Nonhomogeneous Boundary Value Problems Research Articles

Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green's functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green's functions inside the quadrant and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the basic adjoint relationship which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the moment generating function in terms of contour integrals and a conformal mapping.

An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects

Rubberlike materials exhibit strong rate-dependent mechanical response which manifests itself in creep and relaxation tests as well as in the hysteresis curves under cyclic loading. Unlike linear viscoelasticity, creep and relaxation response of rubber is nonlinear and amplitude-dependent. Within this context, the contribution of this work is three fold. (i) On the experimental side, the characterization of equilibrium and non-equilibrium responses are carried out by means of uniaxial and equibiaxial extension tests. Also performed are the creep and relaxation experiments at various stress and strain levels. (ii) On the theoretical side, we extend the well-known eight-chain model via incorporating a simple yet instrumental tube-constraint term composed of the second invariant into the non-affine network contribution reflecting the ground state equilibrium response. For the non-equilibrium response, we propose a new evolution equation for the creep/relaxation behavior of rubberlike materials based on a relaxation kinetics of a single polymer chain. The geometric non-linearity is incorporated via the finite deformation kinematics based on the multiplicative split of the deformation gradient into elastic and viscous parts, whereas the volumetric and isochoric split of the deformation gradient is entirely discarded. The rheology uses a nonlinear spring responsible for equilibrium elastic response in parallel to n number of Maxwell elements, leading to a generalized Maxwell-Wiechert viscoelastic solid. (iii) The algorithmic implementation of the model features the spectral decomposition of the respective terms and is demonstrated within the context of the finite element method. The developed model is validated by fitting both the elastic and viscoelastic model responses with respect to the experimental data in the sense of uniaxial, (equi)biaxial extensions, and pure shear tests. Relaxation and creep behavior of the model are thoroughly assessed. Also presented in the manuscript is the capability and the performance of the model in the face of a relevant non-homogeneous boundary value problem. The quality of the findings earns the model vast utilization areas from an engineering perspective.

Perturbed rigid body motions of an elastic rectangle

Plane and anti-plane dynamic problems for an elastic rectangle loaded along its sides are considered. Low-frequency perturbations to rigid body translations are calculated. The derivation involves the solutions of non-homogeneous boundary value problems for harmonic and bi-harmonic equations. The explicit solution for the harmonic problem for transverse anti-plane translation is expressed through Fourier series. The bi-harmonic problem corresponding to the longitudinal in-plane translation is studied in greater detail for an elongated rectangle, which also may be treated using the elementary theory for plate extension. The derived perturbations incorporate the variations of displacements and stresses over the interior of the rectangle, including the case of self-equilibrated loading. The latter is obviously outside the range of validity of the classical rigid body framework.

On a Problem of Heat Equation with Fractional Load

In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.

Numerical solution of singular boundary value problems using Green’s function and Sinc-Collocation method

This study is the construction of the Green’s function and Sinc function for a class of nonhomogeneous singular boundary value problems (SBVPs). The equivalent Volterra-Fredholm integral equations can be derived from SBVPs by applying Green’s function. This can be approximated by Sinc-Collocation method. Convergence analysis is given. Our approach applied on three various examples. Errors in the solution are demonstrated in the tables. We conclude that our approach converge rapidly with the exponential order.

Low regularity solutions of non-homogeneous boundary value problems of a higher order Boussinesq equation in a quarter plane

In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions:{utt−uxx+βuxxxx−uxxxxxx+(u2)xx=0,x>0, t>0,u(x,0)=φ(x),ut(x,0)=ψ″(x),x>0,u(0,t)=h1(t),uxx(0,t)=h2(t),uxxxx(0,t)=h3(t),t>0, where β=±1. It is shown that the problem is locally well-posed in C(Rloc+,Hs(R+)) with initial condition (φ,ψ)∈Hs(R+)×Hs−1(R+) and boundary condition (h1,h2,h3) in the product space Hs+13(R+)×Hs−13(R+)×Hs−33(R+) for −12<s≤0.

Two Nonhomogeneous Boundary Value Problems for a Rectangle: Exact Solutions

In the paper, for the first time we give exact solutions to two nonhomogeneous boundary value problems of the theory of elasticity for a rectangle with free long sides. Inside the rectangle there are applied two equal concentrated forces directed oppositely along the horizontal axis (even-symmetric deformation). The method of solution is based on the use of the solution to the biharmonic problem for a smooth semi-strip and the method of the integral Fourier transform. In the first problem, the short sides of the rectangle are free; in the second, they are rigidly clamped. The solutions to both problems are constructed on the superposition principle in the form of the sum of integrals and series in trigonometric functions and Papkovich–Fadle eigenfunctions. The coefficients of these expansions are determined by simple formulas as the Fourier integrals of given boundary functions.

A spectral method for solving nonhomogeneous Neumann boundary value problems on quadrilaterals

In this paper, we develop a spectral method with essential imposition of nonhomogeneous Neumann boundary condition on quadrilaterals. Results on irrational orthogonal approximation are established, which facilitate the numerical solutions of partial differential equations of the sort. As applications, we provide spectral schemes for two model problems, and prove their spectral accuracy. Efficient numerical implementations are described. Numerical results demonstrate high efficiency of the suggested algorithms.

