Nonhomogeneous Boundary Value Problems Research Articles

A non-homogeneous boundary-value problem for the KdV–Burgers equation posed on a finite domain

We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval [a,b]. This problem features non-homogeneous boundary conditions applied at x=a and x=b and is known to have unique global smooth solution.

On a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

ABSTRACTIn this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.

Nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a domain with a cusp

The nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a two-dimensional domain with a cusp point on the boundary is studied. The case when the flux of the boundary value $$\mathbf{a }$$ is nonzero, i.e., when there is a source or sink in the cusp point, is considered. The existence of at least one weak solution having infinite Dirichlet integral is proved without any restrictions on the size of the flux $$F=\int \limits _{\partial \Omega }\mathbf{a }\cdot \mathbf{n }dS$$ .

Similar constructing method for solving nonhomogeneous mixed boundary value problem of n‐interval composite ode and its application

This paper studies a simple method—Similar Constructing Method (SCM)—for constructing the exact solutions of the nonhomogeneous mixed boundary value problem for sets of n‐interval composite second‐order ordinary differential equation (ODE) with variable coefficient. Then this paper proves the correctness of the solution obtained by SCM. After that, this paper has done simulation experiment. This section uses the SCM to solve the nonhomogeneous boundary value problem of three‐interval composite Bessel equation. Solutions are presented in graphical form for various parameter values, and the influence of parameters on the solution is analyzed. The example shows that using SCM to solve the class of nonhomogeneous mixed boundary value problems of n‐interval composite second‐order linear ODE is easy, convenient, and effective.

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

Non-homogeneous boundary value problems of the fifth-order KdV equations on a bounded interval

We investigate well-posedness of initial boundary value problem for the fifth-order KdV equation (or Kawahara equation) posed on a finite interval ∂ t u − ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 with general non-homogeneous boundary conditions. Firstly, all possible boundary conditions are found while searching enough dissipative effects to the initial boundary value problem. Then, boundary smoothing effect is established through estimating explicit solution established by using Laplacian transformation. Finally, based on smoothing effect, local well-posedness of the initial boundary value problem was proved by applying Banach's fixed point theorem.

Solutions for stationary Navier–Stokes equations with non-homogeneous boundary conditions in symmetric domains of [formula omitted

We study the non-homogeneous boundary value problem for the stationary Navier–Stokes equations in a multi-connected bounded domain of Rn, n≥4. This problem was fully solved by Korobkov, Pileckas and Russo in 2015 for domains in R2 and partially solved for symmetric domains in R3. In this paper, we prove the existence of solutions in higher dimensional symmetric domains, under the necessary conditions of zero total flux through the boundary.

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

ABSTRACTThis work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.

Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for $$p >1$$ . Under some sufficient conditions, we establish local interior, local boundary, and global $$W^{1,q}$$ -regularity estimates for weak solutions with $$q>p$$ , assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.

Modelling electro-active polymers with a dispersion-type anisotropy

We propose a novel constitutive framework for electro-active polymers (EAPs) that can take into account anisotropy with a chain dispersion. To enhance actuation behaviour, particle-filled EAPs become promising candidates nowadays. Recent studies suggest that particle-filled EAPs, which can be cured under an electric field during the manufacturing time, do not necessarily form perfect anisotropic composites, rather they create composites with dispersed chains. Hence in this contribution, an electro-mechanically coupled constitutive model is devised that considers the chain dispersion with a probability distribution function in an integral form. To obtain relevant quantities in discrete form, numerical integration over the unit sphere is utilized. Necessary constitutive equations are derived exploiting the basic laws of thermodynamics that result in a thermodynamically consistent formulation. To demonstrate the performance of the proposed electro-mechanically coupled framework, we analytically solve a non-homogeneous boundary value problem, the extension and inflation of an axisymmetric cylindrical tube under electro-mechanically coupled load. The results capture various electro-mechanical couplings with the formulation proposed for EAP composites.

