Nonlinear Boundary Conditions Research Articles

Insightful inspection of the nonlinear instability of an azimuthal disturbance separating two rotating magnetic liquid columns

The nonlinear stability examination of two revolving magnetized liquid columns connecting two completely submerged fluids in a permeable region is the aim of the existing paper. Two endless vertical cylinders occupied with two magnetic fluids make up the present structure. Significantly, the disturbance at the border displays an azimuthal behavior. The entire structure is activated by an azimuthal unchanging magnetic field (MF). The increasing interest in the atmospheric and oceanic dynamics is the primary motivation in exploring this problem. To relax the complication of the mathematical processes, the viscous potential theory (VPT) is established. The motion is assessed using three basic coexistent field formulations: Maxwell's formula, Brinkman's formula, and the continuity condition, in the construction of the Coriolis force and centrifugal implications. The explanations of the linearized formula of motion produce a nonlinear categorizing diffusion structure because of the implications of the nonlinear boundary conditions (BCs). The non-perturbative approach (NPA) based on the He's frequency formulation (HFF) is employed to transform the nonlinear characteristic ordinary differential equation (ODE) into a linear one. A short description of the NPA is also presented. The nonlinear ODE with real and imaginary coefficients is exposed by the stability analysis. The stability requirements are implemented using only a nonlinear analysis. As demonstrated, as an unusual state, it is exposed that ignoring the Weber number removes all complex items of the nonlinear formulation. Physically, this means the absence of the angular velocities from the physical model. For both the real and complex situations of the original equation, the stability remains unchanged. It is found that the azimuthal MF, rotating parameter, and Darcy’s numeral have a maintenance impact. On the other hand, the azimuthal wave numeral has a destabilizing one. Several polar designs are drawn to agreement the stability situations.

Global dynamics of a diffusive competitive Lotka–Volterra model with advection term and more general nonlinear boundary condition

We investigate a competitive diffusion–advection Lotka–Volterra model with more general nonlinear boundary condition. Based on some new ideas, techniques, and the theory of the principal spectral and monotone dynamical systems, we establish the influence of the following parameters on the dynamical behavior of system (1.2): advection rates αu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{u}$\\end{document} and αv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{v}$\\end{document}, interspecific competition intensities cu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{u}$\\end{document} and cv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{v}$\\end{document}, the resources functions ru\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{u}$\\end{document} and rv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{v}$\\end{document} of the two competitive species, and nonlinear boundary functions g1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{1}$\\end{document} and g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{2}$\\end{document}. The models of (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018) are particular cases of our results when gi≡const\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{i}\\equiv const$\\end{document} for i=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$i=1,2$\\end{document}, and hence this paper extends some of the conclusions from (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018).

New exact solutions of the (3+1) dimensional Charney - Obukhov equation for the ocean in the case of “separation” of variables

Exact solutions describing Rossby waves and vortices in ocean propagating along the zonal direction at a constant velocity are considered for the (3 + 1) -dimensional nonlinear Charney – Obukhov equation. We give examples of solutions of the Charney-Obukhov equation that satisfy the nonlinear boundary conditions for the ocean, for a special case that corresponds to the “separation” of variables. The following example is considered: a lattice of vortices with 3D flow in lattice cells, the inclination of vortex cells in the meridional plane, and the dependence of the inclination angle on depth.

A Priori Error Analysis and Finite Element Approximations for a Coupled Model Under Nonlinear Slip Boundary Conditions

A Priori Error Analysis and Finite Element Approximations for a Coupled Model Under Nonlinear Slip Boundary Conditions

Azer Sign-changing radial solutions for a semilinear problem on exterior domains with nonlinear boundary conditions

In this paper we are interested to the existence and multiplicity of sign changing radial solutions of problem of elliptic equations $\Delta U(x)+\varphi(|x|)f(U)=0$ with a nonlinear boundary conditions on exterior of the unite ball centered at the origin in $\mathbb{R}^{N}$ such that $ u(x) \rightarrow 0$ as $ |x|\to \infty $, with any given number of zeros where the nonlinearity $ f(u) $ is odd, superlinear for $ u $ lager enough and $ f0 $ on $(0,\beta)$, $ f0$ on $(\beta,\infty) $. The function $\varphi0$ is $ C^{1} $ on $ [R,\infty) $ where $ 0\varphi(|x|)\leq c_0\,|x|^{-\alpha}$ with $ \alpha2(N-1) $ and $ N2 $ for large $ |x| $.

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