Nonlinear Robin Boundary Conditions Research Articles

Hydrodynamics for SSEP with Non-reversible Slow Boundary Dynamics: Part I, the Critical Regime and Beyond

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The hydrodynamic limit is given by the heat equation with non-linear Robin boundary conditions or Neumann boundary conditions, the latter being in the case when the reservoirs are too slow. The proof goes through the entropy method of Guo, Papanicolaou and Varadhan. We also derive the hydrostatic limit for this model, whose proof is based on the method developed by Landim and Tsunoda. We observe that we do not make use of correlation estimates in none of our results.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition

<p style='text-indent:20px;'>We consider a coupled bulk–surface Allen–Cahn system affixed with a Robin-type boundary condition between the bulk and surface variables. This system can also be viewed as a relaxation to a bulk–surface Allen–Cahn system with non-trivial transmission conditions. Assuming that the nonlinearities are real analytic, we prove the convergence of every global strong solution to a single equilibrium as time tends to infinity. Furthermore, we obtain an estimate on the rate of convergence. The proof relies on an extended Łojasiewicz–Simon type inequality for the bulk–surface coupled system. Compared with previous works, new difficulties arise as in our system the surface variable is no longer the trace of the bulk variable, but now they are coupled through a nonlinear Robin boundary condition.

Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

AbstractLetΩ⊂ ℝ2be a bounded domain with smooth boundary andb(x) &gt; 0 a smooth function defined on∂Ω. We study the following Robin boundary value problem:$$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &amp;{\it\Delta} u+u^p=0 &amp;\quad&amp; \text{in }{\it\Omega},\\ &amp;u&gt;0 &amp;\quad&amp; \text{in }{\it\Omega},\\ &amp;\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &amp;\quad&amp; \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$whereνdenotes the exterior unit vector normal to∂Ω, 0 &lt;λ&lt; +∞ andp&gt; 1 is a large exponent. We construct solutions of this problem which exhibit concentration asp→ +∞ and simultaneously asλ→ +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.

Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

Asymptotic Behaviour of the Energy Integral of a Two-Parameter Homogenization Problem with Nonlinear Periodic Robin Boundary Conditions

AbstractWe consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameterδ. The relative size of each periodic perforation is determined by a positive parameterε. Under suitable assumptions, such a problem admits a family of solutions which depends onεandδ. We analyse the behaviour the energy integral of such a family as (ε,δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.

Control of the radiative heating of a semi-transparent body

In a preceding paper, we have studied the radiative heating of a semi-transparent body $$\varOmega $$ (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time $$t_\mathrm{f}$$ , the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution $$(x,t) \mapsto T(x,t)$$ inside the semi-transparent body $$\varOmega $$ near a desired temperature distribution $$T_\mathrm{d}(\cdot ,\cdot )$$ during the time interval of radiative heating $$]0,t_\mathrm{f}[$$ by acting on u, the purpose being to deform $$\varOmega $$ to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls $$U_\mathrm{ad}$$ , for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control $$u:t \mapsto u(t)$$ to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.

Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source term and boundary flux are integrated into the model. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study its initial value problem subject to a general nonlinear Robin boundary condition. The growth rates in the source term and the boundary condition are arbitrarily large. The maximum of the solution, for positive time, is estimated in terms of certain Lebesgue norms of the initial and boundary data. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications. In dealing with the complexity and generality of the equation and boundary condition, suitable trace theorems and Sobolev’s inequalities are utilized, and a well-adapted Moser’s iteration is implemented.

The radiative transfer equation with a nonhomogeneous reflectivity boundary condition coupled with the heat conduction equation

In this paper, we make a rigorous mathematical analysis of the radiative heating of a semitransparent body made of a glass by a black radiative source surrounding it. This requires the study of the coupling between quasi‐steady radiative‐transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semitransparent electromagnetic spectrum of the glass) and a nonlinear heat‐conduction evolution equation with a nonlinear Robin boundary condition, which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution and give also uniform bounds on the solution, ie, on the absolute temperature distribution inside the body and on the radiative intensities.

