Asymptotically expansible solutions of the Helmholtz equation

Sampling and reconstruction of solutions to the Helmholtz equation

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain Ω from their values at scattered points x1, . . . , xn ⊂ Ω. This problem typically arises when sampling acoustic fields with n microphones for the purpose of reconstructing this field over a region of interest Ω contained in a larger domain D in which the acoustic field propagates. In ma...

Reconstruction of solutions to the Helmholtz equation from punctual measurements

We analyze the sampling of solutions to the Helmholtz equation (e.g. sound fields in the harmonic regime) using a least-squares method based on approximations of the solutions by sums of Fourier-Bessel functions or plane waves. This method compares favorably to others such as Orthogonal Matching Pursuit with a Fourier dictionary. We show that using a significant proportion of samples on the bor...

Heteroclinic Solutions to an Asymptotically Autonomous Second-Order Equation

We study the differential equation ẍ(t) = a(t)V ′(x(t)), where V is a double-well potential with minima at x = ±1 and a(t) → l > 0 as |t| → ∞. It is proven that under certain additional assumptions on a, there exists a heteroclinic solution x to the differential equation with x(t)→ −1 as t→ −∞ and x(t) → 1 as t → ∞. The assumptions allow l − a(t) to change sign for arbitrarily large values of |...

WAVELET SOLUTIONS OF THE KLEIN-GORDON EQUATION

The Helmholtz Equation on LipschitzDomainsChangmei

We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ((+k 2)u = 0 on a Lipschitz domain for all wave number k 2C with Im k 0. Following the approach for the case of smooth boundary 3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothne...