Nonlinear Mixed Boundary Conditions Research Articles

Finite element approximation of Hamilton–Jacobi–Bellman equations with nonlinear mixed boundary conditions

Abstract We show strong uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton–Jacobi–Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretized via a lower Dini derivative. In time, the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard’s algorithm.

Blow-up conditions of nonlinear parabolic equations and systems under mixed nonlinear boundary conditions

In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations ut=∇⋅[ρ(u)∇u]+f(x,t,u),in Ω×(0,t∗),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_{t}=\ abla \\cdot \\bigl[\\rho (u)\ abla u \\bigr]+f(x,t,u),\\quad \ ext{in }\\Omega \ imes \\bigl(0,t^{*}\\bigr), $$\\end{document} under mixed nonlinear boundary conditions frac{partial u}{partial n}+theta (z)u=h(z,t,u) on Gamma _{1}times (0,t^{*}) and u=0 on Gamma _{2}times (0,t^{*}), where Ω is a bounded domain and Gamma _{1} and Gamma _{2} are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued C^{1}-functions and ρ is a positive C^{1}-function. To obtain the blow-up solutions, we introduce the following blow-up conditions: (Cρ):(2+ϵ)∫0uρ(w)f(x,t,w)dw≤uρ(u)f(x,t,u)+β1u2+γ1,(2+ϵ)∫0uρ2(w)h(z,t,w)dw≤uρ2(u)h(z,t,u)+β2u2+γ2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (C_{\\rho})\\,:\\, \\begin{aligned} &(2+\\epsilon ) \\int _{0}^{u}\\rho (w)f(x,t,w)\\,dw\\leq u\\rho (u)f(x,t,u)+ \\beta _{1}u^{2}+\\gamma _{1}, \\\\ &(2+\\epsilon ) \\int _{0}^{u}\\rho ^{2}(w)h(z,t,w)\\,dw \\leq u\\rho ^{2}(u)h(z,t,u)+ \\beta _{2}u^{2}+ \\gamma _{2}, \\end{aligned} $$\\end{document} for xin Omega , zin partial Omega , t>0, and uin mathbb{R} for some constants ϵ, beta _{1}, beta _{2}, gamma _{1}, and gamma _{2} satisfying ϵ>0,β1+λR+1λSβ2≤ρm2λR2ϵand0≤β2≤ρm2λS2ϵ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\epsilon >0,\\quad \\beta _{1}+\\frac{\\lambda _{R}+1}{\\lambda _{S}}\\beta _{2} \\leq \\frac{\\rho _{m}^{2}\\lambda _{R}}{2}\\epsilon \\quad \ ext{and}\\quad 0 \\leq \\beta _{2}\\leq \\frac{\\rho _{m}^{2}\\lambda _{S}}{2}\\epsilon , $$\\end{document} where rho _{m}:=inf_{s>0}rho (s), lambda _{R} is the first Robin eigenvalue and lambda _{S} is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.

Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems

The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.

Analysis of Stokes system with solution-dependent subdifferential boundary conditions

We study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

New boundary conditions for the approximate flux-limited diffusion radiative transfer in circumstellar environments

Context. In order to constrain the models describing circumstellar environments, it is necessary to solve the radiative transfer equation in the presence of absorption and scattering, coupled with the equation for radiative equilibrium. However, solving this problem requires much CPU time, which makes the use of automatic minimisation procedures to characterise these environments challenging. Aims. In this context, the use of approximate methods is of primary interest. One promising candidate method is the flux-limited diffusion (FLD), which recasts the radiative transfer problem into a non-linear diffusion equation. One important aspect for the accuracy of the method lies in the implementation of appropriate boundary conditions (BCs). We present new BCs for the FLD approximation in circumstellar environments that we apply here to spherically symmetric envelopes. Methods. At the inner boundary, the entering flux (coming from the star and from the envelope itself) may be written in the FLD formalism and provides us with an adequate BC. At the free outer boundary, we used the FLD formalism to constrain the ratio of the mean radiation intensity over the emerging flux. In both cases we derived non-linear mixed BCs relating the surface values of the mean specific intensity and its gradient. We implemented these conditions and compared the results with previous benchmarks and the results of a Monte Carlo radiative transfer code. A comparison with results derived from BCs that were previously proposed in other contexts is presented as well. Results. For all the tested cases, the average relative difference with the benchmark results is below 2% for the temperature profile and below 6% for the corresponding spectral energy distribution or the emerging flux. We point out that the FLD method together with the new outer BC also allows us to derive an approximation for the emerging flux. This feature avoids additional formal solutions for the radiative transfer equation in a set of rays (ray-tracing computations). Conclusions. The FLD approximation together with the proposed new BCs performs well and captures the main physical properties of the radiative equilibrium in spherical circumstellar envelopes.

Numerical solution of third-order Emden–Fowler type equations using artificial neural network technique

This paper presents the application of artificial neural network technique for solving a class of third-order linear and nonlinear boundary value problems with mixed nonlinear boundary conditions. This technique overcomes the singular behavior of problems and outlines the approximations of high exactness with a vast viable region of convergence. The proposed method has been tested for various examples; acquired outcomes exhibit the effectiveness and robustness of the proposed strategy and are compared with the other existing numerical techniques.

Existence of Positive Solution to Schrödinger-type Semipositone Problems with Mixed Nonlinear Boundary Conditions

We studied the existence of a positive solution to Schrodinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with $C^{1,\alpha}(\overline{\Omega})$-regularity for the superlinear case.

Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions

Abstract This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions

AbstractWe consider an elliptic problem of the type $$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$ where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and $\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem $$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$ For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).

