A class of quasilinear equations with-1 powers

This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ^(N)with N≥1.

Singular Perturbation of Two Points Boundary Value Problems for a Class of Third Order Quasilinear Differential Equations

In this paper, the fixed-point Theorem i s used to estimate an asymptotic solution of boundary value problems for a class o f third order quasilinear differential equation and the uniformly valid asymptot ic expansion of solution of any orders including boundary layer is obtained.

Limit Cycles in the Stability Analysis of a Chemical System

Aris and Amundson studied a chemical reactor and obtained the two equationsDaoud showed that at most one limit cycle may exist in the region of interest. Itis showed in this paper that other singular points exist and that a stable limitt cycle existsaround the singularity (1/2, 2) when K∈(9-δ, 9).

A Class of the Singularly Perturbed Boundary Value Problems for Elliptic Equation with a Super Surface of Turning Point in n-dimensional Space

In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theorem. The uniformly valid asymptotic approxmations of solutions for the boundary value problem is constructed.

Singular Solutions of the Cauchy Problem for SemilinearParabolic Equations with Singular Potentions

In this paper we consider the Cauchy problem for the singular semilinear parabolic equation u t-Δu+V 1(x)u=V 2(x)u p,x∈R n\{0},t>0, where V 1(x),V 2(x) may have singularities at the origin. Using functions of the Kato class and the Green tight functions we got the existence of the positive solution being singular at the origin.

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

New Explicit and Exact Travelling Wave Solutions for Burgers-Kolmogorov-Petrovskii-Piscounov Equations

In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.

Self-similar very singular solution of a p-Laplacian equation with gradient absorption: existence and uniqueness

This paper investigates the self-similar singular solution of the p-Laplacian evolution equation with the nonlinear gradient absorption terms u t=div(u p-2u)-uq for 1<p<2 and q>in Rn× (0, ∞). It has been proved that when 1<q<p-n/(n+1) there exists a unique self-similar very singular solution.

Novel Solutions of KdV Equation

Some novel solutions of the KdV equation are obtained through the modified bilinear B?cklund transformation.

The Nonlinear Predator-Prey Singularly Perturbed Robin Initial Boundary Value Problems for Reaction Diffusion System

The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.

Analysis of Singularity for Reducible Quasi-linear Hyperbolic Systems

In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.

A Class of Singularly Perturbed Problems for Nonlinear Elliptic Equations

A class of singularly perturbed problems for the nonlinear elliptic equations is considered. Under suitable conditions, using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems are studied, which reduced equations possess two intersecting solutions.

Singular Perturbation of a Class of Dirichlet Exterior Problems for Elliptic Equations

In this paper, we consider a class of singularly perturbed Dirichlet exterior problems for elliptic equations. Under the appropriate conditions we construct the formally asymptotic solution of the problem described. Using differential inequaltiy theory we prove the existence of the solution of original problem and the uniforly validity of the formal solution.

New Exact Travelling Wave Solutions for Zakharov-Kuznetsov Equation

In this paper, the nonlinear dispersive Zakharov- Kuznetsov equation is solved by using the generalized auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.

Wave Interaction with Dual Circular Porous Plates

In this paper we have investigated the reflection and the transmission of a system of two symmetric circular-arc-shaped thin porous plates submerged in deep water within the context of linear theory. The hypersingular integral equation technique has been used to analyze the problem mathematically. The integral equations are formulated by applying Green＇s integral theorem to the fundamental potential function and the scattered potential function into a suitable fluid region, and then using the boundary condition on the porous plate surface. These are solved approximately using an expansion-cure-collocation method where the behaviour of the potential functions at the tips of the plates have been used. This method ultimately produces a very good numerical approximation for the reflection and the transmission coefficients and hydrodynamic force components. The numerical results are depicted graphically against the wave number for a variety of layouts of the arc. Some results are compared with known results for similar configurations of dual rigid plate systems available in the literature with good agreement.

Integral Representation of the Manifold Solution for New Class of the Volterra Type Integral Equation with a Boundary Singularity in the Case, When Kernel Contain Logarithmic Singularity and its Power

In this work we suggestion new methods investigation the model Volterra type integral equation with logarithmic singularity, kernel which consisting from composition polynomial function with logarithmic singularity and function with singular point. The problem investigation this type integral equation at n = 2m reduce to problem investigate the Volterra type integral equation （1） for n = 2 the theory for which was constructed in [2]. In this work, we investigation integral equation （1） at = 2m ＋ 1 In this case, we investigate integral equation （1） reduction it＇s to m integral equation type [2] φ（x）＋∫xa[p1＋p2 ln（x-a/t-a）]φ（t）/t-a dt=f（x）and one the following integral equation [1] ω（x）＋p3∫xω（t）/ a t-adt=g（x）.

Framework Definitions of Effects and Phenomena and Examples in Differential and Difference Equations

Framework definitions of earlier informally used notions of effect and phenomenon are proposed, examples of their role in developing mathematics are given. Phenomena of singular cycle and deepening boundary layer in the theory of singularly perturbed differential equations and ancient popular synergetic process described by system of random difference equations are presented.

POSITIVE ENTIRE SOLUTIONS TO SINGULAR SEMILINEAR ELLIPTIC EQUATIONS OF MIXED TYPE

In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.