Homogeneous Boundary Value Problem Research Articles

Global Existence Structure of Parameters for Positive Solutions of a Singular $$(p_1,p_2)$$ ( p 1 , p 2 ) -Laplacian System

We study the homogeneous Dirichlet boundary value problem of a singular \((p_1,p_2)\)-Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains.

Tensor Method to the Weak Solution of the Second-Order System of Ordinary Differential Equations with Boundary Value Problems (I)

In this paper, a tensor method for variational approach is established based on Lax-Milgram theorem. Some new results are obtained for the existence of weak solution of ODE system with homogeneous boundary value problems. By seeking the approximate solutions using Matlab, it is proved that the proposed method is effective to the system of ODE with boundary value problems in practice.

Homogeneous Hilbert boundary-value problem with several points of turbulence

We consider Riemann–Hilbert boundary value problem with infinite index in unit disk. Its coefficient is Holder-continuous everywhere on the unit circle excluding a finite set of points, where its argument has power discontinuities of order less one. The present article is the first research of this version of Hilbert boundary-value problem with infinite index. We obtain formulas for its general solution, investigate existence ad uniqueness of solutions, and describe the set of solutions in the case of non-uniqueness. Our technique is based on theory of entire functions and geometric theory of functions.

A new approach to solving homogeneous Riemann boundary-value problem on a ray with infinite index

We consider a Riemann boundary-value problem with infinite Gakhov’s index. The boundary data are defined on positive ray of the real axis. We solve the problem by means of removal of singularity of boundary data at the infinity. This approach is analogous to Gakhov’s method of elimination of singularities in the problemswith finite indices, but we use another eliminating factors.

Connection between holomorphic vector bundles and cohomology on a Riemann surface and conjugation boundary value problems

This paper studies interconnections between holomorphic vector bundles on compact Riemann surfaces and the solution of the homogeneous conjugation boundary value problem for analytic functions on the one hand, and cohomology and the solution of the inhomogeneous problem on the other. We establish that constructing the general solution to the homogeneous problem with arbitrary coefficients in the boundary conditions is equivalent to classifying holomorphic vector bundles. Solving the inhomogeneous problem is equivalent to checking the solvability of 1-cocycles with coefficients in the sheaf of sections of a bundle; in particular, the solvability conditions in the inhomogeneous problem determine obstructions to the solvability of 1-cocycles, i.e. the first cohomology group. Using this connection, we can apply the methods of boundary value problems to vector bundles. The results enable us to elucidate the role of boundary value problems in the general theory of Riemann surfaces.

Multiple solutions for a polyharmonic homogeneous boundary value problem with critical exponent

In this paper, we analyse the polyharmonic equation in a bounded smooth domain with homogeneous Dirichlet conditions on the boundary and critical exponents. We will obtain a relationship between the topological domain and the multiplicity of solutions. We also give symmetry properties of the solutions and a result about breaking symmetry.

РЕШЕНИЕ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ РИМАНА С БЕСКОНЕЧНЫМ ИНДЕКСОМ ЛОГАРИФМИЧЕСКОГО ПОРЯДКА НА ЛУЧЕ НОВЫМ МЕТОДОМ

РЕШЕНИЕ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ РИМАНА С БЕСКОНЕЧНЫМ ИНДЕКСОМ ЛОГАРИФМИЧЕСКОГО ПОРЯДКА НА ЛУЧЕ НОВЫМ МЕТОДОМ

On the non-uniqueness of solution to the homogeneous boundary value problem for the heat conduction equation in an angular domain

On the non-uniqueness of solution to the homogeneous boundary value problem for the heat conduction equation in an angular domain

Solving the Third Homogeneous Boundary-Value Problem of the Deformation of a Transversely Isotropic Plate with a Curved Hole Under Uniform Tension

The problem of the stress state of an unbounded transversely isotropic plate with a curved (noncircular) hole is solved by expanding the unknown functions into Fourier–Legendre series and using the boundary-shape perturbation method. The plate is subject to uniform tension at infinity and the normal displacement and tangential stresses are zero at the hole edge. The numerical data are analyzed

Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness

The present article is dedicated to the numerical solution of homogeneous Neumann boundary value problems on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on the random domain can be transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Ventcell boundary condition and yields a second order accurate solution in the scale parameter ε of the layer's thickness. With the help of the Karhunen–Loève expansion, we transform this random boundary value problem into a deterministic, parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter ε. Numerical results validate our theoretical findings.

The blow-up of solutions for [formula omitted]-Laplacian equations with variable sources under positive initial energy

This paper deals with homogeneous Dirichlet boundary value problem to a class of m-Laplace equations with variable reaction ∂u∂t−div(|∇u|m−2∇u)=uq(x),x∈Ω,t>0, the bounded domain Ω⊂RN(N≥1) with a smooth boundary. We prove that the weak solutions of the above problems blow up in finite time for all q−>m−1(m≥2), when the initial energy is positive and initial data is suitably large. This result improves the recent result by Zhou and Yang (2015), which asserts the blow-up of solutions for N>m, provided that q+<Nm−m−NN−m.

On singularly weighted generalized Laplacian systems and their applications

Abstract We study the homogeneous Dirichlet boundary value problem of generalized Laplacian systems with a singular weight which may not be integrable. Some explicit intervals which correspond to the existence and nonexistence of positive solutions for the system with the finite asymptotic behaviors of the nonlinearities at 0 and ∞ {\infty} are obtained.

On the second order equations with nonlinear impulses - Fredholm alternative type results

We consider the semilinear homogeneous Dirichlet boundary value problem for the second order equation on a finite interval with nonlinear impulses in the derivative at prescribed points. We introduce Landesman-Lazer type necessary and sufficient conditions for resonance problems and generalize the Fredholm alternative results for linear operators. An interaction between nonlinear restoring force and nonlinear impulses is presented.

