Showing papers on "Boundary value problem published in 2015"

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis

Abstract: Isotropic and Anisotropic Function Spaces Lebesgue and Sobolev Spaces with Variable Exponent History of function spaces with variable exponent Lebesgue spaces with variable exponent Sobolev spaces with variable exponent Dirichlet energies and Euler-Lagrange equations Lavrentiev phenomenon Anisotropic function spaces Orlicz spaces Variational Analysis of Problems with Variable Exponents Nonlinear Degenerate Problems in Non-Newtonian Fluids Physical motivation A boundary value problem with nonhomogeneous differential operator Nonlinear eigenvalue problems with two variable exponents A sublinear perturbation of the eigenvalue problem associated to the Laplace operator Variable exponents versus Morse theory and local linking The Caffarelli-Kohn-Nirenberg inequality with variable exponent Spectral Theory for Differential Operators with Variable Exponent Continuous spectrum for differential operators with two variable exponents A nonlinear eigenvalue problem with three variable exponents and lack of compactness Concentration phenomena: the case of several variable exponents and indefinite potential Anisotropic problems with lack of compactness and nonlinear boundary condition Nonlinear Problems in Orlicz-Sobolev Spaces Existence and multiplicity of solutions A continuous spectrum for nonhomogeneous operators Nonlinear eigenvalue problems with indefinite potential Multiple solutions in Orlicz-Sobolev spaces Neumann problems in Orlicz-Sobolev spaces Anisotropic Problems: Continuous and Discrete Anisotropic Problems Eigenvalue problems for anisotropic elliptic equations Combined effects in anisotropic elliptic equations Anisotropic problems with no-flux boundary condition Bifurcation for a singular problem modelling the equilibrium of anisotropic continuous media Difference Equations with Variable Exponent Eigenvalue problems associated to anisotropic difference operators Homoclinic solutions of difference equations with variable exponents Low-energy solutions for discrete anisotropic equations Appendix A: Ekeland Variational Principle Appendix B: Mountain Pass Theorem Bibliography Index A Glossary is included at the end of each chapter.

Solving Boundary Integral Problems with BEM

Abstract: Many important partial differential equation problems in homogeneous media, such as those of acoustic or electromagnetic wave propagation, can be represented in the form of integral equations on the boundary of the domain of interest. In order to solve such problems, the boundary element method (BEM) can be applied. The advantage compared to domain-discretisation-based methods such as finite element methods is that only a discretisation of the boundary is necessary, which significantly reduces the number of unknowns. Yet, BEM formulations are much more difficult to implement than finite element methods. In this article, we present BEMpp, a novel open-source library for the solution of boundary integral equations for Laplace, Helmholtz and Maxwell problems in three space dimensions. BEMpp is a Cpp library with Python bindings for all important features, making it possible to integrate the library into other Cpp projects or to use it directly via Python scripts. The internal structure and design decisions for BEMpp are discussed. Several examples are presented to demonstrate the performance of the library for larger problems.

Theoretical and numerical aspects of the open source BEM solver NEMOH

Abstract: The aim of this paper is to provide the key theoretical and numerical aspects of the NEMOH software. NEMOH is a numerical solver for computations of first order hydrodynamic coefficients in frequency domain. It is based on linear free surface potential flow theory. It has been developed in Ecole Centrale de Nantes for 30 years. It has been released in open source in January 2014. This paper starts with recalling the assumptions and the resulting boundary value problem for the free surface flow around a body with an arbitrary condition for the normal velocity on the body surface. Then, it is recalled how the 3D flow problem can be transformed into the 2D problem of a source distribution on the body surface using Green's second identity and the appropriate Green function. Discretization of the mathematical problem using the constant panel method leads to a linear matrix problem whose coefficients are the influence coefficients. Depending on the distance between the source and field point, the influence coefficients are calculated analytically or approximately. Irregular frequencies are removed by adding constraints on the interior flow problem. Hydrodynamic coefficients for usual diffraction and radiation problems can be obtained by specifying to the solver the appropriate body conditions. The flexibility in specifying the body conditions allows dealing with complex structures, e.g deformable wave energy converters. Eventually, it is explained how far field coefficients and free surface elevation can be obtained from the solver outputs.

Effect of uniform suction on nanofluid flow and heat transfer over a cylinder

Abstract: The aim of the present paper is to study the nanofluid flow and heat transfer over a stretching porous cylinder. The effective thermal conductivity and viscosity of the nanofluid are calculated by KKL (Koo–Kleinstreuer–Li) correlation. In KKL model, the effect of Brownian motion on the effective thermal conductivity is considered. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions using similarity transformation, which is then solved numerically by the fourth-order Runge–Kutta integration scheme featuring a shooting technique. Numerical results for flow and heat transfer characteristics are obtained for various values of the nanoparticle volume fraction, suction parameter, Reynolds number and different kinds of nanofluids. Results show that inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern. It is found that Nusselt number is an increasing function of nanoparticle volume fraction, suction parameter and Reynolds number.

The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

Abstract: We present the result for the finite part of the two-loop sunrise integral with unequal masses in four space-time dimensions in terms of the O(e0)-part and the O(e1)-part of the sunrise integral around two space-time dimensions. The latter two integrals are given in terms of elliptic generalisations of Clausen and Glaisher functions. Interesting aspects of the result for the O(e1)-part of the sunrise integral around two space-time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms.

A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory

Abstract: A new non-classical sinusoidal plate model is developed on the basis of modified strain gradient theory. This model takes into account the effects of shear deformation without any shear correction factors and also can capture the size effects due to additional material length scale parameters. The governing equations and corresponding boundary conditions for bending, buckling, and free vibration analysis of the microplate are derived by implementing Hamilton’s principle. Analytical solutions based on the Fourier series solution are presented for simply supported square microplates. A detailed parametric study is performed to demonstrate the influences of thickness-to-length scale parameter ratio, length-to-thickness ratio, and shear deformation on deflection, critical buckling load, and fundamental frequencies of microplates. It is observed that the effect of shear deformation becomes more significant for smaller values of length-to-thickness ratio.

A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

Abstract: This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton\'s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

Variational formulation of problems involving fractional order differential operators

Abstract: In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space Hα/2 0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

Abstract: This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.

Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

Abstract: This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.

Methods of Nonlinear Analysis: Applications to Differential Equations

Abstract: Preface.- 1 Preliminaries.- 2 Properties of Linear and Nonlinear Operators.- 3 Abstract Integral and Differential Calculus.- 4 Local Properties of Differentiable Mappings.- 5 Topological and Monotonicity Methods.- 6 Variational Methods.- 7 Boundary Value Problems for Partial Differential Equations.- Summary of Methods.- Typical Applications.- Comparison of Bifurcation Results.- List of Symbols.- Index.- Bibliography.

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium

Abstract: Energy transfer in mixed convection unsteady magnetohydrodynamic (MHD) flow of an incompressible nanofluid inside a channel filled with saturated porous medium is investigated. The channel with non-uniform walls temperature is taken in a vertical direction under the influence of a transverse magnetic field. Based on the physical boundary conditions, three different flow situations are discussed. The problem is modelled in terms of partial differential equations with physical boundary conditions. Four different shapes of nanoparticles of equal volume fraction are used in conventional base fluids, ethylene glycol (EG) (C2H6O2) and water (H2O). Solutions for velocity and temperature are obtained discussed graphically in various plots. It is found that viscosity and thermal conductivity are the most prominent parameters responsible for different results of velocity and temperature. Due to higher viscosity and thermal conductivity, C2H6O2 is regarded as better convectional base fluid compared to H2O.