Showing papers on "Boundary value problem published in 2013"

Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results

Abstract: Preface. List of Notations. List of Acronyms. Part I: Theory. 1.Introduction. 2. Auxiliary Results. 3. Algorithms of Nonsmooth Optimization. 4. Generalized Equations. 5. Stability of Solutions to Perturbed Generalized Equations. 6. Derivatives of Solutions to Perturbed Generalized Equations. 7. Optimality Conditions and a Solution Method. Part II: Applications. 8. Introduction. 9. Membrane with Obstacle. 10. Elasticity Problems with Internal Obstacles. 11. Contact Problem with Coulomb Friction. 12. Economic Applications. Appendices: A. Cookbook. B. Basic Facts on Elliptic Boundary Value problems. C. Complementarity Problems. References. Index.

Stabilization of a System of $n+1$ Coupled First-Order Hyperbolic Linear PDEs With a Single Boundary Input

Abstract: We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and one leftward convecting transport PDE. We design a controller, which requires a single control input applied on the leftward convecting PDE's right boundary, and an observer, which employs a single sensor on the same PDE's left boundary. We prove exponential stability of the origin of the resulting plant-observer-controller system in the spatial L2-sense.

Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

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Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations

Abstract: We consider the initial boundary value problem for a homogeneous time-fractional diffusion equation with an initial condition $v(x)$ and a homogeneous Dirichlet boundary condition in a bounded convex polygonal domain $\Omega$. We study two semidiscrete approximation schemes, i.e., the Galerkin finite element method (FEM) and lumped mass Galerkin FEM, using piecewise linear functions. We establish almost optimal with respect to the data regularity error estimates, including the cases of smooth and nonsmooth initial data, i.e., $v \in H^2(\Omega)\cap H^1_0(\Omega)$ and $v \in L_2(\Omega)$. For the lumped mass method, the optimal $L_2$-norm error estimate is valid only under an additional assumption on the mesh, which in two dimensions is known to be satisfied for symmetric meshes. Finally, we present some numerical results that give insight into the reliability of the theoretical study.

Local exponential H2 stabilization of 2x2 quasilinear hyperbolic systems using backstepping

Abstract: In this work, we consider the problem of boundary stabilization for a quasilinear $2\times2$ system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves $H^2$ exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type $4\times4$ system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.

A Fast and Well-Conditioned Spectral Method

Abstract: A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions. The method leads to matrices that are al...

A Qualitative Approach to Inverse Scattering Theory

Abstract: 1 Functional Analysis and Sobolev Spaces- 2 Ill-Posed Problems- 3 Scattering by Imperfect Conductors- 4 Inverse Scattering Problems for Imperfect Conductors- 5 Scattering by Orthotropic Media- 6 Inverse Scattering Problems for Orthotropic Media- 7 Factorization Methods- 8 Mixed Boundary Value Problems- 9 Inverse Spectral Problems for Transmission Eigenvalues- 10 A Glimpse at Maxwell's Equations

Disturbance rejection in 2 x 2 linear hyperbolic systems

Abstract: Many interesting problems in the oil and gas industry face the challenge of responding to disturbances from afar. Typically, the disturbance occurs at the inlet of a pipeline or the bottom of an oil well, while sensing and actuation equipment is installed at the outlet, only, kilometers away from the disturbance. The present paper develops an output feedback control law for such cases, based on modelling the transport phenomenon as a 2 $\times$ 2 linear partial differential equation (PDE) of hyperbolic type and the disturbance as a finite-dimensional linear system affecting the left boundary of the PDE. Sensing and actuation are co-located at the right boundary of the PDE. The design provides a separation principle, allowing a disturbance attenuating full-state feedback control law to be combined with an observer. The results are applied to a relevant problem from the oil and gas industry and demonstrated in simulations.

Buckling analysis of functionally graded microbeams based on the strain gradient theory

Abstract: The buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) for different boundary conditions is investigated on the basis of Bernoulli–Euler beam and modified strain gradient theory. The higher-order governing differential equation for buckling with all possible classical and non-classical boundary conditions is obtained by a variational statement. The effects of the power of the material property variation function, boundary conditions, slenderness ratio, ratio of additional material length scale parameters for two constituents, beam thickness-to-additional material length scale parameter ratio on the buckling response of FGM microbeams are investigated. Some comparative results are presented in tabular and graphical form in order to show the differences between the results obtained by the present model and those predicted by modified couple stress and classical continuum models.

