Nonlinear Boundary Conditions Research Articles

Multiplicity of Solutions for the Noncooperative Kirchhoff-Type Variable Exponent Elliptic System with Nonlinear Boundary Conditions

Considering the solutions of a class of noncooperative Kirchhoff-type p(x)-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (PS)c* condition using another method, diverging from the approaches cited in the previous literature.

A new methodology in evaluating nonlinear electrohydrodynamic azimuthal stability between two dusty viscous fluids

The objective of the current paper is to scrutinize the azimuthal stability of a circular interface between two viscous fluids across porous medium in the presence of fine dust. Dusty fluid models are employed to comprehend the dynamics of aerosols in the atmosphere, a crucial aspect in the investigation of air quality, climate change, and the dissemination of pollutants. Furthermore, the comprehension and management of azimuthal nonlinear instabilities are essential in diverse engineering applications, where turbulence is a prominent factor. This includes aerospace engineering, where it is necessary for developing more efficient aircraft, as well as fluid dynamics, where it is critical for optimizing industrial processes. The system is subjected to the influence of a uniform electric field (EF) that is directed in the azimuthal direction. Electrospray trustees utilized in spaceship propulsion systems make use of the principles of Electrohydrodynamics (EHD). These representatives employ the process of ionization on a liquid propellant and utilize EFs to accelerate the resulting ions, so producing thrust. Assuming a simplified situation, the prototype is assumed to have a two-dimensional structure. Both fluids are presumed as rotating. An examination of rotating fluids is crucial for comprehending the dynamics of the atmosphere and oceans. Obtaining this knowledge is essential for accurately predicting weather, simulating climate patterns, and investigating oceanic currents. The mathematical approach is simplified by considering the viscous potential theory (VPT). The nonlinear technique relies on the use of linear fundamental partial differential equations (PDEs) and the application of the appropriate nonlinear boundary conditions (BCs). This method produces a nonlinear PDE that controls the displacement of the interface. By disregarding the nonlinear elements, a dispersion equation that is linear in nature is obtained. The stability criteria are graphed using several dimensionless physical quantities. The non-perturbative analysis (NPA) achieves nonlinear stability. As well-known, the primary objective of NPA is to convert a linear ordinary differential equation (ODE) into a linear equation. The NPA distinguishes itself from conventional perturbation methods by its ability to accurately analyze the dynamics of highly nonlinear oscillators. This novel methodology is developed utilizing the He's frequency formula (HFF) for the purpose of designing the displacement of the interface. The computational computations demonstrated that the NPA is efficacious, encouraging, and powerful. The linear and nonlinear techniques demonstrate that the effects of various dimensionless physical parameters remain constant. It is found that the Weber number We and Capillary number Cabecome zero as crucial consequences. The Ohnesorge number, Oh which represents the ratio of viscosity and the mass concentration in dusty particles, has a destabilizing effect.

Global dynamics of a Lotka-Volterra competition-diffusion system with advection and nonlinear boundary conditions

Global dynamics of a Lotka-Volterra competition-diffusion system with advection and nonlinear boundary conditions

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

Chaos of Multi-dimensional Weakly Hyperbolic Equations with General Nonlinear Boundary Conditions

Chaos of Multi-dimensional Weakly Hyperbolic Equations with General Nonlinear Boundary Conditions

Analysis of bulk-surface reaction-sorption-diffusion systems with Langmuir-type adsorption

We consider a class of bulk-surface reaction-adsorption-diffusion systems, i.e. a coupled systems of reaction-diffusion systems on a bounded domain Ω⊆Rd (bulk phase) and its boundary Σ=∂Ω (surface phase), which are coupled via nonlinear normal flux boundary conditions. In particular, this class includes a heterogeneous catalysis model with Fickian bulk and surface diffusion and nonlinear adsorption of Langmuir type, i.e. transport from the bulk phase to the active surface, and desorption. For this model, we obtain well-posedness, positivity and global-in-time existence of solutions under some realistic structural conditions on the chemical reaction network and the sorption model. We work in appropriate Sobolev-Slobodetskii settings, where we aim for a wide range for the integrability index, including in particular values p<d.

