Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [25]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number k, when the convergence of KMF algorithm's [25] is ensured, our algorithm can be used as an acceleration of convergence.In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.We approach our algorithm using finite element method to obtain an accurate numerical results , to affirm theoretical results and to prove it's effectiveness.1. Introduction. The Helmholtz equation is act as a time-independent form of the wave equation. Therefore, it can be arises in wide range for the many branches in the science and engineering which depend on stationary oscillating processes. Especially, in physical phenomena such as aeroacoustics, vibration phenomena, wave

This paper is concerned with the Cauchy problem for the Helmholtz equation. Recently, some new works asked the convergence of the well‐known alternating iterative method. Our main result is to propose a new alternating algorithm based on relaxation technique. In contrast to the existing results, the proposed algorithm is simple to implement, converges for all choice of wave number, and it can be used as an acceleration of convergence in the case where the classical alternating algorithm converges. We present theoretical results of the convergence of our algorithm. The numerical results obtained using our relaxed algorithm and the finite element approximation show the numerical stability, consistency and convergence of this algorithm. This confirms the efficiency of the proposed method.

In this paper, we focus on obtaining an approximate solution of the two types of two- dimensional linear Volterra-Fredhom integral equations of the second kind. Series   solution method is reformulated and applied with different bases functions for finding an approximate solution (sometimes the exact solution) for the above two types of integral equations. This is done by computer program with the aid of the Maple code program version 13 for all the above prescribed methods. Furthermore, we proved some theoretical results on the convergence analysis of the presented methods.