# A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem

Authors: Musa Çakır; Gabil M. Amiraliyev
Journal: Journal of Applied Mathematics
 (2010)

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## Abstract

The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameterε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.


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## Body

## 1. Introduction

This paper is concerned withε-uniform numerical method for the singularly perturbed semilinear boundary-value problem (BVP):
(1.1)Lu:=ε2u′′+εa(x)u′-f(x,u)=0,0<x<ℓ,(1.2)L0u:=-εu′(0)+ψ(u(0))=0,(1.3)u(ℓ)-φ(u(ℓ1))=0,0<ℓ1<ℓ,
where ε is a small positive parameter, the functions a(x)≥0,f(x,u), and ψ(u),φ(u) are sufficiently smooth on [0,ℓ],[0,ℓ]×ℝ, and ℝ, respectively, and furthermore
(1.4)0<β≤∂f∂u≤β*<∞,dψdu≥δ>0,|dφdu|≤κ<1.
The solution u generally has boundary layers near x=0 and x=ℓ.Singularly perturbed differential equations are characterized by the presence of a small parameterε multiplying the highest-order derivatives. Such problems arise in many areas of applied mathematics. Among these are the Navier-Stokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, control theory, reaction-diffusion processes, quantum mechanics, and electrical networks. The solutions of singularly perturbed differential equations typically have steep gradients, in thin regions of the domain, whose magnitude depends inversely on some positive power of ε. Such regions are called either interior or boundary layers, depending on whether their location is in the interior or at the boundary of the domain. An overview of some existence and uniqueness results and applications of singularly perturbed equations can be found in [1–6].It is known that these problems depend on a small positive parameterε in such away that the solution exhibits a multiscale character; that is, there are thin transition layers where the solution varies rapidly, while away from layers it behaves regularly and varies slowly. The treatment of singularly perturbed problems presents severe difficulties that have to be addressed to ensure accurate numerical solutions. Therefore it is important to develop suitable numerical methods for solving these problems, whose accuracy does not depend on the value of parameter ε, that is, methods that are convergent ε-uniformly. These include fitted finite-difference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special piecewise uniform grids which condense in the boundary layers in a special manner. One of the simplest ways to derive parameter-uniform methods consists of using a class of special piecewise uniform meshes, such as Shishkin meshes (see [3, 6–15] for the motivation for this type of mesh), which are constructed a priori and depend on the parameter ε, the problem data, and the number of corresponding mesh points. The various approaches to numerical solution of differential equations with stepwise continuous solutions can be found in [2, 3, 6].There is also an increasing interest in the application of Shishkin meshes to singularly perturbed convection-diffusion problems (see [16, 17] and references cited therein). However, much of the Shishkin mesh literature is concerned with linear or quasilinear singularly perturbed two-point problems with first-order reduced equations. In [18] has been obtained a result ε-uniform for the two-point boundary value problem of (1.1), by using a fitted operator method on uniform meshes.In the present paper, we analyse a fitted finite-difference scheme on a Shishkin type mesh for the numerical solution of the semilinear nonlocal boundary value problem (1.1)–(1.3). The origin of the fitted finite difference method can be traced to [19]; for subsequent work on fitted operator method and its applications, see [2, 3]. Nonlocal boundary value problems have also been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of nonlocal problems see [20–26] and the references cited therein. Some approaches to approximating this type of problem have also been considered [20, 21, 26–28]. However, the algorithms developed in the papers cited above are mainly concerned with regular cases (i.e., when boundary layers are absent). In [27] has been studied the fitted difference schemes on an equidistant mesh for the numerical solution of the linear three-point reaction-diffusion problem.The numerical method presented here comprises a fitted difference scheme on a piecewise uniform mesh. We have derived this approach on the basis of the method of integral identities using interpolating quadrature rules with the weight and remainder terms in integral form. This results in a local truncation error containing only first-order derivatives of exact solution and hence facilitates examination of the convergence. A summary of paper is as follows. Section2 contains results concerning the exact solutions of problem (1.1)–(1.3). In Section 3, we describe the finite-difference discretization and construct a piecewise uniform mesh, which is fitted to the boundary layers. In Section 4, we present the error analysis for the approximate solution. Uniform convergence is proved in the discrete maximum norm. In the following section numerical results are presented, which are in agreement with the theoretical results. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [18, 27–29].


