ANALYSIS OF BOUNDARY LAYER SINGULARITYIN A NONLINEAR DIFFUSION PROBLEM

In this paper, the author analyzes the singularity of a boundary layer in a nonlinear diffusion problem. Results show when the limiting solution satisfies the boundary condition, there is no boundary singularity. Otherwise, the boundary layer exists, and its thickness is proportional to epsilon(1/2), here epsilon is a small positive real parameter.

Existence of Non-trivial Nonnegative Periodic Solutions for a Nonlinear Diffusion System

In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite propagation property and the L^(1)−L^(∞)smoothing effect,we find that the complicated asymptotic behavior of the rescaled solutions t^(μ/2)u(t^(β)⋅,t)for 0<μ<2 N/(N[m(p−1)−1]+p)andβ>(2−μ[m(p−1)−1])/(2 p)can take place.

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.

ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.

A Class of New Special Solution of Nonlinear Diffusion Equation

One of the most interesting problems of nonlinear differential equations is the construction of partial solutions. A new method is presented in this paper to seek special solutions of nonlinear diffusion equations. This method is based on seeking suitable function to satisfy Bernolli equation. Many new special solutions are obtained.

EXACT SOLUTIONS A COUPLED NONLINEAR REACTION-DIFFUSION SYSTEM

In this paper we shall present, certain type of exact solutions for a coupled partial differential equation by using hyperbolic tangent method.

A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS

A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.

SINGULAR PERTURBATION FOR REACTION DIFFUSION EQUATIONS

The singularly perturbed initial boundary value problems for reaction diffusion equations are considered.Under suitable conditions and by using the theory of differential inequality,the asymptotic behavior of solution for initial boundary value problems are studied,where the reduced problems possess two intersecting solutions.

Optical solitons supported by finite waveguide lattices with diffusive nonlocal nonlinearity

We investigate the properties of fundamental,multi-peak,and multi-peaked twisted solitons in three types of finite waveguide lattices imprinted in photorefractive media with asymmetrical diffusion nonlinearity.Two opposite soliton selfbending signals are considered for different families of solitons.Power thresholdless fundamental and multi-peaked solitons are stable in the low power region.The existence domain of two-peaked twisted solitons can be changed by the soliton self-bending signals.When solitons tend to self-bend toward the waveguide lattice,stable two-peaked twisted solitons can be found in a larger region in the middle of their existence region.Three-peaked twisted solitons are stable in the lower(upper)cutoff region for a shallow(deep)lattice depth.Our results provide an effective guidance for revealing the soliton characteristics supported by a finite waveguide lattice with diffusive nonlocal nonlinearity.

The Approximate Analytical Solution of Non-Linear Equation for Simultaneous Internal Mass and Heat Diffusion Effects

For the first time a mathematical modelling of porous catalyst particles subject to both internal mass concentration gradients as well as temperature gradients, in endothermic or exothermic reactions has been reported. This model contains a non-linear mass balance equation which is related to rate expression. This paper presents an approximate analytical method (Modified Adomian decomposition method) to solve the non-linear differential equations for chemical kinetics with diffusion effects. A simple and closed form of expressions pertaining to substrate concentration and utilization factor is presented for all value of diffusion parameters. These analytical results are compared with numerical results and found to be in good agreement.

Non-simultaneous quenching for a slow diffusion system coupled at the boundary

This paper deals with the quenching behavior of positive solutions to the Newton filtration equations coupled with boundary singularities.We determine quenching rates for non-simultaneous quenching at first,and then establish the criteria to identify the simultaneous and non-simultaneous quenching in terms of the parameters involved.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Height

The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.

Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production

In this paper,we consider the quasilinear chemotaxis system of parabolicelliptic type■.under homogeneous Neumann boundary conditions in a smooth bounded domain■n≥1.The nonlinear diffusivity D(ζ)and chemosensitivity f(ζ)as well as nonlinear signal production g(ζ)are supposed to extend the prototypes■We proved that if m+k+l>1+2/n,then there exists nonnegative radially symmetric initial data uo such that the corresponding solutions blow up in finite time.However,the system admits a global bounded classical solution for arbitrary initial datum when m+k+l<1+2/n.

Solvability for Nonlinear Reaction Diffusion Equation with Boundary Perturbation

In this paper, the nonlinear reaction diffusion equation with boundary perturbation is considered. Using discussions on solvability, the perturbed solution of original problem is obtained, and the uniform validity of the solution is proved.

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equa- tions arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion （the equivalence of the nonlinear stability, the linear stability and the spectral sta- bility of the standing pulse solutions） and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equa- tions. The Evans functions for the standing pulse solutions are constructed explicitly.

Image Magnification Method Using Joint Diffusion

In this paper a new algorithm for image magnification is presented. Because linear magnification/interpolation techniques diminish the contrast and produce sawtooth effects, in recent years, many nonlinear interpolation methods, especially nonlinear diffusion based approaches, have been proposed to solve these problems. Two recently proposed techniques for interpolation by diffusion, forward and backward diffusion (FAB) and level-set reconstruction (LSR), cannot enhance the contrast and smooth edges simultaneously. In this article, a novel Partial Differential Equations (PDE) based approach is presented. The contributions of the paper include: firstly, a unified form of diffusion joining FAB and LSR is constructed to have all of their virtues; secondly, to eliminate artifacts of the joint diffusion, soft constraint takes the place of hard constraint presented by LSR; thirdly, the determination of joint coefficients, criterion for stopping time and color image processing are also discussed. The results demonstrate that the method is visually and quantitatively better than Bicubic, FAB and LSR.