Homogeneous Boundary Value Problem Research Articles

Impulsive Boundary Value Problems for Planar Hamiltonian Systems

We give an existence and uniqueness theorem for solutions of inhomogeneous impulsive boundary value problem (BVP) for planar Hamiltonian systems. Green's function that is needed for representing the solutions is obtained and its properties are listed. The uniqueness of solutions is connected to a Lyapunov type inequality for the corresponding homogeneous BVP.

TE AND TM MODES OF A VANE-LOADED CIRCULAR CYLINDRICAL WAVEGUIDE FOR GYRO-TWT APPLICATIONS

The dispersion equation governing the guided propagation of TE and TM fast wave modes of a circular cylindrical waveguide loaded by metal vanes positioned symmetrically around the wave-guide axis is derived from the exact solution of a homogeneous boundary value problem for Maxwell's equations. The dispersion equation takes the form of the solvability condition for an inflnite system of linear homogeneous algebraic equations. The approximate dispersion equation corresponding to a truncation of the inflnite-order coe-cient matrix of the inflnite system of equations to the coe-cient matrix of a flnite system of equations of su-ciently high order is solved numerically to obtain the cut-ofi wave numbers of the various propagating modes. Each cut-ofi wave number gives rise to a unique dispersion curve in the shape of a hyperbola in the !-fl plane.

Stroh formalism in analysis of skew-symmetric and symmetric weight functions for interfacial cracks

The focus of the paper is on the analysis of skew-symmetric weight functions for interfacial cracks in two-dimensional anisotropic solids. It is shown that the Stroh formalism proves to be an efficient tool for this challenging task. Conventionally, the weight functions, both symmetric and skew-symmetric, can be identified as non-trivial singular solutions of a homogeneous boundary-value problem for a solid with a crack. For a semi-infinite crack, the problem can be reduced to solving a matrix Wiener–Hopf functional equation. Instead, the Stroh matrix representation of displacements and tractions, combined with a Riemann–Hilbert formulation, is used to obtain an algebraic eigenvalue problem, which is solved in a closed form. The proposed general method is applied to the case of a quasi-static semi-infinite crack propagating between two dissimilar orthotropic media: explicit expressions for the weight functions are evaluated and then used in the computation of the complex stress intensity factor corresponding to a general distribution of forces acting on the crack faces.

Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace

We consider a second-order hyperbolic equation on an open bounded domain Ω in ℝ n for n ≥ 2, with C 2-boundary , Γ0 ∩ Γ1 = ∅, subject to non-homogeneous Dirichlet boundary conditions on the entire boundary Γ. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ0 = Γ\\Γ1, T > 0, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and Lipschitz stability (at the H θ-level, 0 < θ ≤ 1, ). The latter (ii) is the main result of this article. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H 1 × L 2-level for second-order hyperbolic equations [I. Lasiecka, R. Triggiani, and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Contemp. Math. 268 (2000), pp. 227–325] (b) a correspondingly implied continuous observability inequality at the same energy level Lasiecka et al.; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [I. Lasiecka, J.L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), pp. 149–192; I. Lasiecka and R. Triggiani, A cosine operator approach to modelling L 2(0, T; L 2(Ω)) boundary input hyperbolic equations, Appl. Math. Optimiz., 7 (1981), 35–83; I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under L 2(0, T; L 2(Ω))-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), pp. 243–290]. The proof of the linear uniqueness result (Section 4, Step 5) also takes advantage of a convenient tactical route ‘post-Carleman estimates’ suggested by Isakov in [V. Isakov and M. Yamamoto, Carleman estimate with Neumann B.C. and its application to the observability inequality and inverse hyperbolic problems, Contemp. Math. 268 (2000), pp. 191–225, Theorem 8.2.2, p. 231].

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p − 1) < 1, r ≤ 1, and . By using Lp-integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.

Nonzero positive weak solutions of systems of [formula omitted]-Laplace equations

We obtain new results on the existence of nonzero positive weak solutions of systems of p-Laplace equations under some sublinear conditions. These sublinear conditions employ the principal eigenvalues of the corresponding homogeneous Dirichlet boundary value problems involving p-Laplacian operators. Our results generalize and improve some previous results on the existence of nonzero positive solutions of systems of Laplace equations and on some eigenvalue problems of systems of p-Laplace equations. Applications of our results are given to systems of p-Laplace equations with specific nonlinearities.