## 2. Continuous Solution

In this section, we give uniform bounds for the solution of the BVP (1.1)–(1.3), which will be used to analyze properties of appropriate difference problem.Lemma 2.1.
Leta,f∈C1[0,ℓ]. Then the solution u(x) of problem (1.1)–(1.3) satisfies the inequalities
(2.1)∥u∥∞≤C0,
where
(2.2)C0=(1-k)-1{|B|+k(δ-1|A|+β-1∥F∥∞)}F(x)=f(x,0),A=-ψ(0),B=φ(0),∥u∥∞=max[0,ℓ]|u(x)|,(2.3)|u′(x)|≤C{1+1ε(e-μ1x/ε+e-μ2(ℓ-x)/ε)},0≤x≤ℓ,
with
(2.4)μ1=12(a2(0)+4β+a(0)),μ2=12(a2(ℓ)+4β-a(ℓ)),
providing that (∂f/∂x)(x,u) is bounded for x∈[0,ℓ] and |u|≤C0.Proof.
We rewrite the problem (1.1)–(1.3) as
(2.5)Lu≡ε2u′′(x)+εa(x)u′-b(x)u(x)=F(x),0<x<ℓ,(2.6)L0u≡-εu′(0)+ρu(0)=A,(2.7)u(ℓ)-γu(ℓ1)=B,
where
(2.8)b(x)=∂f∂u(x,ξu),0<ξ<1,ρ=dψdu(η1u(0)),0<η1<1,γ=dφdu(η2u(ℓ1)),0<η2<1.
Here we use the Maximum Principle:  letLandL0be the differential operators in (2.5)-(2.6) andv∈C2[0,ℓ]. IfL0v≥0,v(ℓ)≥0,andLv≤0for all, thenv(x)≥0for allx<0<ℓ. Then, from (2.5)-(2.6) we have the following inequality:
(2.9)|u(x)|≤δ-1|A|+|u(ℓ)|+β-1∥F∥∞,x∈[0,ℓ].
Next, from boundary condition (2.7), we get
(2.10)|u(ℓ)|≤|B|+k|u(ℓ1)|.
By setting the value x=ℓ1 in the inequality (2.9), we obtain
(2.11)|u(ℓ1)|≤δ-1|A|+|u(ℓ)|+β-1∥F∥∞.
From (2.10) and (2.11), then we have
(2.12)|u(ℓ)|≤(1-k)-1{|B|+k(δ-1|A|+β-1∥F∥∞)},
which along with (2.9) leads to (2.1).
After establishing (2.1) the further part of the proof is almost identical to that of [28].


