Two-dimensional Nonlinear Equation Research Articles

Transverse compactlike pulse signals in a two-dimensional nonlinear electrical network.

We investigate the compactlike pulse signal propagation in a two-dimensional nonlinear electrical transmission network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and can reduce from the continuum limit approximation to a two-dimensional nonlinear Burgers equation governing the propagation of the small amplitude signals in the network. This equation has only the mass as conserved quantity and can admit as solutions cusp and compactlike pulse solitary waves, with width independent of the amplitude, according to the sign of the product of its nonlinearity coefficients. In particular, we show that only the compactlike pulse signal may propagate depending on the choice of the realistic physical parameters of the network, and next we study the dissipative effects on the pulse dynamics. The exactness of the analytical analysis is confirmed by numerical simulations which show a good agreement with results predicted by the Rosenau and Hyman K(2,2) equation.

A Fokker-Planck-Landau collision equation solver on two-dimensional velocity grid and its application to particle-in-cell simulation

An approximate two-dimensional solver of the nonlinear Fokker-Planck-Landau collision operator has been developed using the assumption that the particle probability distribution function is independent of gyroangle in the limit of strong magnetic field. The isotropic one-dimensional scheme developed for nonlinear Fokker-Planck-Landau equation by Buet and Cordier [J. Comput. Phys. 179, 43 (2002)] and for linear Fokker-Planck-Landau equation by Chang and Cooper [J. Comput. Phys. 6, 1 (1970)] have been modified and extended to two-dimensional nonlinear equation. In addition, a method is suggested to apply the new velocity-grid based collision solver to Lagrangian particle-in-cell simulation by adjusting the weights of marker particles and is applied to a five dimensional particle-in-cell code to calculate the neoclassical ion thermal conductivity in a tokamak plasma. Error verifications show practical aspects of the present scheme for both grid-based and particle-based kinetic codes.

Bifurcations of Tumor-Immune Competition Systems with Delay

A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

The behavior of solutions of the nonlinear biharmonic equation in an unbounded domain

Periodic (in one variable) solutions in the half-plane of the two-dimensional nonlinear biharmonic equation with exponential nonlinearity on the right-hand side are considered. The power-law and logarithmic asymptotics of the solutions at infinity are obtained.

A Lie-group Approach for Solving Backward Two-dimensional Nonlinear Klein-gordon Equation

In this paper, the backward two-dimensional nonlinear Klein-Gordon equation is solved by using a Lie-group approach. Because the solution does not continuously depend on the presented data, the inverse problem is famous for seriously ill-posed. A numerical example of the backward nonlinear Klein-Gordon equation demonstrates that this method is applicable, and the computational cost is quite cheap. Although the recovery of a presently wave is from the final time and under a large noise imposed on the final time data, this algorithm still works effectively and accurately. These numerical results are rather important in the computations of backward nonlinear Klein-Gordon equation.

DANGEROUS ZONE FORMATION BEHIND FINITE-LENGTH COASTAL FOREST FOR TSUNAMI MITIGATION

After the Indian Ocean Tsunami in 2004, several studies quantitatively investigated the effects of coastal vegetation on tsunami mitigation, but the effects of a limited forest with a small aspect ratio on tsunami mitigation were not yet elucidated. Therefore, the objective of this study was to estimate numerically the effect of the width-length ratio (aspect ratio) of a coastal forest on tsunami mitigation. Numerical simulations were performed using two-dimensional nonlinear long-wave equations that included bed resistance, drag, and turbulence-induced shear forces due to interaction with the forest. When a limited dense forest exists, the tsunami at the edge of the forest diffracts and collides behind the forest, and the fluid force becomes larger than the case without a forest. In particular, when the aspect ratio is from 1 to 4, the effect of a collision behind the forest becomes very great. However, if the aspect ratio is 4 or larger, the effect of a collision becomes smaller.

Numerical solution of nonlinear elliptic equation with nonlocal condition

Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation.

