Boundary Value Problems of Electroelasticity with Concentrated Singularities

1994 ◽  
Vol 1 (5) ◽  
pp. 459-467
Author(s):  
T. Buchukuri ◽  
D. Yanakidi
Keyword(s):  
Singular Point ◽  
Boundary Value Problems ◽  
Power Function ◽  
Boundary Value ◽  
Linearly Independent

Abstract We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

Restoration of the linear potential in the Sturm-Liouville problem

2017 ◽  
Vol 12 (2) ◽  
pp. 152-156 ◽  
Author(s):  
A.M. Akhtyamov ◽  
I.M. Utyashev
Keyword(s):  
Boundary Value Problems ◽  
Variable Coefficient ◽  
Natural Frequencies ◽  
Boundary Value ◽  
Liouville Problem ◽  
Frequency Equation ◽  
Linear Potential ◽  
Linearly Independent ◽  
Sturm Liouville ◽  
Coefficient Of Elasticity

The problem of identifying the variable coefficient of elasticity of a medium with respect to natural frequencies of a string oscillating in this medium is considered. A method for solving the problem is found, based on the representation of linearly independent solutions of the differential equation in the form of Taylor series with respect to two variables, substituting them into the frequency equation, and determining the unknown coefficients of the linear function from this frequency equation. An analytical method has also been developed that allows one to prove the uniqueness or nonuniqueness of the restored polynomial coefficient of elasticity of a medium by a finite number of natural frequencies of oscillations of a string, and also to find a class of isospectral problems, that is, boundary value problems for which the eigenvalue spectra coincide. The latter is based on the method of variation of an arbitrary constant. We consider examples of finding isospectral classes, and also unique boundary value problems having a given spectrum.

scholarly journals DEGENERATE BOUNDARY-VALUE PROBLEMS WITH A PERTURBING MATRIX FOR A DERIVATIVE

2021 ◽  
pp. 29-37
Author(s):  
L. M. Shehda
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Boundary Value Problems ◽  
Differential System ◽  
Laurent Series ◽  
Boundary Value ◽  
Linear Vector ◽  
Linearly Independent ◽  
Bifurcation Of Solutions

In the paper, there is considered degenerated Noether boundary value problem with a perturbing matrix for a derivative, in which the boundary condition is given by a linear vector functional. We have proposed an algorithm to consrtuct a set of linearly independent solutions of boundary value problems with a small parameter in the general case, when the number of boundary conditions given by a linear vector functional does not match with the number of unknowns in a degenerate differential system. There is used the technique of pseudoinverse Moore-Penrose matrices. Applying the Vishik-Lyusternik method, the solution of the boundary value problem is obtained as part of the Laurent series in powers of small parameter. We obtain conditions for the bifurcation of solutions of linear degenerated Noether boundary-value problems with a small parameter under the assumption that the unperturbed degenerated differential system can be reduced to central canonical form.

scholarly journals Convergence of collocation schemes for boundary value problems in nonlinear index 1 DAEs with a singular point

2012 ◽  
Vol 82 (282) ◽  
pp. 893-918 ◽  
Author(s):  
Alexander Dick ◽  
Othmar Koch ◽  
Roswitha März ◽  
Ewa Weinmüller
Keyword(s):  
Singular Point ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlinear Index

On the boundary value problems of electro- and magnetostatics

1982 ◽  
Vol 92 (1-2) ◽  
pp. 165-174 ◽  
Author(s):  
Rainer Picard
Keyword(s):  
Hilbert Space ◽  
Euclidean Space ◽  
Boundary Value Problems ◽  
Vector Fields ◽  
Boundary Value ◽  
Dimensional Euclidean Space ◽  
Space Theory ◽  
3 Dimensional ◽  
Linearly Independent ◽  
Topological Characteristics

SynopsisThe classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.

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