## 3. Discretization and Mesh

LetωN be any nonuniform mesh on [0,ℓ]:
(3.1)ωN={0<x1<⋯<xN-1<ℓ},
and ω̅N=ωN∪{x0=0,xN=ℓ}. For each i≥1 we set the stepsize hi=xi-xi-1. Before describing our numerical method, we introduce some notation for the mesh functions. For any mesh function g(x) defined on ω̅N we use
(3.2)gi=g(xi),gx̅,i=(gi-gi-1)hi,gx,i=(gi+1-gi)hi+1,gx0,i=(gx̅,i+gx,i)2,gx̂,i=(gi+1-gi)ℏi,ℏi=(hi+hi+1)2,gx̅x̂,i=(gx,i-gx̅,i)ℏ,∥w∥∞≡∥w∥∞,ω̅N:=max0≤i≤N|wi|.The difference scheme we will construct follows from the identity(3.3)χi-1ℏi-1∫0ℓLuφi(x)dx=0,i=1,2,…,N-1
with the basis functions {φi(x)}i=1N-1 having the form
(3.4)φi(x)={φi(1)(x),xi-1<x<xi,φi(2)(x),xi<x<xi+1,0,x∉(xi-1,xi+1),
where φi(1)(x) and φi(2)(x), respectively, are the solutions of the following problems:
(3.5)εφ′′-aiφ′=0,xi-1<x<xi,φ(xi-1)=0,φ(xi)=1,εφ′′-aiφ′=0,xi<x<xi+1,φ(xi)=1,φ(xi+1)=0.
The functions φi(1)(x) and φi(2)(x) can be explicitly expressed as follows:
(3.6)φi(1)(x)=eai(x-xi-1)/ε-1eaihi/ε-1,φi(2)(x)=1-e-ai(xi+1-x)/ε1-e-aihi+1/ε,forai≠0,φi(1)(x)=x-xi-1hi,φi(2)(x)=xi+1-xhi+1,forai=0.
The coefficient χi in (3.3) is given by
(3.7)χi=ℏi-1∫xi-1xi+1φi(x)dx={ℏi-1(hi1-eaihi/ε+hi+11-e-aihi+1/ε),ai≠0,1,ai=0.
Rearranging (3.3) gives
(3.8)-ε2ℏi-1χi-1∫xi-1xi+1φi′(x)u′(x)dx+εaiℏi-1χi-1∫xi-1xi+1φi(x)u′(x)dx-f(xi,ui)+Ri=0,i=1,N-1¯
with
(3.9)Ri=εℏi-1χi-1∫xi-1xi+1[a(x)-a(xi)]u'(x)φi(x)dx-ℏi-1χi-1∫xi-1xi+1dxφi(x)×∫xi-1xi+1ddxf(ξ,u(ξ))K0,i*(x,ξ)dξ,K0,i*(x,ξ)=T0(x-ξ)-T0(xi-ξ),1≤i≤N-1,T0(λ)={1,λ≥0,0,λ<0.
As consistent with [26, 27], we obtain the precise relation:
(3.10)-ε2ℏi-1χi-1∫xi-1xi+1φi′(x)u′(x)dx+εaiℏi-1χi-1∫xi-1xi+1φi(x)u′(x)dx=ε2{χi-1(1+0.5ε-1ℏiai(χ2,i-χ1,i))}ux̅x̂,i+εaiux0,i,
where
(3.11)χ1,i=ℏi-1∫xi-1xiφi(1)(x)dx={ℏi-1(εai+hi1-eaihi/ε),ai≠0,hiℏ-12,ai=0,χ2,i=ℏi-1∫xixi+1φi(2)(x)dx={ℏi-1(hi+11-e-aihi+1/ε-εai),ai≠0,hi+1ℏ-12,ai=0.It then follows from (3.8) that
(3.12)ℓui+Ri≡ε2θiux̅x̂,i+εaiux0,i-f(xi,ui)+Ri=0,i=1,N-1¯,
where
(3.13)θi=χi-1[1+0.5ε-1aiℏi(χ2,i-χ1,i)]
and after a simple calculation
(3.14)θi={aiℏi2ε(hi+1(eaihi/ε-1)+hi(1-e-aihi+1/ε)hi+1(eaihi/ε-1)-hi(1-e-aihi+1/ε)),ai≠0,1,ai=0.To define an approximation for the boundary condition (1.2), we proceed our discretization process by
(3.15)∫0x1Luφ0(x)dx=0,
with
(3.16)φ0(x)={1-e-a0(x1-x)/ε1-e-a0h1/ε,x0<x<x1,a0≠0,x1-xh1,x0<x<x1,a0=0,0,x∉(x0,x1).