A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane

<p style='text-indent:20px;'>The paper is concerned with an initial-boundary-value problem of the sixth order Boussinesq equation posed on a quarter plane with non-homogeneous boundary conditions: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation}\label{0}\begin{cases}u_{tt}-u_{xx}+β u_{xxxx}-u_{xxxxxx}+(u^2)_{xx} = 0, \, \, \, \, \,\,\,\,\, \mbox{for }x&gt;0\mbox{, }t&gt;0, \\u(x, 0) = \varphi (x), u_t(x, 0) = ψ "(x), \\u(0, t) = h_1(t), u_{xx}(0, t) = h_2(t), u_{xxxx}(0, t) = h_3(t), \end{cases}\, \, \, \, \, \, \, \, \, \, (1)\end{equation}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$β = ± 1$\end{document}</tex-math></inline-formula> . It is shown that the problem is locally well-posed in the space $H^s(\mathbb{R}^+)$ for any 0≤s&lt;<inline-formula><tex-math id="M2">\begin{document}$\frac{13}{2}$\end{document}</tex-math></inline-formula> with the initial data <inline-formula><tex-math id="M3">\begin{document}$ (\varphi, ψ)$\end{document}</tex-math></inline-formula> in the space <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$H^s(\mathbb{R}^+)× H^{s-1}(\mathbb{R}^+)$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>and the naturally compatible boundary data <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$\mbox{ $h_1∈ H_{loc}^{\frac{s+1}{3}}(\mathbb{R}^+)$, $h_2∈ H_{loc}^{\frac{s-1}{3}}(\mathbb{R}^+) \text{and}\,\,\, h_3∈ H_{loc}^{\frac{s-3}{3}}(\mathbb{R}^+)$}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'> with optimal regularity.

Nonhomogeneous Boundary Value Problems of Nonlinear Schrödinger Equations in a Half Plane

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schr\"odinger equations posed in a half plane $\mathbb{R} \times \mathbb{R}^+$ with nonhomogeneous Dirichlet boundary conditions. For any given $s \ge 0$, if the initial data $\varphi (x, y)$ are in Sobolev space $H^s(\mathbb{R}\times \mathbb{R}^+) $ with the boundary data $ h ( x, t) $ in an optimal space ${\cal H}^s(0,T)$ as defined in the introduction, which is slightly weaker than the space $$H^{(2s+1)/4}_{t} ([0, T]; L_x^2(\mathbb{R} ) ) \cap L^2_t ( [ 0, T]; H^{s+ 1/2} _x ( \mathbb{R} ) ),$$ the local well-posedness of the IBVP in $ C ( [0, T] ; H^s ( \mathbb{R}\times \mathbb{R}^+ ) )$ is proved. The global well-posedness is also discussed for $s = 1$. The main idea of the proof is to derive a boundary integral operator for the corresponding nonhomogeneous boundary condition and obtain the Strichartz's estimates for this operator. The results presented in the paper hold for the IBVP posed in a half space $ \mathbb{R}^n\times \mathbb{R}^+$ with any $n>1$.

Properties of Consistent Grid Operators for Grid Functions Defined Inside Grid Cells and on Grid Faces

Using grids (meshes) formed from polyhedra (polygons in the two-dimensional case), we consider differential and boundary grid operators that are consistent in the sense of satisfying the grid analog of the integral identity – a corollary of the formula for the divergence of the product or a scalar by a vector. These operators are constructed and applied in the Mimetic Finite Difference (MFD) method, in which grid scalars are defined inside the grid cells and grid vectors are defined by their local normal coordinates on the planar faces of the grid cells. We show that the basic grid summation identity is a limit of an integral identity written for piecewise-smooth approximations of the grid functions. We also show that the MFD formula for the reconstruction of a grid vector field is obtained by approximation analysis of the summation identity. Grid embedding theorems are proved, analogous to well-known finite-difference embedding theorems that are used in finite-difference scheme theory to derive prior bounds for convergence analysis of the solutions of finite-difference nonhomogeneous boundary-value problems.

