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Journal of Computational and Applied Mathematics 307 (2016) 93–105

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Journal of Computational and AppliedMathematics

journal homepage: www.elsevier.com/locate/cam

A rational spectral collocation method for third-ordersingularly perturbed problemsSuqin Chen a, Yingwei Wang b,∗a Department of Mathematics, Tongji University, Shanghai 200092, Chinab Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:Received 16 July 2015Received in revised form 22 January 2016

Keywords:Third-order singularly perturbed problemsRational spectral collocation methodBoundary layerAsymptotic expansion

a b s t r a c t

Anewnumericalmethod is developed for solving a class of third-order singularly perturbedboundary value problems. First of all, the given problem is transformed into a systemof twoordinary differential equations (ODEs) subject to suitable initial and boundary conditions.Then, the rational spectral collocationmethod in barycentric formwith sinh transformationis applied to solve the system of ODEs. According to the asymptotic analysis, the locationand width of boundary layer of the given problem, which are chosen as parameters in thesinh transformation, can be determined. Ample numerical experiments are presented toillustrate the computational efficiency and accuracy of the our method.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

Differential equations with a small parameter ε multiplying the highest-order derivative terms are said to be singularlyperturbed, which occur frequently in the mathematical models describing the phenomena with a rapid transition of theobservable quantity. In this paper, a class of third-order singularly perturbed boundary value problems (TSPBVP) is consideredas follows:

εu′′′(x) + a(x)u′′(x) + b(x)u′(x) + c(x)u(x) = f (x), x ∈ (0, 1), (1.1)

u(0) = p, u′(0) = q, u′(1) = r, (1.2)

where ε ∈ (0, 1] is a given small positive parameter, a(x), b(x), c(x) and f (x) are sufficiently continuously differentiablefunctions satisfying the conditions a(x) > α > 0, b(x) 6 −β, β > 0. Under these assumptions, the problem (1.1)–(1.2) hasa unique solution u = u(x, ε), in general, displaying a boundary layer of width O(ε) near x = 0 as ε tends to zero. Besides,since the boundary conditions are Neumann type [1], the boundary layer is less severe.

The second-order singularly perturbed boundary value problems (SSPBVP) have been studied extensively [2–7] whileonly few results on TSPBVP are reported in the literature. For the discussion of existence, uniqueness and asymptotic esti-mates of the solutions to TSPBVP see [8–13] and the references therein. The numerical analysis of TSPBVPhas always sufferedfrom more difficulties than SSPBVP due to the following facts: (i) the third-order differential operator lacks the symmetryof the second-order one; (ii) direct discretization for high-order differential equations usually leads to a linear system withlarge condition number. In particular, a known drawback on classical spectral collocation methods is the fast growth of thecondition numbers of differentiation matrices. More precisely, the condition number of Chebyshev differentiation matrices

∗ Corresponding author.E-mail addresses: [email protected] (S. Chen), [email protected] (Y. Wang).

http://dx.doi.org/10.1016/j.cam.2016.03.0090377-0427/© 2016 Elsevier B.V. All rights reserved.

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94 S. Chen, Y. Wang / Journal of Computational and Applied Mathematics 307 (2016) 93–105

scales likeO(N2k), where k is the order of differentiation. Hence, classical collocationmethods for third and higher odd-orderboundary value problems lead to very large condition numbers and often exhibit unstable modes if the collocation pointsare not properly chosen (see, for instance, [14–17]).

Recent years have witnessed substantial progress in the development of numerical methods for solving TSPBVP.Valarmathi et al. [18–20] developed an asymptotic numerical method in which the boundary value technique andspecial-purposed finite difference schemeswere employed; Cui et al. [21] solved such problem in reproducing kernel space;Temsah [22] proposed two kinds of collocation methods based on differential Chebyshev polynomial (CM-I) and integratedChebyshev polynomial (CM-II) respectively. In this paper, we present a new kind of numerical method based on rationalspectral collocation in barycentric form with sinh transformation and some results from asymptotic theory.