The Kuramoto-Sivashinsky equation for adatoms interacting through quasi-Rayleigh waves and formation of ordered adatom structures

It is shown that the defect-deformational (DD) cooperative nucleation of ordered ensembles of nanoparticles on isotropic surfaces of solids with the participation of quasi-Rayleigh waves can be described by a closed two-dimensional nonlinear DD equation of the Kuramoto-Sivashinsry (KS) type. A solution to the linearized DDKS equation describes the threshold appearance of the periodic surface strain modulation accompanied by the simultaneous formation of adatom (surface defect) piles at extrema of the surface strain. Numerical solutions to the DDKS equation in linear and nonlinear regimes describe the formation of three types of surface structures of adatom piles (clusters): lamellar-like structures, cellular disordered and hexagonal ordered ones. In the well-developed nonlinear regime, generated ensembles of nanoparticles become trimodal regarding their size distribution due to the generation of the second harmonics, degenerate parametric decay and summation of wavevectors of DD gratings taking part in the DD selforganization. The selforganizing periodic cellular DD surface structure can serve as a selforganized template for the subsequent growth of nanoparticles in the processes of atoms deposition.

Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation

The present paper is devoted to the study of the global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. In the absence of horizontal dissipation, we establish a growth estimate on vertical component of velocity, that is, supp⩾2‖u2(t)‖Lpplogp which is close to ‖u2(t)‖L∞ and is bounded via the low-high decomposition technique. This together with the smoothing effect in vertical direction enables us to obtain the H1-estimate for velocity. Based on this, we prove the existence and uniqueness of classical solution without smallness assumptions. In addition, we also discuss the global well-posedness result for the rough initial data.

Compact alternating direction implicit method to solve two-dimensional nonlinear delay hyperbolic differential equations

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously in L2- and H1-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes.

Modeling of Shallow-Water Equations by Using Implicit Higher-Order Compact Scheme with Application to Dam-Break Problem

The paper deals with the unsteady two-dimensional (2D) non-linear shallow-water equations (SWE) in conservation-law form to capture the fluid flow in transition. Numerical simulations of dam-break flood wave in channel transitions have been performed for inviscid and incompressible flow by using two new implicit higher-order compact (HOC) schemes. The algorithm is second order accurate in time and fourth order accurate in space, on the nine-point stencil using third order non-centered difference at the wall boundaries. To solve the algebraic system, bi-conjugate gradient stabilized method (BiCGStab) with preconditioning has been employed. Although, both the schemes are able to capture both transient and steady state solution of shallow water equations, the scheme expressed in conservative law form is unconditionally stable. The model results have been validated for dam-break problem and compared with the experimental data for dry and wet bed conditions. The model results are found to be in good agreement with the experimental observations. The proposed scheme is useful to solve to capture flow transition with minimal number of nodal points, particularly for hyperbolic system.

Travelling wave solution of two-dimensional nonlinear KdV-Burgers equation

Travelling wave solution of two-dimensional nonlinear KdV-Burgers equation

Improvement of effectiveness of existing Casuarina equisetifolia forests in mitigating tsunami damage

Coastal vegetation can play a significant role in reducing the severity of a tsunami because the energy associated with the tsunami is dissipated when it passes through coastal vegetation. Field surveys were conducted on the eastern coastline of Sri Lanka to investigate which vegetation species are effective against a tsunami and to evaluate the effectiveness of existing Casuarina equisetifolia forests in tsunami mitigation. Open gaps in C. equisetifolia forests were identified as a disadvantage, and introduction of a new vegetation belt in front or back of the existing C. equisetifolia forest is proposed to reduce the disadvantages of the open gap. Among the many plant species encountered during the field survey, ten species were selected as effective for tsunami disaster mitigation. The selection of appropriate vegetation for the front or back vegetation layer was based on the vegetation thickness per unit area (dNu) and breaking moment of each species. A numerical model based on two-dimensional nonlinear long-wave equations was applied to explain the present situation of open gaps in C. equisetifolia forests, and to evaluate the effectiveness of combined vegetation systems. The results of the numerical simulation for existing conditions of C. equisetifolia forests revealed that the tsunami force ratio (R = tsunami force with vegetation/tsunami force without vegetation) was 1.4 at the gap exit. The species selected for the front and back vegetation layers were Pandanus odoratissimus and Manilkara hexandra, respectively. A numerical simulation of the modified system revealed that R was reduced to 0.7 in the combined P. odoratissimus and C. equisetifolia system. However, the combination of C. equisetifolia and M. hexandra did not effectively reduce R at the gap exit. Therefore, P. odoratissimus as the front vegetation layer is proposed to reduce the disadvantages of the open gaps in existing C. equisetifolia forests. The optimal width of P. odoratissimus (W1) calculated from the numerical simulation was W1 = 10 m. R at the exit of a 15-m-wide open gap was 0.8, and therefore the proposed system was appropriate for cases with the highest velocity at the gap exit as well. Establishment of a new front vegetation layer except for open gaps that are essential, such as access roads to the beach, is proposed.

