Generalized Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation
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This paper considers the Cauchy problem of a semi-linear elliptic equation and uses a generalized Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness, and stability for regularized solution are proven. Under an a priori bound assumption for exact solution, we derive the convergence estimate of Hölder type for this method. An application of this method to the Cauchy problem of Helmholtz equation is discussed, and we investigate the stability and convergence estimates for different wave numbers. Finally, an iterative scheme is constructed to calculate the regularization solution, numerical results show that this method is stable and feasible.
KeywordsIll-posed problem Cauchy problem Semi-linear elliptic equation Generalized Tikhonov-type regularization method Convergence estimate
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The authors would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper.
The work was supported by the YSRP (2017KJ33), NSF of China (11761004, 11371181), and SRP (2017SXKY05) at North Minzu University.
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