Spectral methods are among the best choices in solving singularly perturbed problems due to the fact that spectralcollocation points are naturally well-adapted to the representation of boundary layers. For example, for the classicalChebyshev–Gauss–Lobatto points xk = cos(kπ/N), we have |x1 − x0| = |xN − xN−1| = |cos(π/N) − 1| ≈ 12

πN

2≈

5N2

.It follows that the spacing between the points near two endpoints −1 and 1 is of order O(N−2), in contrast with O(N−1) infinite differences or finite elements. However, for the problems like (1.1) with extremely small ε, still a largeN is required toobtain numerical solutionswith high accuracy. In order to capture thin boundary layers accurately by a reasonable number ofN , many experts have put forth a variety of modified spectral methods [23–29]. An effective approach is that by introducinga certain transformation x → xt = t(x), the transformed collocation points xt aremore located in the boundary layer region.

Rational interpolants in barycentric form have recently emerged as promising tools for the development of spectralmethods for singularly perturbedboundary valueproblems [30–33]. Their success is partly due to the fact that theunderlyingproblem is not required to be transformed into new coordinates. A rational spectral collocation method integrated withsinh transformation is applied to solve blow-up and steep-front problems by Tee and Trefethen [34], in which the sinhtransformation is constructed as:

g(x) = λ + µ sinh

sinh−11 − λ

µ

+ sinh−1

1 + λ

µ

x − 12

+ sinh−11 − λ

µ

, (1.3)

which maps the original Chebyshev grid into the one clustered near the steep gradients of the solution (singular lines).However, the parameters λ and µ in (1.3) respectively represent the location and width of the boundary layer, which

are usually not known exactly when applying the sinh transformation. Hence, it is necessary to approximate them usingsome singularity location technique. Tee et al. [34] obtained the parameters λ, µ of the n + 1 time step depended on thenumerical results of the n time step,which is only suitable for time-dependent problems. It is still difficult to determine theseparameters for non time-dependent problems like (1.1)–(1.2). We established a framework of rational spectral collocationmethod based on asymptotic theory for first order parameterized problems [35] and second-order coupled systems [36].In this work, we will apply this methodology to TSPBVP.

In our proposed method, the problem (1.1)–(1.2) is firstly transformed into a system of two ODEs subject to suitableinitial and boundary conditions. Then with the help of asymptotic analysis theory, we can calculate the location and widthof boundary layer for the given problem, which are related to the parameters λ, µ in the sinh transformation (1.3). Finally,the rational spectral collocation method is applied to solve the system of two ODEs. This method is called rational spectralcollocation method combined with asymptotic theory (abbreviated as RSCAT). The advantages of our method are (i) it enjoysimproved exponential convergence rates; (ii) it does not require the tedious transformation of underlying differentialequation into new coordinates; (iii) boundary conditions can be easily incorporated with.

The organization of the paper is the following. The analytic analysis for the solutions of the TSPBVP is presented inSection 2. The algorithmic details of RSCAT method for TSPBVP are proposed in Section 3. In Section 4, we focus on thenon-linear TSPBVP. Section 5 illustrates the performance of the proposed schemes via several examples, which demonstratethe high accuracy and efficiency of our method. Some concluding remarks are given in Section 5.

2. Analytic analysis

2.1. Asymptotic expansion approximation

The TSPBVP (1.1)–(1.2) can be transformed into an equivalent problem including a system of two ODEs as follows:u′1(x) − u2(x) = 0,εu′′2(x) + a(x)u

′

2(x) + b(x)u2(x) + c(x)u1(x) = f (x), x ∈ (0, 1),u1(0) = p, u2(0) = q, u2(1) = r.

(2.1)

The asymptotic expansion solution of ODE system (2.1) is in the form

u(x, ε) = (ũ0 + ṽ0) + ε(ũ1 + ṽ1) + O(ε2).