Numerical simulation of Urumqi Glacier No. 1 in the eastern Tianshan, central Asia from 2005 to 2070

Due to climate changes, most of the alpine glaciers have retreated dramatically during the past decades. Thus it is significant to predict the alpine glacier variability in the future for a better understanding of the impact of climate changes on water resource. In this paper, we perform the numerical simulation on Urumqi Glacier No.1 in the eastern Tianshan, central Asia (hereafter Glacier No.1 for short) by considering both the mass balance and ice flow. Given the shape of the Glacier No.1, the velocity of the glacier is obtained by solving a two-dimensional nonlinear Stokes equation and simulated result is in agreement with the observation. In order to predict the variability of Glacier No.1 in the next decades, a climatic scenario is constructed with a temperature rise rate as 0.17°C/10 a and precipitation as constant during the period of 2005–2070. The simulation shows that, the glacier terminus will retreat slowly and the glacier will thin dramatically before 2040, while after year 2040, the glacier terminus retreat will accelerate. This study confirms the increasing retreat rate of alpine glaciers under global warming.

Solitary waves solution in atmospheric inhomogeneous quantum plasma

For an inhomogeneous quantum magnetoplasma system in the atmospheric environment with density and temperature gradient, a two-dimensional nonlinear fluid dynamic perturbation equation is studied in the case where the collision frequency between ions and neutrals is small. The approximate solution of the potential in the dense astrophysical environment is obtained.

Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

A two‐dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is exponentially decayed if the memory kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).

Tsunami mitigation by coastal vegetation considering the effect of tree breaking

Damage to vegetation by tsunami moment and reduction of potential tsunami force are discussed based on a numerical simulation. A numerical model based on two-dimensional nonlinear long-wave equations that include drag forces and turbulence-induced shear force due to the presence of vegetation was developed to estimate tree breaking. The numerical model was then applied to a coastal forest where two dominant tropical vegetation species, Pandanus odoratissimus and Casuarina equisetifolia, were considered. The threshold water depth for tree breaking increased with increasing forest width, and the analysis was consistent with the field investigation results that the critical tsunami water depth for breaking is around 80% of the tree height for P. odoratissimus. C. equisetifolia is stronger than P. odoratissimus against tsunami action, but P. odoratissimus can reduce a greater tsunami force than C. equisetifolia due to its complex of aerial root structures. Even if breakage occurs, P. odoratissimus still has high potential to reduce the tsunami force due to its dense aerial root structures. Previous numerical models that do not include the breaking phenomena may overestimate the vegetation effect for reducing tsunami force. The combination of P. odoratissimus and C. equisetifolia is recommended as a vegetation bioshield to protect coastal areas from tsunami hazards.

L ∞-estimates of mixed finite element methods for general nonlinear optimal control problems

This paper investigates L∞-estimates for the general optimal control problems governed by two-dimensional nonlinear elliptic equations with pointwise control constraints using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. The authors derive L∞-estimates for the mixed finite element approximation of nonlinear optimal control problems. Finally, the numerical examples are given.

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson–Boltzmann equation and diffusion–reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.

A compact discretization of O(h4) for two-dimensional nonlinear triharmonic equations

We report a new finite-difference approximation of O(h4) for two-dimensional nonlinear triharmonic partial differential equations on a nine-point compact stencil where the values of u, ∂ 2u/∂n2 and ∂ 4u/∂n4 are prescribed on the boundary. In this method, there is no need to discretize the derivative boundary conditions. The Laplacian and the biharmonic of the solution are obtained as a by-product of the method. We require only a system of three equations to obtain the solution. We compare the advantages and implementation of the proposed method with the corresponding central difference approximations of O(h2) in the context of iterative methods. Numerical results are given to verify the fourth-order convergence rate of the method.