Here the function φ0(x) is the solution of the following problem:
(3.17)εφ0′′-a0φ0′=0,x0<x<x1,φ0(x0)=1,φ0(x1)=0.
In the analogous way, as in construction of (3.12), we obtain
(3.18)-εθ0(0)ux,0+ψ(u0)+θ0(1)f(x0,u0)-r(0)=0,
where
(3.19)θ0(0)={a0ρ1-e-a0ρ,a0≠0,1,a0=0,ρ=h1ε,(3.20)θ0(1)={ρ1-e-a0ρ-a0-1,a0≠0,h12ε,a0=0,(3.21)r(0)=∫x0x1[a(x)-a0]u′(x)φ0(x)dx-ε-1∫x0x1dxφ0(x)∫x0x1ddξf(ξ,u(ξ))T0(x-ξ)dξ,ξ∈(x0,x1).Now, it remains to define an approximation for the second boundary condition (1.3). Let xN0 be the mesh point nearest to ℓ1. Then
(3.22)uN-φ(uN0)+r(1)=0,
where
(3.23)r(1)=(u(ℓ1)-u(xN0))φ′(ξ),
being ξ-intermediate point between u(xN0) and u(ℓ1).Based on (3.12), (3.18), and (3.22), we propose the following difference scheme for approximating (1.1) and (1.3):
(3.24)ε2θyx̅x̂+εayx0-f(x,y)=0,x∈ωN,(3.25)-εθ0(0)yx,0+ψ(y0)+θ0(1)f(x0,y0)=0,(3.26)y(ℓ)-φ(yN0)=0,
where θ,θ0(0), and θ0(1) are defined by (3.14), (3.19), and (3.20), respectively.The difference scheme (3.24)–(3.26) in order to be ε-uniform convergent, we will use the Shishkin mesh. For a divisible by 4 positive integer N, we divide each of the intervals [0,σ1] and [ℓ-σ2,ℓ] into N/4 equidistant subintervals and also [σ1,ℓ-σ2] into N/2 equidistant subintervals, where the transition points σ1 and σ2, which separate the fine and coarse portions of the mesh, are obtained by taking
(3.27)σ1=min{ℓ4,μ1-1εlnN},σ2=min{ℓ4,μ2-1εlnN},
where μ1 and μ2 are given in Lemma 2.1. In practice one usually has σi≪ℓ(i=1,2); so the mesh is fine on [0,σ1],[ℓ-σ2,ℓ], and coarse on [σ1,ℓ-σ2]. Hence, if we denote by h(1),h(2), and h(3) the step-size in [0,σ1],[σ1,ℓ-σ2], and [ℓ-σ2,ℓ], respectively, we have
(3.28)h(1)=4σ1N,h(2)=2(ℓ-σ2-σ1)N,h(3)=4σ2N,12(h(1)+h(3))=2ℓN,h(k)≤ℓN-1,k=1,3,ℓN-1≤h(2)<2ℓN-1,
and so
(3.29)ω̅N={xi=ih(1),i=0,1,…,N4;xi=σ1+(i-N4)h(2),i=N4+1,…,3N4;xi=ℓ-σ2+(i-3N4)h(3),i=3N4+1,…,N,h(1)=4σ1N,h(2)=2(ℓ-σ2-σ1)N,h(3)=4σ2N}.
In the rest of the paper we only consider this mesh.We note that on this mesh the coefficientθi which is defined by (3.14) simplifies to
(3.30)θi={aih(1)2εcothaih(1)2ε,for1≤i≤N4-1,aih(2)2εcothaih(2)2ε,forN4+1≤i≤3N4-1,aih(3)2εcothaih(3)2ε,for3N4+1≤i≤N.
For the evaluation of the rest values θN/4 and θ3N/4 we will use the form (3.14).


