scholarly journals Integral and Variational Formulations for the Helmholtz Equation Inverse Source Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Marcelo L. Rainha ◽  
Nilson C. Roberty
Keyword(s):  
Functional Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Helmholtz Equation ◽  
Green's Function ◽  
Equivalence Theorem ◽  
Functional Structure ◽  
Inverse Source Problem ◽  
Source Reconstruction ◽  
Variational Formulations

The purpose of this paper is to explore the Hilbert space functional structure of the Helmholtz equation inverse source problem. An integral equation for the sources reconstruction based on the composition of the trace and Green's function operators is introduced and compared with the reciprocity source reconstruction methodologies. An equivalence theorem comparing the integral inverse source equation with the variational weak reciprocity gap functional equation is then demonstrated. Some examples on applications to the unitary disk are presented.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

2011 ◽  
Vol 255-260 ◽  
pp. 1830-1835 ◽  
Author(s):  
Gang Cheng ◽  
Quan Cheng ◽  
Wei Dong Wang
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Free Vibration ◽  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
Integral Equation Method ◽  
Free Vibrations ◽  
Equation Method ◽  
Concentrated Mass

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

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