Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667

Hongwu Zhang; Xiaoju Zhang

doi:10.3390/math7100888

and

¹

School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China

²

Development Center of Teachers’ Teaching, North Minzu University, Yinchuan 750021, China

^*

Author to whom correspondence should be addressed.

Mathematics2019, 7(10), 888;https://doi.org/10.3390/math7100888

This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications

Version Notes

Order Reprints

The authors wish to make the following corrections to this paper [1]:

1. In the original paper, the value of the parameter

γ

is greater than 0. Throughout the article, the value of the parameter

γ

should be

γ \geq 1

.

2. For the definition of function

H_{2} (n)

in (23), the original expression is

H_{2} (n) = \frac{e^{- (2 T - y) \sqrt{n^{2} + k^{2}}}}{\sqrt{K} (\frac{β {(n^{2} + k^{2})}^{γ}}{K} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} + e^{- 2 T \sqrt{n^{2} + k^{2}}})} .

We would like to make the following corrections:

H_{2} (n) = \frac{e^{- (2 T - y) \sqrt{n^{2} + k^{2}}}}{\sqrt{K} (β {(n^{2} + k^{2})}^{γ - 1} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} + e^{- 2 T \sqrt{n^{2} + k^{2}}})} .

3. In the proof of Theorem 4, the original value of

ν

is

ν = \frac{β {(n^{2} + k^{2})}^{γ}}{K} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} .

It is corrected as:

ν = β {(n^{2} + k^{2})}^{γ - 1} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} .

4. The original mathematical expression

\frac{e^{- (2 T - y) \sqrt{n^{2} + k^{2}}}}{\sqrt{K} (\frac{β {(n^{2} + k^{2})}^{γ}}{K} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} + e^{- 2 T \sqrt{n^{2} + k^{2}}})}

in line three of (49), and line six of (50) is inappropriate.

They should both be:

\frac{e^{- (2 T - y) \sqrt{n^{2} + k^{2}}}}{\sqrt{K} (β {(n^{2} + k^{2})}^{γ - 1} {(\frac{1 - e^{- 2 \sqrt{K} T}}{2})}^{2} + e^{- 2 T \sqrt{n^{2} + k^{2}}})} .

The authors would like to apologize for any inconvenience caused to the readers by these changes.

References

Zhang, H.; Zhang, X. Generalized Tikhonov method and convergence estimate for the Cauchy problem of modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum. Mathematics 2019, 7, 667. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667

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