Some existence results on nonlinear fractional differential equations

Abstract: In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η

Nonlinear Second Order Elliptic Equations Involving Measures

Abstract: This book presents a comprehensive study of boundary value problems for linear and semilinear second order elliptic equations with measure data,especially semilinear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role.

Isogeometric divergence-conforming b-splines for the steady navier–stokes equations

Abstract: We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method

Abstract: SUMMARY Enforcing essential boundary conditions plays a central role in immersed boundary methods. Nitsche's idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints. We formulate an extension of Nitsche's method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the B-spline and non-uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries. Furthermore, we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3D geometries. With several benchmark problems, we demonstrate the overall good convergence behavior of the proposed method and its good accuracy. We provide extensive studies on the stability of the method, its influence parameters and numerical properties, and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two. A newly composed boundary integration concept further enhances the modeling process and allows a flexible, discretization-independent introduction of boundary conditions. Finally, we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non-uniform rational basis spline geometries. Copyright © 2013 John Wiley & Sons, Ltd.

Nanofluid flow and heat transfer due to a stretching cylinder in the presence of magnetic field

Abstract: In this paper, flow and heat transfer of a nanofluid over a stretching cylinder in the presence of magnetic field has been investigated. 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. Different types of nanoparticles as copper (Cu), silver (Ag), alumina (Al2O3) and titanium oxide (TiO2) with water as their base fluid has been considered. The influence of significant parameters such as nanoparticle volume fraction, nanofluids type, magnetic parameter and Reynolds number on the flow and heat transfer characteristics is discussed. It was found that the Nusselt number increases as each of Reynolds number or nanoparticles volume fraction increase, but it decreases as magnetic parameter increase. Also it can be found that choosing copper (for small of magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem.

Three-particle scattering amplitudes from a finite volume formalism

Abstract: We present a quantization condition for the spectrum of a system composed of three identical bosons in a finite volume with periodic boundary conditions. This condition gives a relation between the finite volume spectrum and infinite volume scattering amplitudes. The quantization condition presented is an integral equation that in general must be solved numerically. However, for systems with an attractive two-body force that supports a two-body bound state, a diboson, and for energies below the diboson breakup, the quantization condition reduces to the well-known L\"uscher formula with exponential corrections in volume that scale with the diboson binding momentum. To accurately determine infinite volume phase shifts, it is necessary to extrapolate the phase shifts obtained from the L\"uscher formula for the boson-diboson system to the infinite volume limit. For energies above the breakup threshold, or for systems with no two-body bound state (with only scattering states and resonances) the L\"uscher formula gets power-law volume corrections and consequently fails to describe the three-particle system. These corrections are nonperturbatively included in the quantization condition presented.

New Boundary Conditions for AdS3

Abstract: New chiral boundary conditions are found for quantum gravity with matter on AdS3. The associated asymptotic symmetry group is generated by a single right-moving U(1) Kac-Moody-Virasoro algebra with $ {c_R}=\frac{{3\ell }}{2G } $ . The Kac-Moody zero mode generates global left-moving translations and equals, for a BTZ black hole, the sum of the total mass and spin. The level is positive about the global vacuum and negative in the black hole sector, corresponding to ergosphere formation. Realizations arising in Chern-Simons gravity and string theory are analyzed. The new boundary conditions are shown to naturally arise for warped AdS3 in the limit that the warp parameter is taken to zero.

An Efficient Shear Deformation Beam Theory Based on Neutral Surface Position for Bending and Free Vibration of Functionally Graded Beams

Abstract: In this article, an efficient shear deformation beam theory based on neutral surface position is developed for bending and frees vibration analysis of functionally graded beams. The theory accounts for hyperbolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The neutral surface position for a functionally graded beam in which its material properties vary in the thickness direction is determined. Based on the present higher order shear deformation beam theory and the neutral surface concept, the equations of motion are derived from Hamilton's principle. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and vibration behavior of functionally graded beams.

Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

Abstract: We compute the partition function on the hemisphere of a class of twodimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section

Abstract: This paper presents a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation. Moreover, all physical quantities can be expressed in terms of the solution of the resulting equation. Making use of power series for unknown function, we can transform the single equation to a system of linear algebraic equations and will get a characteristic equation in natural frequencies for different boundary conditions. An advantage of the suggested approach is that the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-roots. Several examples of estimating natural frequencies for axially grade beams and non-uniform beams are presented, which show that our method has fast convergence and obtained numerical results have high accuracy.