A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

Lyapunov functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano’s method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.

Burgers equation with no-flux boundary conditions and its application for complete fluid separation

Burgers equation in a one-dimensional bounded domain with no-flux boundary conditions at both ends is proven to be exactly solvable. Cole–Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v¯ is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v¯. The problem arises naturally as a continuum limit of a network of certain micro-devices. Each micro-device imperfectly separates a target fluid component from a mixture of more than one component, and its input–output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macro-separator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.

A nonlinear thermoelastic system with nonlinear boundary conditions

In this paper, we give a mathematical treatment of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to nonlinear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear nonlocal thermoelastic coupled system under nonlinear boundary conditions.

Existence of solutions for fractional q-difference equation with mixed nonlinear boundary conditions

Abstract In this paper, we study the boundary value problem for a class of nonlinear fractional q-difference equation with mixed nonlinear conditions involving the fractional q-derivative of Riemann-Liouuville type. By means of the Guo-Krasnosel’skii fixed point theorem on cones, some results concerning the existence of solutions are obtained. Finally, examples are presented to illustrate our main results. MSC:39A13, 34B18, 34A08.

Subsonic Irrotational Flows in a Finitely Long Nozzle with Variable end Pressure

In this paper, we characterize a set of physically acceptable boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a finitely long flat nozzle. Our results show that if the incoming flow is horizontal at the inlet and an appropriate pressure is prescribed at the exit, then there exist two positive constants m 0 and m 1 with m 0 < m 1, such that a global smooth irrotational subsonic flow exists uniquely in the nozzle, provided that the incoming mass flux m ∈ [m 0, m 1). The horizontal velocity of the flow is always positive in the whole nozzle and the maximum speed of the flow will approach the sonic speed as the mass flux m tends to m 1. The flow is governed by the steady compressible irrotational Euler equations with nonlocal and nonlinear mixed boundary conditions. A new key issue is that the Bernoulli's constant of the irrotational flow is not given a priori, which can be determined uniquely by the end pressure and the incoming mass flux. To handle the nonlocal boundary condition raised from the mass flux, we introduce an auxiliary problem with Bernoulli's constant as a parameter, instead of the mass flux. Furthermore, the monotonicity between the mass flux and the Bernoulli's constant is established for given end pressure.

Nonlinear mixed boundary conditions in BVPs of Logistic type with spatial heterogeneities and a nonlinear flux on the boundary with arbitrary sign. The case [formula omitted

This paper analyzes the structure of the set of positive solutions of a general class of BVPs of Logistic type, with spatial heterogeneities and nonlinear mixed boundary conditions containing a flux with arbitrary sign. The main results obtained depend on the nodal behavior of the potential appearing in the partial differential equation, on the sign of the nonlinear flux on the boundary and on the relationship the exponents p>1, from the reaction in the partial differential equation, and q>1, from the nonlinear flux on the boundary. This paper is focused to the case p>2q−1. The main technical tools used are the Characterization of the Strong Maximum Principle given by H. Amann and J. López-Gómez [3], monotonicity technics, blow-up arguments and bifurcation. The results obtained in this paper are the natural extensions or complements of the previous ones obtained by S. Cano-Casanova [7,8], C. Morales-Rodrigo and A. Suárez [17] and J. García-Melián, C. Morales-Rodrigo, J.D. Rossi and A. Suárez [11].

The Solution of a Class of Third‐Order Boundary Value Problems by the Reproducing Kernel Method

This paper expands the application of reproducing kernel method to a class of third‐order boundary value problems with mixed nonlinear boundary conditions. The analytical solution is represented in the form of series in the reproducing kernel space. The n‐term approximation is obtained and is proved to converge to the analytical solution. The numerical examples are given to demonstrate the computation efficiency of the presented method. Results obtained by the method indicate that the method is simple and effective.

MIXED BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH DIFFERENT DEVIATED ARGUMENTS

This paper deals with second order differential equations with different deviated arguments (t) and (t, (t)). We investigate the existence of solutions of such problems with nonlinear mixed boundary conditions. To obtain corresponding results we apply the monotone iterative technique and the lower-upper solutions method. Two examples demonstrate the application of our results.

Inhomogeneous shear of orthotropic incompressible non-linearly elastic solids: Singular solutions and biomechanical interpretation

The paper presents a detailed study of rectilinear shear deformation in the framework of orthotropic nonlinear elasticity, under Dirichlet and mixed boundary conditions. Here the shear takes place in a slab made of a soft matrix reinforced with two families of extensible fibers, along the bisectrix of the angle between the two privileged directions aligned with the fibers. An analytic approach coupled to a careful computational treatment reveal that if the two families of parallel fibers are mechanically equivalent, then only smooth solutions are possible, whereas if the mechanical differences between the two families of fibers are pronounced, then strain singularities may develop. For the standard reinforcing orthotropic model, it is possible to determine the precise conditions for the existence of singular solutions. For an orthotropic constitutive law used for artery modeling, the existence of singular solutions can have repercussions in biomechanical applications.

Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems

We establish uniqueness of viscosity solutions for some boundary value problems arising from stochastic optimal control problems with unbounded, possibly singular, controls. They involve a nonlinear degenerate second order Bellman-Isaacs equation and mixed boundary conditions (Dirichlet, generalized Dirichlet and state constrained conditions).

Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions

We investigate the uniqueness of solutions forboundary value problems in bounded and unbounded domains involving nonlinear degenerate second orderBellman-Isaacs equations and mixed boundary conditions(Dirichlet, generalized Dirichlet and state constrained conditions).These boundary value problems arise from exit or stopping time stochastic differential games or optimal control problems with constraints, such as state and integral constraints.