Acta Mathematica Vietnamica
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A NEW REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION WITH NONHOMOGENEOUS CAUCHY DATA
PHAM HOANG QUAN, PHAN TRUNG HIEUpp. 419-430
Abstract
In this paper, we investigate the Cauchy problem for the Helmholtz equation in the infinite strip $\{(x, y) \,\,|\,\, x \in \mathbb R,\,\, 0 < y < 1\}$ with nonhomogeneous Cauchy data given at $y = 0$. The problem is severely ill-posed. We shall use the Fourier transform to get an integral equation and give a regularized solution by directly perturbing this equation in combination with truncating high frequencies. The error estimate between the regularization solution and the exact solution is given. Finally, a numerical example shows the effectiveness of the proposed method.