We present an iterative solution method for the discrete high wavenumber Helmholtz equation. The basic idea of the solution method, already presented in [18], is to develop a preconditioner which is based on a Helmholtz operator with a complex-valued shift, for a Krylov subspace iterative method. The preconditioner, which can be seen as a strongly damped wave equation in Fourier space, can be a...

We consider the following Cauchy problem for the Helmholtz equation with Dirichlet boundary conditions at x = 0 and x = π    ∆u− ku = 0, (x, y) ∈ (0, π)× (0, 1) u(0, y) = u(π, y) = 0, y ∈ (0, 1) uy(x, 0) = f(x), (x, y) ∈ (0, π)× (0, 1) u(x, 0) = φ(x), 0 < x < π (1) The problem is shown to be ill-posed, as the solution exhibits unstable dependence on the given data functions. Using a modifi...

In this paper, the Cauchy problem for the modified Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a new regularization method to solve it based on the solution given by the method of separation of variables. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effec...

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time ...

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We present a general method for solving the modified Helmholtz equation without shape approximation an arbitrary periodic charge distribution, whose solution is known as Yukawa potential or screened Coulomb potential. The extension of Weinert’s pseudo-charge [Weinert M, J Math Phys , 1981, 22:2433–2439] Poisson same class density distributions. inherent differences between and are in their resp...