Pre-fracture zone modeling for an interface crack in an isotropic bimaterial

An interface crack in an infinite bimaterial space under remote combining loading is considered. The complex potentials approach is applied and an exact analytical solution of this problem is presented. To remove the oscillating singularities which occur in this solution a model based on the introduction of the shear stress pre-fracture zones at the crack tips is suggested. In the framework of this assumption the nonhomogeneous combined Dirichlet-Riemann boundary value problem with the conditions at infinity is formulated and its analytical solution is presented. The length of this zone is found from the condition of restriction of the shear stress at the end point of the zone. In this case the shear stress becomes finite in the right hand side of the pre-fracture zone while the normal stress has a square root singularity at the crack tip. The energy release rates (ERR) at the crack tip and also along the pre-fracture zone are found and their total values are compared the ERR of the classical model. For the case of a similar problem, but for finite sized body the finite element method is applied. The crack length is assumed to be much smaller then characteristic body size. The finite element net with two levels of concentration is constructed. The first level provides the uniform concentration from the boundaries of the body to the crack and the second level assumes the similar concentration at the singular points of the pre-fracture zone. The different values of the shear stress in the pre-fracture zone are considered and the local ERR at the singular points as well as the global energy release rate is found. It is shown that for different values of the mentioned shear stress the global energy release rate remains almost invariable and is in a a good agreement with the analytical solution.

Nonhomogeneous boundary value problem for the time periodic linearized Navier–Stokes system in a domain with outlet to infinity

The time periodic linearized Navier–Stokes system (Stokes system) with nonhomogeneous boundary conditions is studied in a domain Ω which has a “paraboloidal” outlet to infinity. The time periodic boundary value a(x,t) is assumed to have a compact support and it is supposed that the flux of a over ∂Ω is nonzero. The existence and uniqueness of a weak solution is proved. The solution can have either finite or infinite Dirichlet integral depending on geometrical properties of the outlet to infinity.

The existence of solutions for nonlinear fractional boundary value problem and its Lyapunov-type inequality involving conformable variable-order derivative

We consider a nonlinear fractional boundary value problem involving conformable variable-order derivative with Dirichlet conditions. We prove the existence of solutions to the considered problem by using the upper and lower solutions’ method with Schauder’s fixed-point theorem. In addition, under some assumptions on the nonlinear term, a new Lyapunov-type inequality is given for the corresponding boundary value problem. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and a method to prove uniqueness for the nonhomogeneous boundary value problem. These new results are illustrated through examples.

A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval

This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval (0, L) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the space Hs(0, L) for any s &gt; −2 with the initial data in Hs(0, L) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the space Hs(0, L) for any s &lt; −2 in the sense that the corresponding solution map fails to be in C2.

超几何方程的一类非齐次边值问题解的相似结构

超几何方程的一类非齐次边值问题解的相似结构

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii's Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

A Crack between Orthotropic Materials with a Shear Yield Zone at the Crack Tip

A plane problem for a crack between two anisotropic semi-infinite spaces under remote tensile-shear loading is considered. In the framework of the assumption that the crack faces are free of stresses an exact analytical solution of the problem is given on basis of the complex potentials approach. This solution possesses oscillating square root singularities in stresses and in the derivatives of the displacement jumps at the crack tips. To remove these singularities a new model founded on the introduction of the shear yield zones at the crack tips is suggested. This model is appropriate for the cases where interface adhesive layer is softer than the surrounding matrixes. Under this assumption the problem is reduced to the nonhomogeneous combined Dirichlet-Riemann boundary value problem with the conditions at infinity. An exact analytical solution of this problem is presented for the case of a single yield zone. The length of this zone is found from the finiteness of the shear stress at the end point of the zone. Due to such simulation the shear stress becomes finite at any point and the normal stress possesses only square root singularity at the crack tip. Therefore, the conventional stress intensity factor of the normal stress at the crack tip is used. The numerical illustration of the obtained solution is given.

A RANDOM ACCESS G-NETWORK: STABILITY, STABLE THROUGHPUT, AND QUEUEING ANALYSIS

The effect of signals on stability, stable throughput region, and delay in a two-user slotted ALOHA-based random-access system with collisions is considered. This work gives rise to the development of random access G-networks, which can model security attacks, expiration of deadlines, or other malfunctions, and introduce load balancing among highly interacting queues. The users are equipped with infinite capacity buffers accepting external bursty arrivals. We consider both negative and triggering signals. Negative signals delete a packet from a user queue, while triggering signals cause the instantaneous transfer of packets among user queues. We obtain the exact stability region, and show that the stable throughput region is a subset of it. Moreover, we perform a compact mathematical analysis to obtain exact expressions for the queueing delay by solving a non-homogeneous Riemann boundary value problem. A computationally efficient way to obtain explicit bounds for the expected number of buffered packets at user queues is also presented. The theoretical findings are numerically evaluated and insights regarding the system performance are derived.

Nonhomogeneous initial and boundary value problem for the Caputo-type fractional wave equation

In this paper, we give an analytical solution of a fractional wave equation for a vibrating string with Caputo time fractional derivatives. We obtain the exact solution in terms of three parameter Mittag-Leffler function. Furthermore, some examples of the main result are exhibited.