## 4. Error Analysis

Letz=y-u,x∈ω̅N. Then for the error of the difference scheme (3.24)–(3.26) we get
(4.1)ε2θizx̅x̂,i+εaizx0,i-[f(xi,yi)-f(xi,ui)]=Ri,1≤i≤N-1,(4.2)-εθ0(0)zx,0+[ψ(y0)-ψ(u0)]+θ0(1)[f(x0,y0)-f(x0,u0)]+r(0)=0,(4.3)zN-[φ(yN0)-φ(uN0)]=r(1),
where the truncation errors Ri,r(0), and r(1) are defined by (3.9), (3.21), and (3.23), respectively.Lemma 4.1.
The solutionzi of problem (4.1)–(4.3) satisfies
(4.4)∥z∥∞,ω̅N≤(1-κ)-1((δ+βθ0(1))-1|r(0)|+|r(1)|+β-1∥R∥∞,ωN).Proof.
The problem (4.1)–(4.3) can be rewritten as
(4.5)ε2θizx̅x̂,i+εaizx0,i-b̃izi=Ri,1≤i≤N-1,-εθ0(0)zx,0+(δ̃+b̃θ0(1))z0=-r(0),zN-γ̃zN0=r(1),
where
(4.6)b̃i=∂f∂u(xi,ỹi),δ̃=ψ'(ỹ0),γ̃=φ'(ỹN0),ỹ0,ỹN0,ỹi-intermediate points called for by the mean value theorem.
Since the discrete maximum principle is valid here, we have the proof of (4.4) by analogy with the proof of  Lemma 2.1.Lemma 4.2.
Under the above assumptions of Section1 and Lemma 2.1, for the error functions Ri,r(0), and r(1), the following estimates hold:
(4.7)∥R∥∞,ωN≤CN-1lnN,|r(0)|≤CN-1lnN,|r(1)|≤CN-1lnN.Proof.
From explicit expression (3.9) for Ri, on an arbitrary mesh we have
(4.8)|Ri|≤C{εmax[xi-1,xi+1]|u'(x)|max[xi-1,xi+1]|xi-x|+∫xi-1xi+1|∂f(ξ,u(ξ))∂ξ+∂f∂udu(ξ)dξ|dξ}≤C{hi+hi+1+∫xi-1xi+1(1+|u'(ξ)|)dξ},1≤i≤N.
This inequality together with (2.3) enables us to write
(4.9)|Ri|≤C{hi+hi+1+1ε∫xi-1xi+1(e-μ1x/ε+e-μ2(ℓ-x)/ε)dx},
in which
(4.10)hi={h(1),1≤i≤N4,h(2),N4+1≤i≤3N4,h(3),3N4+1≤i≤N.
We consider first the caseσ1=σ2=ℓ/4, and so ℓ/4<μk-1εlnN,k=1,2, and h(1)=h(2)=h(3)=h=ℓN-1. Hereby, from (4.9) we can write
(4.11)|Ri|≤C{N-1+ε-1h}≤C{N-1+ℓN4μ1-1ℓlnN}=C{N-1+4μ1-1N-1lnN}.
Hence
(4.12)|Ri|≤CN-1lnN,1≤i≤N.
We now consider the caseσ1=μ1-1εlnN and σ2=μ2-1εlnN, and so μk-1εlnN<ℓ/4,k=1,2 and estimate Ri on [0,σ1],[σ1,ℓ-σ2], and [ℓ-σ2,ℓ] separately. In the layer region [0,σ1], the inequality (4.9) reduces to
(4.13)|Ri|≤C(1+ε-1)h(1)≤C(1+ε-1)4μ1-1εlnNN,1≤i≤N4-1.
Hence
(4.14)|Ri|≤CN-1lnN,1≤i≤N4-1.
The same estimate is obtained in the layer region[ℓ-σ2,ℓ] in the similar manner.