Nonlinear Nonhomogeneous Boundary Value Problems with Competition Phenomena

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $\lambda>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $\lambda>0$ varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations

We discuss an initial-boundary problem for a time-fractional diffusion equation with non-zero Dirichlet boundary values which belong to L2 in time t and to a Sobolev space of negative order in space and prove the unique existence of weak solutions and a priori estimates. The proof is based on the Caputo fractional derivative in Sobolev spaces and the transposition method. We show one application to the existence of solution to an optimal control problem.

Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations

This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line R+ or on a bounded interval (0,L) with nonhomogeneous boundary conditions. For any s with 0≤s<5/2 and s≠1/2, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the L2-based Sobolev spaces Hs(R+) in the case of the half line and in Hs(0,L) on a bounded interval, provided the boundary data are selected from Hloc(2s+1)/4(R+) and H(s+1)/2loc(R+), respectively. (For s>12, compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when s≥1. From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on R+ and the IBVP posed on (0,L). The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schrödinger equations posed on the whole line R while the theory on a bounded interval looks more like that of the pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on R+ is consistent with the temporal trace results that obtain for solutions of the pure IVP on R, while the slightly higher regularity of boundary data for the IBVP on (0,L) resembles what is found for temporal traces of spatially periodic solutions.

Towards a thermo-magneto-mechanical coupling framework for magneto-rheological elastomers

Magnetorheological elastomers (MREs) are a relatively new class of smart materials that can undergo large deformations resulting from external magnetic excitation. These are promising candidates in producing sensors and actuators. Due to their inherent chemical compositions, most polymeric materials are highly susceptible to temperature. While performing experiments on MREs that are exposed to magneto-mechanically coupled loads, maintaining a constant temperature profile is a non-trivial task for various reasons, e.g., i) experiments need to be performed in a temperature chamber that can maintain a prescribed temperature throughout a test, and ii) additional temperature gradients can be generated internally. In this paper, a thermo-magneto-mechanically coupled constitutive model is devised that is based on the total energy approach frequently used in MREs modelling and computation. Relevant constitutive equations are derived exploiting basic laws of thermodynamics that result in a thermodynamically consistent formulation. We demonstrate the performance of the proposed thermo-magneto-mechanically coupled framework with the help of two non-homogeneous boundary value problems. In both problems an axisymmetric cylindrical tube is deformed under thermo-magneto-mechanically coupled loads. In the first example the mechanical deformation is a combination of axial stretch and radial inflation whereas in the second example the cylinder is put under a mechanical load of torsion around the cylinder axis combined with an axial stretch. In both examples a circumferential magnetic field and a radial temperature gradient are applied. The results capture various thermo-magneto-mechanical couplings with the formulation proposed for MRE.

Nonhomogeneous boundary value problems for the complex Ginzburg–Landau equation posed on a finite interval

This paper is concerned with the well-posedness of the complex Ginzburg–Landau equation posed on a finite interval (0, L) with nonhomogeneous Dirichlet boundary conditions. When the initial datum belongs to () and the boundary data belong to some subspace of , the complex Ginzburg–Landau equation is shown to be globally well-posed.

The existence theorem for the steady Navier–Stokes problem in exterior axially symmetric 3D domains

We study the nonhomogeneous boundary value problem for the Navier–Stokes equations of steady motion of a viscous incompressible fluid in a three-dimensional exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric domains and data (without any smallness restrictions on the fluxes). Our main tool is a recent version of the Morse–Sard theorem for Sobolev functions obtained by Bourgain et al. (Rev Mat Iberoam 29(1):1–23, 2013).

Existence of solutions to the nonhomogeneous steady Magnetohydrodynamic equations

We study the nonhomogeneous boundary value problem for the steady Magnetohydrodynamic equations in a two-dimensional bounded domain with multiply connected boundary. We prove that this problem has an admissible solution in an admissible domain if the boundary value is admissible. The proof of the main result uses some property for a weak solution to the transport equations in an admissible domain.