Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

ANALYSIS OF BOUNDARY LAYER SINGULARITYIN A NONLINEAR DIFFUSION PROBLEM

In this papcr, the author analyzes the singularity of a boundary layer in anonlinear diffusion problem. Results show when the limiting solution satisfies the boundarycondition, there is no boundary singularity. Otherwise, the boundary layer ecists, and itsthickness is proportional to ,ε here ε is a small positive real parameter.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

MULTIPLE POSITIVE SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR SUPERLINEAR SECOND ORDER ODES

This paper deals with the existence of positive solutions to the singular boundary value problemwhere q(t) may be singular at t = 0 and t = 1, f(t,y) may be superlinear at y =∞ and singular, at y = 0.

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER SEMILINEAR ELLIPTIC EQUATIONS

In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.

RADIAL CONVEX SOLUTIONS OF A SINGULAR DIRICHLET PROBLEM WITH THE MEAN CURVATURE OPERATOR IN MINKOWSKI SPACE

In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.

The Shock Solution for a Class of Quasilinear Singularly Perturbed Boundary Value Problems

The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM FOR QUASILINEAR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS

The singularly perturbed boundary value problem for quasilinear third_order ordinary differential equation involving two small parameters has been considered. For the three cases ε/μ 2→0(μ→0), μ 2/ε→0(ε→0) and ε=μ 2, the formal asymptotic solutions are constructed by the method of two steps expansions and the existences of solution are proved by using the differential inequality method. In addition, the uniformly valid estimations of the remainder term are given as well.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second_order nonlinear boundary value problems u″+g(t)f(u)=0, 0<t<1, αu(0)-βu′(0)=0, γu(1)+ δu′(1)=0under the conditions0≤f + 0<M 1, m 1<f - ∞≤∞ or0≤f+ ∞<M 1, m 1<f- 0≤∞, where f+ 0= lim u→0 f(u)/u, f - ∞= lim u→∞ f(u)/u, f - 0= lim u→0 f(u)/u, f + ∞= lim u→∞ f(u)/u, g may be singular at t=0 and/or t=1 The proof uses a fixed point theorem in cone theory.

A KIND OF NONLOCAL PROBLEMS FOR SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS

The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, usillg the comparison theorem the asymptotic behavior of solution for the initial boundary value problems are studied.

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMILINEAR ELLIPTIC EQUATION

The singularly perturbed boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem, the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations(ODEs) are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Solvability of Inverse Eigenvalue Problem for Dense Singular Symmetric Matrices

Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.

MULTIPLE POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

This paper deals with the singular nonlinear third-order periodic boundary value problem u′″ + ρau = f(t,u), 0 ≤ t ≤ 2π, with u(i)(0) = u(i)(2π), i = 0, 1, 2, where ρ∈ (0, 1/√3) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2π] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.

A Class of Strongly Nonlinear Singular Perturbed Boundary Value Problems

In this paper,a class of strongly nonlinear singular perturbed boundary value problems are considered by the theory of differential inequalities and the correction of boundary layer,under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.

A CLASS OF SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

We consider the growth rate and quenching rate of the following problem with singular nonlinearity ut = △u-v-λ,vt = △v-u-μ,(x,t) ∈ Rn×(0,∞),u(x,0) = u0(x),v(x,0) = v0(x),x ∈ Rn for any n ≥ 1,where λ,μ > 0 are constants.More precisely,for any u0(x),v0(x) satisfying A11(1 + |x|2)α11≤ u0 ≤ A12(1 + |x|2)α12,A21(1 + |x|2)α21≤ v0 ≤ A22(1 + |x|2)α22for some constants α12 ≥α11,α22 ≥α21,A12 ≥ A11,A22 ≥ A21,the global solution(u,v) exists and satisfies A11(1 + |x|2+ b1 t)α11≤ u ≤ A12(1 + |x|2+ b2 t)α12,A21(1 + |x|2+ b1t)α21≤ v ≤ A22(1 + |x|2+ b2t)α22for some positive constants b1,b2(see Theorem 3.3 for the parameters A ij,α ij,b i,i,j = 1,2).When(1-λ)(1-λμ) > 0,(1-λ)(1-λμ) > 0 and 0 < u0 ≤ A1(b1 T + |x|2)(1-λ)/(1-λμ),0 < v0 ≤ A2(b2 T + |x|2)(1-μ)/(1-λμ)in Rnfor some constants A i,b i(i = 1,2) satisfying A λ 2 > 2nA1(1-λ)/(1-λμ),A-μ1 > 2nA2(1-μ)/(1-λμ) and 0 < b1 ≤(1-λμ)A λ 2(1-λ)2nA1(1 λ)A1,0 < b2 ≤(1 λμ)A μ 1(1 μ)2nA2(1 μ)A2,we prove that u(x,t) ≤ A1(b1(T t) + |x|2)1 λ 1 λμ,v(x,t) ≤ A2(b2(T t) + |x|2)1 μ 1 λμin Rn×(0,T).Hence,the solution(u,v) quenches at the origin x = 0 at the same time T(see Theorem 4.3).We also find various other conditions for the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.

MULTIPLE POSITIVE SOLUTIONS FOR SUPERLINEAR SECOND ORDER SINGULAR BOUNDARY VALUE PROBLEMS WITH DERIVATIVE DEPENDENCE

The existence of at least two positive solutions is presented for the singular second-order boundary value problem 1/ p(t)( p(t)x′(t))′+Φ(t)f(t,x(t),p(t)x′(t))=0,0<t<1, lim t→0 p(t)x′(t)=0,x(1)=0, by using the fixed point index,where f may be singular at x=0 and px′=0.