Some remarks on a modified Helmholtz equation with inhomogeneous source

In this paper, we are going to consider the following problemequation(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)(φε,gε), a given function f   is known as the “source function”. We want to determine the solution u(·,y)u(·,y) for 0