Perfectly conducting tape-helix model for guided electromagnetic wave propagation

The homogeneous boundary value problem arising in the propagation of electromagnetic waves guided by an open tape helix modelled to be of infinitesimal tape thickness and infinite tape-material conductivity is shown to be inherently ill posed. It is demonstrated how the ill posed problem may be regularised using the mollification method. The regularised boundary value problem is then solved to yield the approximate dispersion equation which takes the form of the solvability condition for an infinite system of linear homogeneous algebraic equations viz., the determinant of the infinite-order coefficient matrix is zero. For the numerical computation of the dispersion characteristic, all the entries of the symmetrically truncated version of the coefficient matrix are estimated by summing an adequate number of the rapidly converging (after regularisation) series for them. The tape-current distribution is estimated from the null-space vector of the truncated coefficient matrix corresponding to a specified root of the dispersion equation. A comparison of the numerical results with those for the anisotropically conducting model (that neglects the component of the tape-current density perpendicular to the winding direction) of the tape helix reveals that the propagation characteristic computed on the basis of the anisotropically conducting model could be substantially in error even for moderately wide tapes.

Approximate controllability of a parabolic equation with memory

In this paper, we study the approximate controllability of a parabolic equation with memory y t − y x x + ∫ 0 t y ( x , s ) d s = 0 by boundary control. The proof relies on the explicit solution of the corresponding homogeneous initial boundary value problem and a duality method.

Nonlocal boundary value problems for discrete systems

Our goal in this paper is to provide sufficient conditions for the existence of solutions to discrete, nonlinear systems subject to multipoint boundary conditions. The criteria we present depends on the size of the nonlinearity and the set of solutions to the corresponding linear, homogeneous boundary value problems. Our analysis is based on the Lyapunov–Schmidt Procedure and Brouwerʼs Fixed Point Theorem. The results presented extend the previous work of D. Etheridge and J. Rodríguez (1996, 1998) [5,6] and J. Rodríguez and P. Taylor (2007) [18,19].

Extinction properties of solutions for a class of fast diffusive [formula omitted]-Laplacian equations

We consider the extinction properties of solutions for the homogeneous Dirichlet boundary value problem for the p -Laplacian equation u t − div ( ∣ ∇ u ∣ p − 2 ∇ u ) + β u q = λ u r with 1 < p < 2 , q ≤ 1 and r , λ , β > 0 . For β = 0 , it is known that r = p − 1 is the critical extinction exponent for the weak solution. For β > 0 , we show that r = p − 1 is still the critical extinction exponent when q = 1 . Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived. However, extinction can always occur when 0 < q ≤ r < 1 .

Non-convex self-dual Lagrangians and variational principles for certain PDEʼs

We study the concept and the calculus of non-convex self-dual ( Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They yield new variational resolutions for large class of partial differential equations with variety of linear and non-linear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both non-linear and homogeneous boundary value problems.

Trapped modes in a linear problem of the theory of surface water waves

We consider the linear problem of the steady oscillations of an ideal unbounded fluid with a free surface in the presence of a system of submerged horizontal cylindrical bodies of arbitrary cross section. We propose a criterion of the unique solvability of the problem and, based on this criterion, develop an algorithm for finding trapped unforced modes of fluid oscillations (solutions to the homogeneous boundary value problem) for bodies of a given arbitrary shape. The algorithm is applicable to the oscillations frequencies located on the continuous spectrum, as well as outside it. We also discuss the numerical realization and reliability of the obtained numerical results. As an illustration, we give examples of numerical experiments which are compared with known results. Bibliography: 28 titles. Illustrations: 5 figures.

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

We study the concept and the calculus of Non-convex self-dual ( Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler–Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.

Bifurcation for Quasilinear Elliptic Singular BVP

For a continuous function g ≥ 0 on (0, + ∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ∇u, − Δu + g(u)|∇u|2, with a power type nonlinearity, λu p + f 0(x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.

On the Construction of Statistically Similar Representative Volume Elements Based on the Lineal‐Path Function

AbstractIn this contribution we propose a method for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs and which represent quite accurately the mechanical response of the real material. For the design of such SSRVEs an objective function is minimized taking into account least‐square functionals based on suitable statistical measures, that characterize the inclusion morphology. Here, we find that the incorporation of the lineal‐path function leads to promising results in FE2‐simulations of macroscopically homogeneous boundary value problems as well as inhomogenous ones. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15‐19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.

Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems

In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δ ? u=f, ???, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125---142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.

ON THE DISTRIBUTION OF CURRENT ON AN OPEN TAPE HELIX

The approximate distribution of the current density induced on the tape surface by guided electromagnetic waves supported by an inflnite open tape helix is estimated from an exact solution of a homogenous boundary value problem for Maxwell's equations. It is shown that the magnitude of the surface current density component perpendicular to the winding direction is at least three orders of magnitude smaller than that of the surface current density component parallel to the winding direction everywhere on the tape surface. Also, the magnitude and phase distribution for the surface current density components parallel and perpendicular to the winding direction are seen to be nearly uniform at all frequencies corresponding to real values of the propagation constant.

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.

On approximation in the fictitious domain method for a bidimensional transport equation

AbstractIn this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D2. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D2. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010