Differential forms, spectral theory, and boundary value problems

2002 ◽  
pp. 69-92
Author(s):  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Spectral Theory ◽  
Boundary Value ◽  
Differential Forms

Boundary value problems for differential forms on compact Riemannian manifolds, Part II

Analysis ◽  
2007 ◽  
Vol 27 (4) ◽  
Author(s):  
Jürgen Bolik
Keyword(s):  
Boundary Value Problems ◽  
Riemannian Manifolds ◽  
Boundary Value ◽  
Differential Forms ◽  
Second Order ◽  
First Order

SummaryThis paper provides solutions to second order boundary value problems for differential forms by means of the method applied in [3] for first order problems. These

scholarly journals MODELING OF FINITE INHOMOGENEITIES BY DISCRET SINGULARITIES

2021 ◽  
pp. 138-144
Author(s):  
G. M. Zrazhevsky ◽  
V. F. Zrazhevska
Keyword(s):  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Differential Forms ◽  
Elastic Beam ◽  
Convergent Series ◽  
Boundary Integral ◽  
Model Description ◽  
Nonsmooth Coefficients ◽  
The Impact

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

scholarly journals Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Shurong Sun ◽  
Martin Bohner ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Spectral Theory ◽  
Hamiltonian Systems ◽  
Dynamic Systems ◽  
Time Scale ◽  
Boundary Value ◽  
Hamiltonian Dynamic ◽  
Special Cases ◽  
Linear Hamiltonian Systems ◽  
Singular Hamiltonian Systems

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for𝕋=ℝand𝕋=ℤwithin one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i)M(λ)theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

scholarly journals The variational iteration method for solving the Volterra integro-differential forms of the Lane-Emden and the Emden-Fowler problems with initial and boundary value conditions

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Abdul-Majid Wazwaz ◽  
Suheil A. Khuri
Keyword(s):  
Boundary Value Problems ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Differential Forms ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Shape Factors ◽  
Variational Iteration

AbstractIn this paper, the variational iteration method (VIM) is used to examine the Volterra integro-differential forms of the singular Lane–Emden and the Emden–Fowler initial value problems and boundary value problems arising in physics, astrophysics and stellar structures. The Volterra integro-differential forms of the Lane–Emden and the Emden–Fowler equations overcome the singularity behavior at the origin x = 0. The Lagrange multiplier, needed for the VIM, is λ = −1 for the various cases of the specified equations having distinct shape factors. We illustrate our work by analyzing few initial value problems and boundary value problems to emphasize the convergence of the acquired results.

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