On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

2006 ◽  
Vol 58 (4) ◽  
pp. 487-504 ◽  
Author(s):  
V. P. Burskii ◽  
A. S. Zhedanov
Keyword(s):  
Dirichlet Problem ◽  
Abel Equation ◽  
String Equation

Existence of Solutions of an Ill-Posed Problem for the Vibrating String

1982 ◽  
Vol 25 (1) ◽  
pp. 29-36
Author(s):  
L. L. Campbell
Keyword(s):  
Dirichlet Problem ◽  
Simple Formula ◽  
Existence Of Solutions ◽  
Sufficient Condition ◽  
String Equation ◽  
Boundary Values ◽  
Necessary And Sufficient Condition ◽  
Vibrating String ◽  
Ill Posed ◽  
Necessary And Sufficient

AbstractThe Dirichlet problem is examined for the vibrating string equation on a rectangle with commensurable sides. As is well-known, a solution, if it exists, is not unique. A necessary and sufficient condition is obtained on the boundary values for existence of solutions. A simple formula for the solution is obtained.

scholarly journals Three spectra problem for Stieltjes string equation and Neumann conditions

2019 ◽  
Vol 12 (1) ◽  
pp. 41-55 ◽  
Author(s):  
Anastasia Dudko ◽  
Vyacheslav Pivovarchik
Keyword(s):  
Dirichlet Problem ◽  
Neumann Problem ◽  
Dirichlet Condition ◽  
The Other ◽  
Neumann Condition ◽  
String Equation ◽  
Dirichlet Problems ◽  
Transversal Vibrations ◽  
The Right ◽  
The Masses

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

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