Semi-implicit Solver for the Heat Equation with Stefan–Boltzmann Law Boundary Condition

The surface energy balance on an atmosphereless body consists of solar irradiance, subsurface heat conduction, and thermal radiation to space by the Stefan–Boltzmann law. Here we extend the semi-implicit Crank–Nicolson method to this specific nonlinear boundary condition and validate its accuracy. A rapid change in incoming solar flux can cause a numerical instability, and several approaches to dampen this instability are analyzed. A predictor based on the Volterra integral equation formulation for the heat equation is also derived and can be used to improve accuracy and stability. The publicly available implementation provides a fast and robust thermophysical model that has been applied to lunar, Martian, and asteroidal surfaces, on occasion to millions of surface facets or parameter combinations.

Existence of solutions for boundary value problems for first order impulsive difference equations

In this work, we establish sufficient conditions for the existence of nonnegative solutions for a class of first order impulsive difference equations with a family of nonlinear boundary conditions. To prove our main result we use a new topological approach on the fixed point index theory for the sum of two operators in Banach spaces. An example is given to illustrate the main result.

Thin film flow of chemically reactive nonlinear thermally radiative magnetized variable viscosity fluid over a stretchable surface with non-uniform heat source/sink and Fick’s mass flux theory

The study of thin film flow over a stretching sheet holds significant importance from an industrial perspective due to its relevance in various manufacturing and engineering processes. In the present investigation, we delved into the combined influence of non-uniform heat source/sink on the governing heat and mass transfer equations in a thin liquid film on a permeable stretching surface. We have considered the nonlinear thermal radiation and implemented convective-type boundary conditions. Our analysis examined the effects of crucial physical parameters, including non-uniform heat source/sink, thermal radiation, chemical reaction, magnetic parameter , Prandtl number, Schmidt number, and unsteadiness parameter in the range [−3.5, 2.0], [−2, 5], [0, 1], [0.3, 4.0], [1, 7], [0.8, 1.7], [0.7, 2.0], respectively. It is found that a non-uniform heat source/sink mitigated the temperature gradient, while an elevation in the Schmidt number intensified the mass transfer rate. The validation process enhanced the robustness and reliability of our numerical simulations.

Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi-Laplacian and gi:∂Ω×R×R→R are Carathéodory functions that satisfy general growth and structure conditions for i=1,2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of gi near zero related to the first eigenvalue of the pi-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g1,g2)=∇g with a smooth function (s1,s2)↦g(x,s1,s2). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the pi-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

Isogeometric analysis of hyperelastic Timoshenko beams under large deformation and incompressibility constraints

This article investigates the nonlinear analysis of hyperelastic planar curved beams based on the total Lagrangian Timoshenko theory. The fully incompressible neo-Hookean model has been selected as the constitutive relation to accurately capture large deformations. The nonlinear equations and boundary conditions derived from the virtual work principle are solved using the Newton–Raphson method. Employing isogeometric analysis with NURBS functions to discretize the unknown kinematics, including displacements and rotation, and the linearized governing equations. The study accounts for conservative and non-conservative forces, validating results against available experimental and numerical results reported in the literature, confirming the proposed formulation’s effectiveness and precision.

On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions

We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.

Nonlinear ice sheet/liquid interaction in a channel with an obstruction

The interaction between the flow in a channel with an obstruction on the bottom and an elastic sheet representing the ice covering the liquid is considered for the case of steady flow. The mathematical model based on the velocity potential theory and the theory of thin elastic shells fully accounts for the nonlinear boundary conditions at the elastic sheet/liquid interface and on the bottom of the channel. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. This allows one to formulate the coupled ice/liquid interaction problem and reduce it to a system of nonlinear equations in the unknown magnitude of the velocity at the interface. Case studies are carried out for a semi-circular obstruction on the bottom of the channel. Three flow regimes are studied: a subcritical regime, for which the interface deflection decays upstream and downstream; an ice supercritical and channel subcritical regime, for which two waves of different lengths may exist; and a channel supercritical regime, for which the elastic wave is found to extend downstream to infinity. All these regimes are in full agreement with the dispersion equation. The obtained results demonstrate a strongly nonlinear interaction between the elastic and the gravity wave near the first critical Froude number where their lengths approach each other. The interface shape, the bending moment and the pressure along the interface are presented for wide ranges of the Froude number and the obstruction height.

