Regularization and New Error Estimates for a Modified Helmholtz Equation

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 modified regularization method, we regularize the problem and to get some new error estimates. The numerical results show that our methods work effectively. This paper extends the work by T.Wei and H.H.Quin[8].