In this paper, the Cauchy problem for the modified Helmholtz equation is investigated in a rectangle, where the Cauchy data is given for y=0 and boundary data for x=0 and x=π. The solution is sought in the interval 0<y≤1. We propose a quasi-boundary value regularization method to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates. In addition, we also carry out numerical experiments and compare numerical results of our method with Qin's methods [Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation, Math. Comput. Simulation 80 (2009), pp. 352–366] and Tuan's methods [Regularization and new error estimates for a modified Helmholtz equation, An. Stiint Univ. ‘Ovidius' Constanta Ser. Mat. 18(2) (2010), pp. 267–280]. It shows that our quasi-boundary value method give a better results than quasi-reversibility method of Qin and modified regularization method of Tuan.
['Ailin Qian', 'Xiao-Mei Yang', 'Yousheng Wu']
Optimal error bound and a quasi-boundary value regularization method for a Cauchy problem of the modified Helmholtz equation