The Dirichlet Problem for the Propagative Helmholtz Equation in a Cracked Domain without Compatibility Conditions at the Tips of the Cracks

2009 ◽  
Author(s):  
P. A. Krutitskii ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Dirichlet Problem ◽  
Helmholtz Equation ◽  
Compatibility Conditions

scholarly journals The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Helmholtz Equation ◽  
Exterior Domain ◽  
Index Zero ◽  
Compatibility Conditions ◽  
Unique Classical Solution ◽  
Asymptotic Formulae ◽  
Solution Gradient

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

The Dirichlet Problem on Wave Propagation in a 2D Exterior Cracked Domain without Compatibility Conditions at the Tips of the Cracks

2011 ◽  
Author(s):  
P. A. Krutitskii ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Wave Propagation ◽  
Dirichlet Problem ◽  
Compatibility Conditions
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