By the method of stretching variable, the zeroth-order asymptotic expansion can be obtained as

uas = ũ0 + ṽ0,

S. Chen, Y. Wang / Journal of Computational and Applied Mathematics 307 (2016) 93–105 95

where ũ0 is a solution of the reduced problem (2.1) given byu′01(x) − u02(x) = 0,a(x)u′02(x) + b(x)u02(x) + c(x)u01(x) = f (x), x ∈ (0, 1),u01(0) = p, u02(1) = r,

(2.2)

and ṽ0 is the layer correction function given by ṽ0 = (v01, v02)T with

v01 = −ε

a(0)[p − u02(0)] e−

a(0)xε , v02 = [p − u02(0)] e−

a(0)xε . (2.3)

Theorem 2.1 ([20]). The zeroth-order asymptotic expansion approximation uas satisfies the inequality

∥u(x) − uas(x)∥ ≤ Cε,

where u(x) is the solution of ODEs (2.1), and C is a generic constant.

2.2. The location and width of boundary layer

Theorem 2.2. If u(x) is the solution of ODEs (2.1) and ũ0 is a solution of the reduced problem (2.2), there exists an ε0 such thatfor 0 ≤ ε ≤ ε0, the following inequality is satisfiedu(x) − ũ0(x) ≤ C1ε, for x ∈ [βε, 1], (2.4)where β is a constant for a family of small values ε.Proof. According to Theorem 2.1, there exists an ε0 such that for 0 ≤ ε ≤ ε0, the solution u(x) of ODEs (2.1) satisfies thefollowing inequality

∥u(x) − uas(x)∥ ≤ Cε, for x ∈ [0, 1],

where uas = ũ0 + ṽ0. It follows thatu(x) − ũ0(x) = u(x) − uas(x) + uas(x) − ũ0(x)≤ ∥u(x) − uas(x)∥ + ∥ṽ0(x)∥≤ Cε + ∥ṽ0(x)∥ . (2.5)

For the layer correction function ṽ0 = (v01, v02)T defined in (2.3), we can choose a positive constant β , such that β ≥ − ln εa(0)for sufficiently small values ε. For x ∈ [βε, 1], it implies

x ≥ βε ⇒ x ≥ −ln εa(0)

ε ⇒ −a(0)x

ε≤ ln ε ⇒ e−

a(0)xε ≤ eln ε = ε.

Here, β is called the boundary layer parameter.Let D1 = |p − u02(0)|, D2 =

D1|a(0)|ε, then we have

|v02| = D1e−a(0)x

ε ≤ D1ε, for x ∈ [βε, 1],

|v01| =ε

|a(0)|D1e−

a(0)xε ≤

D1|a(0)|

ε2 ≤ D2ε, for x ∈ [βε, 1].

Combining the bounds of (2.5) and taking C1 = C + max(D1,D2) yields the inequality (2.4).

Remark 2.1. The above theorem suggests that the boundary layer region of the solution u(x, ε) to the problem is [0, βε],that is to say, the location of boundary layer is at the left endpoint of the underlying interval [0, 1] and its width is βε, whereβ can be chosen as − ln εa(0) . This result plays an important role in the RSCAT method in next section.

3. Rational spectral collocation method in barycentric form

3.1. The barycentric form of rational interpolation

A rational function pN(x) in barycentric form which interpolates a function u(x) at N + 1 distinct points {xk}Nk=0 can beexpressed as [33]

u(x) ≈ pN(x) =

Nk=0

ωkx−xk

u(xk)

Nk=0

ωkx−xk

, (3.1)


where {ωk}Nk=0 are nonzero numbers called barycentric weights. In particular, for Chebyshev–Gauss–Lobatto points xk =cos(kπ/N), the barycentric weights are chosen as [32]

ω0 =12, ωN =

(−1)N

2, ωk = (−1)k, k = 1, 2, . . . ,N − 1. (3.2)

The derivatives of pN can serve to determine the nth order differentiation matrix {D(n)jk }

Nj,k=0 associated with pN

represented by (3.1) at the point xj:

p(n)N (xj) =N

k=0

dn

dxn

ωk

x−xku(xk)

Nl=0

ωlx−xl

x=xj

=

Nk=0

D(n)jk u(xk), j = 0, . . . ,N.

