Second-order Boundary Research Articles

Existence and nonexistence of positive solutions for the second-order m-point eigenvalue impulsive boundary value problem

This paper is devoted to the study of the existence and nonexistence of positive solutions for a second-order m-point eigenvalue impulsive boundary value problem. We use the fixed point theorems on the cones in order to achieve our results.

Existence and uniqueness criteria for nonlinear quantum difference equations with $ p $-Laplacian

<abstract><p>Q-calculus plays an extremely important role in mathematics and physics, especially in quantum physics, spectral analysis and dynamical systems. In recent years, many scholars are committed to the research of nonlinear quantum difference equations. However, there are few works about the nonlinear $ q- $difference equations with $ p $-Laplacian. In this paper, we investigate the solvability for nonlinear second-order quantum difference equation boundary value problem with one-dimensional $ p $-Laplacian via the Leray-Schauder nonlinear alternative and some standard fixed point theorems. The obtained theorems are well illustrated with the aid of two examples.</p></abstract>

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

A Comparative Study of 3d Cumulant and Central Moments Lattice Boltzmann Schemes with Interpolated Boundary Conditions for the Simulation of Thermal flows in High Prandtl Number Regime

Thermal flows characterized by high Prandtl number are numerically challenging as the transfer of momentum and heat occurs at different time scales. To account for very low thermal conductivity and obey the Courant-Friedrichs-Lewy condition, the numerical diffusion of the scheme has to be reduced. As a consequence, the numerical artefacts are dominated by dispersion errors commonly known as wiggles. In this study, we explore possible remedies in the framework of the lattice Boltzmann method by applying novel collision kernels, lattices with a large number of discrete velocities, namely D3Q27, and second-order boundary conditions. For the first time, the cumulant-based collision operator is utilised to simulate both the hydrodynamic and the thermal field. Alternatively, the advected field is computed using the central moments' collision operator. Different relaxation strategies have been examined to account for additional degrees of freedom introduced by a higher-order lattice. To validate the proposed kernels for a pure advection-diffusion problem, the numerical simulations are compared against an analytical solution of a Gaussian hill. The structure of the numerical dispersion is shown by simulating advection and diffusion of a square indicator function. Finally, a study of steady forced heat convection from a confined cylinder is performed and compared against a FEM solution. It has been found, that the relaxation scheme of the advected field must be adjusted to profit from lattice with a larger number of discrete velocities. Obtained results show clearly that it is not sufficient to assume that only the first-order central moments/cumulants contribute to solving the macroscopic advection-diffusion equation. In the case of central moments, the beneficial effect of the two relaxation time approach is presented.

Dual Solutions of EMHD Nanofluid at Stretching Sheet with Mixed Convection Slip Boundary Condition

In this paper stagnation boundary layer flow of nanofluid with mixed convection heat and mass transfer with Electrical Magneto Hydrodynamic (EMHD) effects over a second-order momentum slip boundary condition have been mathematically analysed. The governing equations are transformed by similarity variable and the problem becomes coupled third-order nonlinear coupled differential equations. We use fourth-order Runge Kuta method and shooting technique to find the solution. The effect of second-order momentum slip condition with linear thermal slip condition has determined. Variation of all nano energy conversion parameters depends on different factors has shown graphically. Some of the parameters possesses dual solution at different values of second-order velocity slip parameter (β1&β2).

On Existence of Symmetric Positive Solutions for SSBVPs on Time Scales

This paper considers symmetric positive solutions for the system of second-order boundary value problems (SSBVPs) on time scales using the Krasnosel’skii fixed point theorem. We then demonstrate two examples. Finally, we discuss the need for further results.

Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions

Nonlinear boundary conditions are difficult to be fulfilled exactly, when one employs numerical methods to treat a highly nonlinear boundary value problem (NBVP). In this paper, a novel iterative algorithm to solve NBVP involved with two coupled nonlinear boundary conditions at two-end of a unit interval is developed, of which the solution can satisfy the nonlinear boundary conditions automatically. By letting the free function in the boundary shape function (BSF) be a new variable, an initial value problem (IVP) is created from the second-order NBVP. While the initial values of the new variable are given, the terminal values are viewed as unknown parameters to be determined iteratively. Therefore, a very accurate solution for the NBVP with nonlinear boundary conditions can be quickly determined through a few iterations. Some numerical examples confirm the efficiency and accuracy of the proposed iterative scheme, wherein the examples with multiple solutions and unique solution are worked out.

Family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method.

In this paper, we propose a family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method (FD-LBM), which consist of convex combinations. The FD-LBM unifies several different numerical schemes for the Navier-Stokes equations, and thereby these boundary schemes are naturally applicable for the standard LBM. The accuracy of the boundary schemes is independent of the boundary location, and it is validated by several numerical experiments, two- and three-dimensional flow problems, with straight and curved boundaries.

