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.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

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.

THE ASYMPTOTIC EXPANSIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

In this paper we study the singularly penurbed boundary value problem: where e is a positive small parameter In the conditions: we prove the existences, and uniformly valid asymptotic expansions of solutions for the given boundary value problems, and hence we improve the existing results.

B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems

We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.

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 PEOBLEMS WITH PARABOLIC LAYERS,I

In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe 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 differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. 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 Dirichlet, Neumann and Robin 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

Quasilinear singularly perturbed problem with boundary perturbation

A class of quasilinear singularly perturbed problems with boundary perturbation is considered. Under suitable conditions, using theory of differential inequalities we studied the asymptotic behavior of the solution for the boundary value problem.

A CLASS OF NONLINEAR SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.

Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Two Parameters and Boundary Perturbation

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with two parameters and boundary perturbation were considered.Under suitable conditions,the existence,uniqueness and asymptotic behavior of solutions for the initial boundary value problems were studied.An example was also given to illustrate our main results.

The Nonlinear Singularly Perturbed Problems for Elliptic Equations with Boundary Perturbation

The nonlinear singularly perturbed problems for elliptic equations with boundary perturbation are considered. Under suitable conditions, by using the theory of differential inequalities the asymptotic behavior of solutions for the boundary value problems is studied.

BOUNDARY AND ANGULAR LAYER BEHAVIOR IN SINGULARLY PERTURBED SEMILINEAR SYSTEMS

Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ?-n+ , of the solutions of scalar boundary value problemsIn this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM OF SYSTEMS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUAT1ONS

In this paper,we study the singular perturbation of boundary value problem of systems for quasilinear ordinary differential equations:x'=j(i,x,y,ε),εy'=g(t,x,y,ε)y'+h(t,x,y,ε),y(0,ε)=A(ε),y(0,ε)=B(ε),y(1,ε)=C(ε)where xf.y,h,A,B and C belong to Rn and a is a diagonal matrix.Under the appropriate assumptions,using the technique of diagonalization and the theory of differential inequalities we obtain the existence of solution and its componentwise uniformly valid asymptotic estimation.

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

Distributed Capacitance of Coaxial Line With Perturbed Walls

Perturbation method of boundary geometry(PMOBG) used in Lapiacian problems is dealt with and the three--term perturbation expression of distributed capacitance of a coaxial line with perturbed walls is obtained. As an example,four-order expression of distributed capacitance of a elliptic coaxial line with small eccentricity is given.

SINGULAR PERTURBATION OF SECOND-ORDER NONLINEAR SYSTEM WITH BOUNDARY PERTURBATION

The singular perturbation of boundary value problem of second-order nonlinear system of differential equations with integral operators and boundary perturbation is discussed. Under the suitable assumed conditions, by the technique of diagonalization, the existence of the solutions is proved and its remainder term is estimated.

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM FOR QUASILINEAR HIGHER ORDINARY DIFFERENTIAL EQUATION

In this paper we study the singular perturbation of boundary value problems with perturbations both in the operator and in the interval ends. So as to prove the existence and uniqueness of solution of perturbed problem, to establish the asymptotic expression involving three parameters. Thus, the iterative equation of finding the asymptotic solution is derived and the estimation of the remainder term is given out. We extend results of [l]-[5].

Quasilinear Robin Problem with Boundary Perturbation

A class of quasilinear Robin problems with boundary perturbation are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the boundary value problem is studied.