An Existence Principle for Nonlocal Difference Boundary Value Problems with φ-Laplacian and Its Application to Singular Problems

AbstractExistence criteria are presented for nonlinear singular initial and boundary value problems. In particular our theory includes a problem arising in the theory of pseudoplastic fluids.

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Exact Solutions for Transverse Vibrations of Beams and Plates of Parabolic Thickness Variation

Design Engineering ◽

10.1115/imece2004-61204 ◽

2004 ◽

Cited By ~ 1

Author(s):

Dumitru I. Caruntu

Keyword(s):

Free Boundary ◽

Boundary Conditions ◽

Exact Solutions ◽

Orthogonal Polynomials ◽

Boundary Value Problems ◽

Boundary Value ◽

Thickness Variation ◽

Singular Problems ◽

Free Boundary Conditions ◽

Transverse Vibrations

This paper presents the class of nonuniform beams and nonuniform axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations, respectively, have been identified to be eigenvalue singular problems of orthogonal polynomials. Recent published results regarding a fourth order differential equation and eigenvalue singular problem of classical orthogonal polynomials allowed this study, which extends the class of nonuniform beams and circular nonuniform plates having exact solutions for the problem of free transverse vibrations. The geometry of the elements belonging to the class presented in this paper consists of beams convex parabolic thickness variation and polynomial width variation with the axial coordinate, and plates of convex parabolic thickness variation with the radius. Two boundary value problems of transverse vibrations of beams are reported: 1) complete beam (sharp at either end) with free-free boundary conditions, and 2) half-beam, i.e. a half of the symmetric complete beam, with the large end hinged and sharp end free. The boundary value problem of circular complete plate (zero thickness at zero and outer radii) with free-free boundary conditions has been also reported. For all these boundary value problems the exact mode shapes were Jacobi polynomials and the exact dimensionless natural frequencies were found from the eigenvalues of the eigenvalue singular problems of orthogonal polynomials.

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General Existence Principle for Singular BVPs and Its Application

Georgian Mathematical Journal ◽

10.1515/gmj.2004.549 ◽

2004 ◽

Vol 11 (3) ◽

pp. 549-565

Author(s):

Irena Rachůnková ◽

Svatoslav Staněk

Keyword(s):

Boundary Value Problems ◽

Large Class ◽

Closed Subset ◽

Boundary Value ◽

Singular Boundary ◽

Singular Boundary Value Problems ◽

Carathéodory Conditions ◽

Existence Principle ◽

General Existence

Abstract We present a general existence principle which can be used for a large class of singular boundary value problems of the form where 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × 𝐷, a set 𝐷 ⊂ is not closed, 𝑓 has singularities in its phase variables on the boundary ∂𝐷 of 𝐷, and 𝑆 is a closed subset in 𝐶𝑛–1([0, 𝑇]). The proof is based on the regularization and sequential techniques. An application of the general existence principle to singular conjugate (𝑝, 𝑛–𝑝) BVPs is also given.

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