel behaviors are obtained.

The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

The history and conceptual development of relativistic two-body formalisms leading up to the Bethe-Salpeter equation are reviewed. The various three dimensional reductions of the Bethe-Salpeter equation, in particular the quasipotential equations, and methods for solving them are discussed. Previous studies which attempted to assess the relative merits of the various equations are examined, and a new criterion for judging the

In the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [C.S. Gardner, G.K. Morikawa, Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant Institute Math. Sci. Report, NYO-9082 (1960) 1–30, T. Taniuti, C.C. Wei, Reductive perturbation method in non-linear wave propagation I, J. Phys. Soc. Jpn., 24 (1968) 941–946]. We obtained the forced Korteweg–de Vries (FKdV) equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The variations of radial displacement and the fluid pressure with the distance parameter are also examined numerically. The results seem to be consistent with Bernoulli’s law for inviscid fluid.

... TA16 10:40 Controllability and motion planning for linear delay systems with an application to a flexible rod* ... uni-Stuttgart .de Abstract: The wave equation describing the tor-sional behavior of a flexible rod with a mass is inter-preted as a linear delay system. ...

Electromagnetism is the branch of physics concerned with electromagnetic phenomena. We discussed the Maxwell's equations in analytic form. The equation of electro-magnetic waves is derived and shown it is invariant to Lorentz transformation. Eventually, it is generalized.

PURPOSE:This Coastal and Hydraulic Engineering Technical Note (CHETN) is a user's guide for the Infra-Gravity Wave Toolbox (IGWT) developed as an activity of the Coastal Inlets Research Program (CIRP) of the US Army Engineer Research and Development Center, Coastal ...

In the framework of Special Relativity and in agreement with the electromagnetic approach used to connect Quantum and Relativistic effects in Bridge Theory, a first order wave-equation able to emulate the wave-particle duality in terms of local electromagnetic interactions is proposed. The equation is able to describe the propagation of a fermion as a de Broglie's wave and simultaneously to characterise the wave in motion with a mass energy as a particle. The solution of the wave-equation proposed is compatible with the de Broglie's idea of an empty pilot wave, carrying information about the dynamical properties of a particle materialising during each local interaction with the observers.

.We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, & (t,x)\in [0,T)\times\Omega,\[6pt] u(t,x)=0, & (t,x)\in [0,T)\times\partial \Omega,\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es& \ea $ where $<\le \infty$, $\Omega$ is a bounded regular open subset of $\mathbb{R}^n$, $n\ge 1$, $b,c>0$, $p>2$, $m>1$ . We prove a global nonexistence theorem for positive initial value of the energy when $ 1<m<p,\quad 2, <p\le \frac{2n}{n-2}. $ We also give applications concerning the classical equations of linear elasticity, the damped clamped plate equation and evolution systems involving the q‐Laplacian operator, $q>1$ .

ABSTRACT Frequency up-conversion of a laser beam causing gas ionization in a cavity is considered in a fully relativistic fashion. A simple ionization model, based on one-photon and multiphoton processes is used. A Vlasov fully nonlinear and ionization-model independent description is used in calculating the current which drives the wave equation for the electric field. The evolution of the wave frequency and its upshift are contrasted with those obtained in the nonrelativistic limit. It is found that the nonrelativistic treatment overestimates the frequency upshift by a factor more than two. Purely relativistic effects, such as a significant frequency modulation and a respective temporal pulse compression, are observed in the exact case

Development of a new approach to the solution of the coplanar problem of a satellite orbiting two primaries. First, the stability of the libration points is discussed, by examining weak variations of a topological surface. Collinear points are shown to reside on local maxima, while triangular points reside on local minima of this surface. Second, a conjecture is made concerning the correctness of the gravity potential, and canonical classification is discussed from the viewpoint of simplifying the governing partial differential equations. A transformation is constructed which reduces these equations to an inhomogeneous Laplace equation describing the satellite&#39;s motion. An inverse procedure, coupled with this approach, generates an unknown gravity potential for a known satellite trajectory. This potential must solve an inhomogeneous wave equation.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2010) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2886 ... Hybrid finite element/volume method for shallow water ...