## A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation.(English, Russian)Zbl 1212.65355

Summary: This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain $$0<x \leqslant 1$$, $$y\in \mathbb R$$. The Cauchy data at $$x = 0$$ is given and the solution is then sought for the interval $$0<x \leqslant 1$$. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under suitable choices of the regularization parameters and the a priori assumption on the bounds of the exact solution. A numerical implementation is considered and numerical examples show that the proposed method is effective and stable.

### MSC:

 65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35R25 Ill-posed problems for PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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