Some remarks on a modified Helmholtz equation with inhomogeneous source

Abstract

In this paper, we are going to consider the following problem(1)Δu=k2u+f(x,y),(x,y)∈[0,π]×[0,1],u(0,y)=u(π,y)=0,0⩽y⩽1,uy(x,0)=g(x),0⩽x⩽π,u(x,0)=φ(x),0⩽x⩽πwith corresponding measured data functions (φε,gε), a given function f is known as the “source function”. We want to determine the solution u(·,y) for 0<y⩽1 from the Cauchy data (u,uy) at y=0. The problem is ill-posed, as the solution exhibits unstable dependence on the given data functions. The aim of this paper is to introduce new efficient regularization methods, such as truncation of high frequency and quasi-boundary-type methods, with explicit error estimates for an extended case (i.e. the inhomogeneous problem with f≠0 in Eq. (1)). In addition, we also carry out numerical experiments and compare numerical results of our methods with Qin and Wei’s methods [1] in homogeneous case, as well as to compare our numerical solutions with exact solutions in inhomogeneous case. It shows that our truncation and Qin and Wei’s truncation methods giving almost same results. However, our quasi-boundary-type methods show a better results than quasi-reversibility method of Qin and Wei in term of error estimation and convergence speed.