It remains to estimateRi for N/4+1≤i≤3N/4-1. In this case we are able to write (4.9) as
(4.15)|Ri|≤C{h(2)+μ1-1(e-μ1xi-1/ε-e-μ1xi+1/ε)+μ2-1(e-μ2(ℓ-xi+1)/ε-e-μ2(ℓ-xi-1)/ε)},N4+1≤i≤3N4-1.
Since xi=μ1-1εlnN+(i-N/4)h(2), it follows that
(4.16)e-μ1xi-1/ε-e-μ1xi+1/ε=1Ne-μ1(i-1-N/4)h(2)/ε(1-e-2μ1h(2)/ε)<N-1.
Also, if we rewrite the mesh points in the form xi=ℓ-σ2-(3N/4-i)h(2), evidently
(4.17)e-μ2(ℓ-xi+1)/ε-e-μ2(ℓ-xi-1)/ε=1Ne-μ2(3N/4-i-1)h(2)/ε(1-e-2μ2h(2)/ε)<N-1.
The last two inequalities together with (4.15) give the bound
(4.18)|Ri|≤CN-1.
It remains to estimateRi for the mesh points xN/4 and x3N/4. For the mesh point xN/4, inequality (4.9) reduces to
(4.19)|RN/4|≤C{(1+ε-1)h(1)+h(2)+1ε∫xN/4xN/4+1(e-μ1x/ε+e-μ2(ℓ-x)/ε)dx}.
Since
(4.20)e-μ1xN/4/ε-e-μ1xN/4+1/ε=1N(1-e-μ1h(2)/ε)<N-1,e-μ2(ℓ-xN/4+1)/ε-e-μ2(ℓ-xN/4)/ε=1Ne-μ2h(1)/ε(1-e-μ2h(1)/ε)<N-1,
it then follows that
(4.21)|RN/4|≤CN-1lnN.
The same estimate is obtained fori=3N/4 in the similar manner.
The same estimate is valid when only one of the valuesσ1 and σ2 is equal to ℓ/4.
Next, we estimate the remainder termr(0). From the explicit expression (3.21), taking into consideration that  (δ+βθ0(1))-1≤δ-1 and |φ0(x)|≤1, we obtain
(4.22)|r(0)|=1δ{h1∫x0x1|u′(x)|dx+h1ε∫x0x1|∂f∂x+∂f∂uu′(x)|dx}≤C{h12ε+(h1+h1ε)∫x0x1|u′(x)|dx}≤C{h12ε+(h1+h1ε)∫x0x1(1+1εe-μ1x/ε)dx}≤C{h12+εh12+h1ε}.
Hence
(4.23)|r(0)|≤CN-1lnN.
Finally, we estimate the remainder termr(1). From the expression (3.23) we obtain
(4.24)|r(1)|≤|φ′(ξ)|∫xN0ℓ1|u′(τ)|dτ≤C{ℓ1-xN0+1ε∫xN0ℓ1(e-μ1τ/ε+e-μ2(ℓ-τ)/ε)dτ}≤C{N-1+1ε∫xN0xN0+1(e-μ1τ/ε+e-μ2(ℓ-τ)/ε)dτ},
where we have assumed that xN0is on the left-hand side of ℓ1 (if xN0 is on right side of ℓ1, the integral will be over (xN0-1,xN0)). In the same manner as above we therefore obtain from here that
(4.25)|r(1)|≤CN-1lnN.
Thus Lemma4.2 is proved.Combining the two previous lemmas gives us the following convergence result.Theorem 4.3.
Letu(x) be the solution of (1.1)–(1.3) and y the solution (3.24)–(3.26). Then
(4.26)∥y-u∥∞,ϖN≤CN-1lnN.