SH waves in a weakly inhomogeneous half space with a nonlinear thin layer coating

We investigate the self-modulation of Love waves propagating in a nonlinear half-space covered by a nonlinear layer. We assume that the constituent material of the layer is nonlinear, homogeneous, isotropic, compressible, and hyperelastic, whereas for the half-space, it is nonlinear, heterogeneous, compressible and a different hyperelastic material. By employing the nonlinear thin layer approximation, the problem of wave propagation in a layered half-space is reduced to the one for a nonlinear heterogeneous half-space with a modified nonlinear homogeneous boundary condition on the top surface. This new problem is analyzed by a relevant perturbation method, and a nonlinear Schrödinger (NLS) equation defining the self-modulation of waves asymptotically is obtained. The dispersion relation is derived for different heterogeneous properties of the half-space and the thin layer. Then the results of the thin layer approximation are compared with the ones for the finite layer obtained in Teymur et al. (Int J Eng Sci 85:150–162, 2014). The solitary solutions of the derived NLS equation are obtained for selected real material models. It has been discussed how these solutions are influenced by the heterogeneity of the semi-infinite space.

On singular φ−Laplacian BVPs of nonlinear fractional differential equation

This paper investigates the existence of multiple positive solutions for a class of φ−Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed-point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 47H10. Received 27 July; Accepted 13 December 2021

An efficient localized Trefftz method for the simulation of two-dimensional sloshing behaviors

This study proposes a new method for effectively and accurately simulating sloshing in a two-dimensional numerical tank. This technique uses the localized Trefftz method (LTM), a meshless numerical approach. Sloshing in a numerical tank is mathematically described as a space-time boundary-value problem rooted in potential flow theory. This occurrence is governed by a second-order partial differential equation and two nonlinear free-surface boundary conditions. In this study, the moving boundary problem undergoes discretization in time and space by using the LTM and the explicit Euler method, respectively. After discretization with the explicit Euler method, the elevation of the free surface is updated, and a boundary-value problem is generated at each time step. This boundary-value problem can be effectively examined using the recently developed LTM, which eliminates the need for complex meshing. Contrary to conventional methods, such as the method of fundamental solutions and the boundary element method, the LTM generates sparse matrices, allowing large-scale numerical simulations. Four numerical examples are presented to demonstrate the clarity and accuracy of the proposed meshless approach.

Meso-scale simulation of moisture transport in concrete during wet-dry cycles using 3D RBSM conduit model with variable diffusion model

This paper introduces the 3D RBSM Conduit model, an improved simulation program based on the Rigid Body Spring Model (RBSM), designed to accurately depict the intricate phenomenon of moisture transfer within concrete. This is achieved by incorporating an expanded diffusion model into the simulation framework. The diffusion coefficient is dynamically adjusted at each location and at each time step to accurately represent the precise moisture transfer processes occurring in the concrete. This dynamic adjustment allows the program to efficiently simulate the rate of moisture transport in concrete based on the moisture degree and the dry or wet state of the material. By adapting the diffusion coefficient to the changing conditions in this way, the program achieves simulated moisture transfer rates that align closely with the real-world behavior of moisture in concrete. It offers a dynamic model of the changes in concrete transmission during the switch from drying and wetting conditions. Further, the program exhibits inherent advantages in addressing water transfer problems involving nonlinear boundary conditions and transfer equations. In this study, the program is used to simulate various processes including drying, wetting, local wetting, and combined wet-dry cycles. The results reveal the lag in wet-dry transitions between the interior and exposed surface during the wet-dry conversion process due to retained interior moisture. The simulation captures the moisture variation in the transitional zone.

Multiple Solutions of a Nonlocal Problem with Nonlinear Boundary Conditions

In this article, we consider a class of nonlocal p(x)-Laplace equations with nonlinear boundary conditions. When the nonlinear boundary involves critical exponents, using the concentration compactness principle, mountain pass lemma, and fountain theorem, we can prove the existence and multiplicity of solutions.

Doubly reflected generalized BSDEs with jumps and an obstacle problem of parabolic IPDEs with nonlinear Neumann boundary conditions

Abstract A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions.

On a nonlinear Robin problem with an absorption term on the boundary and L 1 data

Abstract We deal with existence and uniqueness of nonnegative solutions to: − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}&amp; \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η ≥ 0 \eta \ge 0 and f , λ f,\lambda , and g g are the nonnegative integrable functions. The set Ω ⊂ R N ( N &gt; 2 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν \nu denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of p p -Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.