Abstract
In this paper, we consider the inverse problem of identifying the unknown source for the modified Helmholtz equation. We propose the Landweber iterative regularization method to solve this problem and obtain the regularization solution. Under the a priori and a posteriori regularization parameters choice rules, we all obtain the Hölder type error estimates between the exact solution and the regularization solutions. Several numerical examples are also provided to show that the Landweber iterative method works well for solving this problem.
Highlights
The modified Helmholtz equation or the Yukawa equation which is pointed out in [ ] appears in implicit matching schemes for the heat equation, in Debye-Huckel theory, and in the linearization of the Poisson-Boltzmann equation
The Cauchy problems associated with the modified Helmholtz equation have been studied by using different numerical methods, such as the Landweber method with boundary element method and the conjugate gradient method [ ], the method of fundamental solutions (MFS) [ ], the iteration regularization method [ ], Tikhonov type regularization [ ], Quasi-reversibility and truncation methods [ – ], Quasi-boundary regularization [, ], and so on
In [ ], the authors used the simplified Tikhonov regularization method to identify the unknown source for the modified Helmholtz equation
Summary
The modified Helmholtz equation or the Yukawa equation which is pointed out in [ ] appears in implicit matching schemes for the heat equation, in Debye-Huckel theory, and in the linearization of the Poisson-Boltzmann equation. In [ ], the authors used the simplified Tikhonov regularization method to identify the unknown source for the modified Helmholtz equation. In [ , ], the authors used the Tikhonov regularization method to identify the unknown source for the modified Helmholtz equation. There is a defect for any a priori method, i.e., the a priori choice of the regularization parameter depends on the a priori bound E of the unknown solution. We use the Landweber iterative method to identify the unknown source of the modified Helmholtz equation. We use the Landweber iterative method to obtain the regularization solution for
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