Green’s Functions for Reducible Functional Differential Equations

In this work, we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green’s function of reducible functional differential equations and illustrate it with the case of homogeneous boundary value problems with reflection and several specific examples.

Propagation of electromagnetic waves guided by perfectly conducting model of a tape helix supported by dielectric rods

The homogeneous boundary value problem existing in the electromagnetic wave propagation in a dielectric-loaded perfectly conducting tape helix with infinitesimal tape thickness is investigated in this study. The ill-posed boundary value problem is regularised using the mollification method. The homogeneous boundary value problem is solved for the dielectric loaded perfectly conducting tape helix taking into account the exact boundary conditions for the perfectly conducting dielectric loaded tape helix. The solved approximate dispersion equation takes the form of the solvability condition for an infinite system of linear homogeneous equations namely, the determinant of the infinite order coefficient matrix is zero. For the numerical computation of the dispersion equation, all the entries of the symmetrically truncated version of the coefficient matrix are estimated by summing an adequate number of the rapidly converging 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. The numerical results suggest that the propagation characteristic computed by the anisotropically conducting model (that neglects the component of the tape-current density perpendicular to the winding direction) is only an abstinent approximation to consider for moderately wide tapes.

Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

Optimal-order error estimates in the energy norm and the $$L^2$$L2 norm were previously proved in the literature for finite element methods of Dirichlet boundary-value problems of steady-state fractional diffusion equations under the assumption that the true solutions have desired regularity and that the solution to the dual problem has full regularity for each right-hand side. We show that the solution to the homogeneous Dirichlet boundary-value problem of a one-dimensional steady-state fractional diffusion equation of constant coefficient and source term is not necessarily in the Sobolev space $$H^1$$H1. This fact has the following implications: (i) Up to now, there are no verifiable conditions on the coefficients and source terms of fractional diffusion equations in the literature to ensure the high regularity of the true solutions, which are in turn needed to guarantee the high-order convergence rates of their numerical approximations. (ii) Any Nitsche-lifting based proof of optimal-order $$L^2$$L2 error estimates of finite element methods in the literature is invalid. We present numerical results to show that high-order finite element methods for a steady-state fractional diffusion equation with smooth data and source term fail to achieve high-order convergence rates. We present a preliminary development of an indirect finite element method, which reduces the solution of fractional diffusion equations to that of second-order diffusion equations postprocessed by a fractional differentiation. We prove that the corresponding high-order methods achieve high-order convergence rates even though the true solutions are not smooth, provided that the coefficient and source term of the problem have desired regularities. Numerical experiments are presented to substantiate the theoretical estimates.

Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $${B_{{r_e}}}$$ and $${B_{{r_i}}}$$ , with the difference r e -r i (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and elaborates arguments developed by Aftalion, Busca, and Reichel.

On propagation of interface cracks in bi-layered strip in relation to delamination of coatings from substrates of final thickness

The problem of a coatings delamination from a substrate is considered. Both the delaminated part of the coating and substrate are modelled as plates with a special attention to the boundary conditions, which are considered as generalized elastic clamping. For this type of boundary conditions the angle of rotation and components of the displacement vector at the clamping point are related to the acting bending moment and components of the resulting force vector at that point by means of 3x3 matrix of elastic compliance. Such type of boundary conditions appeared from the solution of the problem of a bilayered strip with the semi-infinite interface crack loaded at infinity. The analytical solution of this problem for the particular relation between the elastic constants of the layers (zero second Dundurs’ parameter) is obtained by reducing it to the homogeneous matrix Riemann boundary value problem solved by matrix factorization for two particular cases: (1) equal thickness of both layers; (2) thickness of one of the layers (corresponding to the substrate) being much more than the thickness of the other, so that this layer may be considered as a half-plane. For the general case of the arbitrary elastic constants and layers thicknesses the approximate solution has been obtained. The approximation involves neglecting the influence of the normal to interface stresses to the shear displacements and shear stresses to the normal displacements. According to the obtained solutions the asymptotics for the relative displacements of the separated layers for the points, remote from the crack tip, correspond to the deflection of plate (beam) with the described above boundary conditions. The fracture parameters (stress intensity factors and energy release rates) have also been obtained. Using the obtained solution the problem on buckling of the delaminated coatings is addressed. It has been shown that for a wide range of parameters the ratio of the critical stress, corresponding to buckling of the delaminated parts of coatings and the critical stress of a rigidly clamped plate is determined by a single nondimensional parameter, which is a combination of the elastic constants of the coating and substrate, and a combination of the geometrical parameters of the model.

Existence of Solutions to a Class of Semilinear Elliptic Problem with Nonlinear Singular Terms and Variable Exponent

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation- div(M(x)∇un)=f(x)/uα(x)withf∈Lm(Ω) (m⩾ 1)andα(x)&gt;0. The results show the dependence of the summability offin some Lebesgue spaces and on the values ofα(x).

Prym differentials as solutions to boundary value problems on Riemann surfaces

Construction of multiplicative functions and Prym differentials, including the case of characters with branch points, reduces to solving a homogeneous boundary value problem on the Riemann surface. The use of the well-established theory of boundary value problems creates additional possibilities for studying Prym differentials and related bundles. Basing on the theory of boundary value problems, we fully describe the class of divisors of Prym differentials and obtain new integral expressions for Prym differentials, which enable us to study them directly and, in particular, to study their dependence on the point of the Teichmuller space and characters. Relying on this, we obtain and generalize certain available results on Prym differentials by a new method.