## 5. Numerical Results

In this section, we present some numerical results which illustrate the present method.Example 5.1.
Consider the test problem:(5.1)a(x)=1+x,f(x,u)=u+tan-1(x+u),0<x<1,ψ(u)=sinu+2u,φ(u)=1+cosπu4,ℓ1=12.
The exact solution of our test problem is unknown. Therefore, we use a double-mesh method [2] to estimate the errors and compute the experimental rates of convergence in our computed solutions; that is, we compare the computed solution with the solution on a mesh that is twice as fine (for details see [13, 28]). The error estimates obtained in this way are denoted by
(5.2)eεN=maxi|yiε,N-ỹiε,2N|,
where ỹiε,2N is the approximate solution of the respective method on the mesh
(5.3)ω̃2N={xi/2:i=0,1,…,2N}
with
(5.4)xi+1/2=xi+xi+12fori=0,1,…,N-1.
The convergence rates are
(5.5)pεN=ln(eεN/eε2N)ln2.
Approximations to the ε-uniform rates of convergence are estimated from
(5.6)eN=maxεeεN.
The corresponding ε-uniform convergence rates are computed using the formula
(5.7)pN=ln(eN/e2N)ln2.
To solve the nonlinear problem (3.24)–(3.26) we use the following iteration technique:
(5.8)ε2θiyx̅x̂,i(n)+εaiyx,i0(n)-f(xi,yi)-∂f∂y(xi,yi(n-1))(yi(n)-yi(n-1))=0,1≤i≤N,(5.9)-εθ0(0)yx,0(n)+ψ(y0(n-1))+∂ψ∂y(y0(n-1))(y0(n)-y0(n-1))+θ0(1)(f(x0,y0(n-1))+∂f∂y(x0,y0(n-1))(y0(n)-y0(n-1)))=0,(5.10)yN(n)=φ(yN0(n-1))+∂φ∂y(yN0(n-1))(yN0(n)-yN0(n-1)),n=1,2,…;yi(0)given1≤i≤N.
To solve (5.8)–(5.10), we take the initial approximation as yi(0)=xi2 and the stopping criterion is
(5.11)maxi|yi(n)-yi(n-1)|≤10-5.
The computed maximum pointwise errorseεN and eε2N, and the orders of uniform convergence  pεN for different values of ε and N, based on the double-mesh principle are presented in Tables 1 and 2. The results established here are that the discrete solution is uniformly convergent with respect to the perturbation parameter  pN and the errors are uniformly convergent with rates of almost unity as predicted by our theoretical analysis.Table 1
For the case ofa(x)≠0, approximate errors  eεN and eN and the computed orders of convergence  pεN on the piecewise uniform mesh  ωN for various values of ε and N.
εN=8N=16N=32N=64N=128N=25610-20.022332200.013506080.009542560.004283370.003100380.001765330.770.820.860.930.980.9910-40.022332150.013768350.008506150.005382130.003154460.001605200.750.800.850.920.970.9910-60.022332100.013760650.008514540.005402030.003143560.001605440.750.800.850.920.970.9910-80.022332050.013760740.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-100.022332090.013760650.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-120.022332090.013760650.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-140.022332090.013760650.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-160.022332130.013760740.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-180.022332130.013760740.008514050.005402030.003143560.001605220.750.800.850.920.970.9910-200.022332130.013760740.008514050.005402030.003143560.001605220.750.800.850.920.970.99eN0.022332200.013768350.009542560.005402030.003154460.00176533pN0.750.800.850.920.970.99Table 2
For the casea(0)=0 of approximate errors eεN and eN and the computed orders of convergence  pεN on the piecewise uniform mesh ωN for various values of ε and N.
εN=8N=16N=32N=64N=128N=25610-20.024532250.017211100.011230850.006524850.004396740.001895430.680.790.830.880.960.9910-40.024532200.017165850.011209730.006511940.004396860.001885150.670.760.800.850.940.9910-60.024532210.017065820.011204030.006510890.0043 92250.001885230.670.760.800.850.940.9910-80.024532150.017065530.011202820.006510460.004392140.001885170.670.760.800.850.940.9910-100.024532150.017065930.011202750.006510930.004392140.001885170.670.760.800.850.940.9910-120.024532160.017064520.011202700.006510250.004392140.001885170.670.760.800.850.940.9910-140.024532090.017064250.011202560.006510250.004392140.001885170.670.760.800.850.940.9910-160.024532090.017064250.011202560.006510250.004392140.001885170.670.760.800.850.940.9910-180.024532090.017064250.011202560.006510250.004392140.001885170.670.760.800.850.940.9910-200.024532090.017064250.011202560.006510250.004392140.001885170.670.760.800.850.940.99eN0.024532250.017165850.011230850.006524850.004396860.00189543pN0.680.790.830.880.960.99Example 5.2.
Consider the test problem:(5.12)a(x)=sin(πx2),f(x,u)=1+x2+u+tanhu,0<x<1,ψ(u)=u-1,φ(u)=12u+2,ℓ1=12.
The exact solution of our test problem is unknown. In the same manner as above we solve this problem.



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*Source: 495184-2010-06-21.xml*