Homogenization of an Elliptic Equation in a Domain with Oscillating Boundary with Non-homogeneous Non-linear Boundary Conditions

While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The asymptotic analysis of boundary value problems in oscillating domains has been extensively studied and involves some key issues such as: finding uniformly bounded extension operators for function spaces on oscillating domains, the choice of suitable sequences of test functions for passing to the limit in the variational formulation of the model equations etc. In this article, we study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. The non-homogeneous Neumann condition or the Robin boundary condition on the oscillating boundary adds a further difficulty to the limit analysis since it involves taking the limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello (Ricerche Mat 43(2):239–292, 1994) and in Mel’nyk (Math Methods Appl Sci 31(9):1005–1027, 2008) by converting the surface term into a volume term using auxiliary boundary value problems. Some problems of this nature have also been studied using an extension of the notion of two-scale convergence (Allaire et al. in Proceedings of the international conference on mathematical modelling of flow through porous media, Singapore, 15–25, 1996, Neuss-Radu in C R Acad Sci Paris Sr I Math 322:899–904, 1996). In this article, we use a different approach to handle of such terms based on the unfolding operator.

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

A meshless Galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

Abstract A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.

Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method

The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

The Laplace operator on the Sierpinski gasket with Robin boundary conditions

We study the Laplace operator on the Sierpinski gasket with nonlinear Robin boundary conditions. We show that for certain Robin boundary conditions the Laplace operator generates a positive, order preserving, L∞-contractive semigroup which is sandwiched (in the sense of domination) between the semigroups generated by the Dirichlet–Laplace operator and the Neumann–Laplace operator. We also characterise all local semigroups which are sandwiched between these two extremal semigroups by showing that their generators are Robin–Laplace operators.

Renormalized solutions of elliptic equations with Robin boundary conditions

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type {-div(B(x,u)∇u)=f in Ω,u=0 on ΓoB(x,u)∇u⋅n→+γ(x)h(u)=g on Γ1where f and g are the element of L1 (Ω) and L1 (Γ1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.

High-order finite-volume solutions of the steady-state advection–diffusion equation with nonlinear Robin boundary conditions

We propose high-order finite-volume schemes for numerically solving the steady-state advection–diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

Approximation of surface–groundwater interaction mediated by vertical streambank in sloping terrains

New analytical solutions are derived to estimate the interaction of surface and groundwater in a stream–aquifer system. The analytical model consists of an unconfined sloping aquifer of semi-infinite extant, interacting with a stream of varying water level in the presence of a thin vertical sedimentary layer of lesser hydraulic conductivity. Flow of subsurface seepage is characterized by a nonlinear Boussinesq equation subjected to mixed boundary conditions, including a nonlinear Cauchy boundary condition to approximate the flow through the sedimentary layer. Closed form analytical expressions for water head, discharge rate and volumetric exchange are derived by solving the linearized Boussinesq equation using Laplace transform technique. Asymptotic cases such as zero slope, absence of vertical clogging layer and abrupt change in stream-stage can be derived from the main results by taming one or more parameters. Analytical solutions of the linearized Boussinesq equation are compared with numerical solution of corresponding nonlinear equation to assess the validity of the linearization. Advantages of using a nonlinear Robin boundary condition, and combined effects of aquifer parameters on the bank storage characteristic of the aquifer are illustrated with a numerical example.

Exact and approximate analytic solutions of the thin film flow of fourth-grade fluids by the modified Adomian decomposition method

Purpose The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications. Design/methodology/approach The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids. Findings The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature. Originality/value This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

An iterative method for solving boundary value problems for second order differential equations

The purpose of this paper is to investigate the application of the Adomian decomposition method (ADM) for solving boundary value problems for second-order differential equations with Robin boundary conditions. We first reformulate the boundary value problems for linear equations as a fixed point problems for a linear Fredholm integral operator, and then apply the ADM. We also extend our approach to include second-order nonlinear differential equations subject Robin boundary conditions.