A lattice Boltzmann model for the Navier-Stokes equation

The lattice Boltzmann model is a vital numerical calculation tool, which has become a useful solution to fluid problems because of the limitations of the Navier-Stokes equation and other macroscopic models in solving such problems. Here, the effectiveness of this model in the nearly incompressible Navier-Stokes equation is investigated as follows. First, the principle and excellent characteristics of the lattice Boltzmann model in numerical calculations are described and analyzed. Second, the approximate incompressible Navier-Stokes equations of the BGK (Bhatnagar-Gross-Krook) model in lattice Boltzmann method are analyzed. Finally, the two-dimensional five discrete velocity (D2N5) model in BGK model is taken as an example, and the vector rebound boundary scheme and the single point second-order boundary scheme based on the Dirichlet boundary condition are derived. Numerical results based on the Poiseuille flow and Taylar-Green vortex show that the BGK-based vector rebound boundary scheme has second-order accuracy only when the proportional distance is 0.5, and in other cases it cannot reach the accuracy. For the single-point second-order boundary scheme, the second-order accuracy can be achieved both in the Poiseuille flow based on the straight boundary region and in the Taylar-Green vortex based on the curved boundary. Through Maxwell iteration, the vector BGK model based on discretization is stable. When the boundary position is in the middle of two lattice points, the vector rebound boundary scheme has second order accuracy, and the single point second order boundary scheme has better performance in second order accuracy. This provides an idea for dealing with boundary conditions in incompressible Navier-Stokes equations.

The Solvability of the Discrete Boundary Value Problem on the Half-Line.

This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form with where , . To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.

Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Gaussian process regression constrained by boundary value problems

We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.

Second order boundary value problems of nonsingular type on time scales

In this study, the existence of positive solutions is considered for second-order boundary value problems on any time scales even in the case when y ≡ 0 may also be a solution.

Nonlinear wave loads on a stationary cylindrical-type oscillating water column wave energy converter

Based on the potential flow theory, a second-order Higher-Order Boundary Element Method model is developed to simulate hydrodynamic loads and pressure on a stationary cylindrical-type oscillating water column (OWC) wave energy converter. Quadratic pneumatic damping and additional viscous damping are introduced to simulate the effects of the power take-off and the viscous dissipation inside the chamber, respectively. The proposed model is verified against a carefully instrumented scaled experiment. The effects of the chamber geometry and the wave environment on the hydrodynamic loads of the OWC device are then explored using the validated model. The resonance excited by the second-order wave component is observed with the fundamental frequency being half of the natural frequency. This resembles a sloshing mode and leads to a significant increase in hydrodynamic pressure and surge force as well as pitch moment. The effects of the second-order waves cannot be ignored under the action of the low-frequency long waves. In addition, the surge force and the pitch moment are not sensitive to the variation in the orifice opening ratios, while the heave force increases with the opening ratio at a resonant heave frequency.

Kinetic boundary schemes of axisymmetric multi-relaxation-time lattice Boltzmann model for microscale gas flows in microtube

Because of its mesoscopic nature and distinctive computing features, lattice Boltzmann method (LBM) is shown as a promising tool to simulate microscale gas flows. However, studies on the microtube flows are rare, and some fundamental problems for LBM to such flows still remain unsolved. In this work, three widely used kinetic boundary schemes, i.e., combined bounce-back and specular-reflection scheme, combined Maxwell-diffusion and specular-reflection scheme, and combined Maxwell-diffusion and bounce-back scheme, for the axisymmetric lattice Boltzmann equation (LBE) with multi-relaxation-time (MRT) are investigated. The discrete effects of the three boundary schemes in the axisymmetric MRT-LBE are observed by analyzing the micro-scale Hagen-Poiseuille flow, and the strategies of adjusting the parameters in the boundary schemes are put forward to implement the no-slip boundary condition and the second-order slip boundary condition. Some numerical tests are implemented to validate the proposed correction strategies for kinetic boundary schemes. Our analysis also shows that the kinetic boundary schemes of the axisymmetric Bhatnagar-Gross-Krook (BGK) LBE cannot realize the no-slip boundary condition and are also inapplicable to microscale gas flows with even some moderate Knudsen numbers (). Therefore, the axisymmetric MRT-LBE is superior to the axisymmetric BGK-LBE in realizing both the no-slip and slip boundary conditions.

On a system of second-order multi-point boundary value problems on time scales

This paper is concerned with the existence and nonexistence of positive solutions for a system of nonlinear second order dynamic equations with multi-point boundary conditions on time scales.

Positive solutions for second-order differential equations o Kirchhoff type on the half-line

The aim of the present paper is to study the existence of nontrivial nonnegative solutions for a second-order boundary value problem of Kirchhoff type on the half-line. Our approach is based on variational methods, a monotonicity trick related to the mountain pass lemma, cut-off functional technique, and a Pohozaev type identity.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

Second-order boundary estimates for large positive solutions to an elliptic system of competitive type

In this paper, we study the second-order boundary asymptotic behaviour for large positive solutions to an elliptic system of competitive type. First, we derive a second-order estimate to a related single weighted equation with boundary blow-up data. Then, by relaxing the system and iterating the estimate of the single equation, we establish second-order estimates of the solutions.