CLASSICAL SCATTERING OF ELECTROMAGNETIC RADIATION Light is an electromagnetic disturbance which is propagated by vector wave equations which are readily derived from Maxwell's equations. The Helmholtz wave equation results from Maxwell's equations. The Helmholtz equation is linear; thus, superposition of solutions is allowed. Huygens' principle is that a point source of light will give rise to ...

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, show that the mCSP is linked to coupled dispersionless type via hodograph transformation, thus, its Lax pair can be deduced. Then bilinearization formulated dependent variable and transformations. One- two-dark soliton solutions found by Hirota’s bilinear method their properties anal...

A continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation in R. The algorithm requires only single-frequency scattering data. Using an initial guess from the Born approximation, each update is obtained via recursive linearization on the spatial frequency of a one-parameter family of plane waves by solving one forward and one adjoint problem o...

Exploiting the similarities between the Helmholtz wave equation and the Klein-Gordon equation, the former is linearized using the Feschbach-Villars procedure used for linearizing the Klein-Gordon equation. Then the Foldy-Wouthuysen iterative diagonalization technique is applied to obtain a Hamiltonian description for a system with varying refractive index. Besides reproducing all the traditiona...

Since supp f ⊂ Ω, uθ satisfies the plain Helmholtz equation with constant wavenumber outside of Ω, the radiation condition and uθ|∂Ω = gθ. So by solving the exterior problem of the Helmholtz equation (analytically if Ω is a circle), uθ can be computed everywhere outside of Ω without prior knowledge of f . In particular, we can compute h = ∂uθ ∂ν on ∂Ω. Thus, our original problem can be rewritte...

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, ...

A hybrid asymptotic-numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with a reflecting boundary. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine t...

This chapter is devoted to the investigation of the Dirichlet problem for the Helmholtz equation in a bounded domain. In Section 1, we briefly explain why and how boundary valued problems for the Helmholtz equation are solved. Using the properties ofDα, Tα, Fα as well as the projections defined by the singular Cauchy operator the Dirichlet problem for classical α−hyperholomophic functions are s...

In this paper we give analytical evidence of increasing stability in the Cauchy Problem for the Helmholtz equation when frequency is growing. This effect depends on convexity properties of the surface where the Cauchy Data are given and on some monotonicity properties of the variable coefficient of the Helmholtz equation. Proofs use Carleman estimates and the theory of elliptic and hyperbolic b...