scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.

Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Bénard layer eigenvalue problems

1990 ◽  
Vol 431 (1883) ◽  
pp. 433-450 ◽  
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Second Order ◽  
Sixth Order ◽  
Asymptotic Estimates ◽  
Eighth Order ◽  
The Family ◽  
Convergent Method

A family of numerical methods is developed for the solution of special nonlinear sixth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order convergence are contained in the family. The problem is also solved by writing the sixth-order differential equation as a system of three second-order differential equations. A family of second- and fourth-order convergent methods is then used to obtain the solution. A second-order convergent method is discussed for the numerical solution of general nonlinear sixth-order boundary-value problems. This method, with modifications where necessary, is applied to the sixth-order eigenvalue problems associated with the onset of instability in a Bénard layer. Numerical results are compared with asymptotic estimates appearing in the literature.

scholarly journals On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Mahir Hasanov
Keyword(s):  
Quantitative Analysis ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Variational Principles ◽  
Travelling Waves ◽  
Boundary Value ◽  
Dispersion Relations ◽  
Duality Principle ◽  
Nonlinear Schrödinger ◽  
Generalized Nonlinear Schrödinger Equation

This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbedp-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbedp-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.

scholarly journals Eigenvalue problems for anisotropic discrete boundary value problems

2009 ◽  
Vol 15 (6) ◽  
pp. 557-567 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Vicenţiu Rădulescu ◽  
Stepan Tersian
Keyword(s):  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Boundary Value

scholarly journals A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

2021 ◽  
Author(s):  
Manting Xie ◽  
Fei Xu ◽  
Meiling Yue
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Multigrid Method ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Positive Definite ◽  
Adaptive Finite Element Method ◽  
Steklov Eigenvalue ◽  
Full Multigrid Method ◽  
Steklov Eigenvalue Problems

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

scholarly journals Existence Theorems for Some Classes of Boundary Value Problems Involving the P(X)-Laplacian

2008 ◽  
Vol 13 (2) ◽  
pp. 145-158
Author(s):  
Ionica Andrei
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Theoretical Approach ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Existence Theorems ◽  
Boundary Value ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We prove an alternative for a nonlinear eigenvalue problem involving the p(x)-Laplacian and study a subcritical boundary value problem for the same operator. The theoretical approach is the Mountain Pass Lemma and one of its variants, which is very useful in the study of eigenvalue problems.

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