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.

Existence Results of Third Order Multi-Point Boundary Value Problem at Resonance

A kind of third order multi-point boundary value problems, x＇＇＇（ ι） = f（ t, x （ t ）, x＂ （ t ）, x＇＇＇ （ t ） ） ＋ m 2 e（t）,t∈（0, 1）,x（0）=ax（ξ）,x＇（0）-0,x（l）= ^m2∑j=1 βjx（ηj）, fεC[0, 1]×R^3, e（t）∈L^1[0, 1],a≥0, is considered, all theβj＇s have not the same sign, 0〈ξ〈 l, 0〈η1〈 η2〈… 〈ηm.2〈 1. By using the coincidence degree theory, some existence theorems for the problems at resonance are obtained.

SOLUTIONS TO A SECOND-ORDER MULTI-POINT BOUNDARY VALUE PROBLEM AT RESONANCE

This article deals with the following second-order multi-point boundary value problem x″（t）=r（t,x（t）,x′（t））＋e（t）,t∈（0,1）x′（0）=m∑i=1aix′（ξi）,x（1）=n∑j=1βjx（ηj）, Under the resonance conditions m∑i=1ai=1,n∑j=1βj=1,n∑j=1βjηj=1 , by applying the coincidence degree theory, some existence results of the problem are established. The emphasis here is that the dimension of the linear operator is two. In this paper, we extend and improve some previously known results like the ones in the references.

Multi-point Boundary Value Problems for Nonlinear Fourth-order Differential Equations with All Order Derivatives

By using fixed point theorem, multiple positive solutions for some fourth- order multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained. The associated Green＇s functions are also given.

Analytic decomposition and numerical procedure for solving the singular boundary value problem arising in viscous flows

An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.

Existence of Two Solutions of Nonlinear m-Point Boundary Value Problems

Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.

Solvability of 2n-order m-point boundary value problem at resonance

The existence of solutions at resonance is obtained by using the an example to demonstrate our result. noncontinuous. for the 2n-order m-point boundary value problem coincidence degree theory of Mawhin. We give The interest is that the nonlinear term may be

POSITIVE SOLUTIONS TO AN m-POINT BOUNDARY VALUE PROBLEM

The existence of positive solutions for second order m-point boundary value problemx″-q(t)f(x,x′)x′=0, x(0)= m i=2 b ix(ξ i),x′(1)=αx′(0)are investigated,where ξ i,b i and α are constants satisfying 0=ξ 1<ξ 2<...<ξ m-1 <ξ m=1,b i≥0 for i=2,...,m with β∶= m i=2 b i∈[0,1), and α>1. Our approach is based on the fixed point theorem in cones.

Solvability of Third-order Three-point Boundary Value Problems with Caratheodory Nonlinearity

A class of third-order three-point boundary value problems is considered, where the nonlinear term is a Caratheodory function. By introducing a height function and considering the imtegration of this height function, an existence theorem of solution is proved when the limit growth function exists. The main tools are the Lebesgue dominated convergence theorem and the Schauder fixed point theorem.

A Class of Singularly Perturbed Boundary Value Problems Arising from Catalytic Reactions

A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.

Numerical Solution of Singularly Perturbed Two-Point Boundary Value Problem via Liouville-Green Transform

In this paper, authors describe a Liouville-Green transform to solve a singularly perturbed two-point boundary value problem with right end boundary layer in the interval [0, 1]. They reply Liouville-Green transform into original given problem and finds the numerical solution. Then they implemented this method on two linear examples with right end boundary layer which nicely approximate the exact solution.

MULTIPLE BOUNDARY LAYER PHENOMENAOF SOLUTION OF BOUNDARY VALUE PROBLEM FOR THIRD ORDER DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper, we study the multiple boundary layer phenomena of solution ofboundary value problem for third order semilinear ordinary differential equationwith two small parameters. Under suitable conditions, using the method of differential inequalities, we prove the existence of solution and give its estimation of remmainder term.

Singular Nonlinear Boundary Value Problems Arising in the Boundary Layer Behind Expansion Wave

A class of singular nonlinear boundary value problems arising in the boundary layer behind expansion wave are studied.Sufficient conditions for the existence and uniqueness of positive solutions to the problems are established by utilizing the monotonic approaching tech- nique.And a theoretical estimate formula for skin friction coefficient is presented.The numerical solution is presented by using the shoot method.The reliability and efficiency of the theoretical prediction are verified by numerical results.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS, III

In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions

(k, n-k) CONJUGATE MULTI-POINT BOUNDARY VALUE PROBLEMS FOR A CLASS OF HIGHER-ORDER DIFFERENTIAL INCLUSIONS

In this paper a fixed point theorem for contracting maps is used to investigate the existence of solutions to a class of higher-order differential inclusions with (k, n-k) conjugate multi-point boundary value problem.

EXISTENCE OF TRIPLE POSITIVE SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM

We apply a fixed point theorem to verify the existence of at least three positive solutions to a multi-point boundary value problem with p-Laplacian. Existence criteria which ensure the existence of triple positive solutions are established.

SOLVABILITY FOR SECOND-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS ON A HALF-LINE

In this paper, we are concerned with the existence of a second-order differential equation with m-point boundary conditions on the half line. The existence results are proved by the Leray-Scha¨uder point theorem in a special Banach space. Green function plays an important role in the proof and its discussion is very interesting. The results obtained generalize those in the previous references.

A THEOREM OF TRIPLE POSITIVE SOLUTIONS FOR MULTI-POINT BOUNDARY VALUE PROBLEMS

In this paper we prove a new fixed point theorem in cones and then obtain the existence of triple positive solutions for a class of multi-point boundary value problem.