The advantage of the barycentric representation is the simplicity of the formula for the entries of first and second orderdifferentiation matrices [37]

D(1)jk =

ωk

ωj(xj − xk), j ≠ k,

−

i≠k

D(1)ji , j = k,(3.3)

D(2)jk =

2D(1)jk

D(1)jj −

1xj − xk

, j ≠ k,

−

i≠j

D(2)ji , j = k.(3.4)

The convergence rate of the above rational interpolation based on barycentric form depends primarily on the analyticregion of u in the complex plane. When a real function u can be continued analytically to the closed ellipse Eρ (ellipsewith foci at ±1, the sum of its semi-major axis length and semi-minor axis length equal to ρ), then the approximation error|pN(x) − u(x)| decreases as a rateO(ρ−N) asN → ∞. Such a convergencemeans that a fewdegrees of freedomare sufficientto achieve a high degree of accuracy. Unfortunately, if the function u has some singularities in the complex plane close to[−1, 1] so that ρ ≈ 1, the convergence could be too slow for the method to be effective. As suggested in Ref. [31], one couldchoose a conformal map g such that the ellipse of analyticity of u ◦ g is larger than the original ellipse of analyticity of u, andapply g into the spectral collocation method based on rational interpolant (3.1). Then we can obtain a better approximationof u which is more accurate than that obtained using the Chebyshev spectral method with the same number of gridpoints.

Tee and Trefethen [34] considered the solutions with one relevant front to be resolved, and thus construct the conformalmap g as follows:

g(x) = λ + µ sinh

sinh−11 − λ

µ

+ sinh−1

1 + λ

µ

x − 12

+ sinh−11 − λ

µ

, (3.5)

where sinh−1 is the inverse of sin hyperbolic, λ and µ are the parameters related to the location and width of the boundarylayers, respectively. Once the grid adapted as xk = g (cos (kπ/N)), the rational interpolant of u in barycentric form is usedwith differentiationmatrices given by (3.3)–(3.4). However, it is still a difficultwork to determine analytically the parametersλ andµ. Fortunately, to the class of TSPBVP like (1.1)–(1.2), the asymptotic results of Theorem2.2 give us enough informationabout the location and width of the boundary layer of the solution which are chosen as λ and µ respectively.

Emphasis should be laid on that the differentiation matrices (3.3) and (3.4) merely rely on the barycentric weights ωkgiven in (3.2) and new grid points x̂k = g(xk), where g is shown in (1.3). That is why the transformation of the derivativesin underlying equation into new coordinates is not required after mapping g .

3.2. Numerical discretization of the ODE system (2.1)

Now, we have all the ingredients to solve the ODEs (2.1) using rational spectral collocation method. First, by introducingthe transformation x = 0.5(y + 1) and defining û(y) = u(x) = u (0.5(y + 1)), then u′(x) = 2û′(y), u′′(x) = 4û′′(y), we canrewrite (2.1) as2û

′

1(y) − û2(y) = 0,4εû′′2(y) + 2â(y)û

′

2(y) + b̂(y)û2(y) + ĉ(y)û1(y) = f̂ (y), y ∈ (−1, 1),û1(−1) = p, û2(−1) = q, û2(1) = r.

(3.6)


According to Remark 2.1, the solution of (3.6) has a boundary layer which occurs at y = −1 with the width 2βε. Let thetransformed Chebyshev collocation points be

Y = {yk}Nk=0 = {g (cos (kπ/N))}Nk=0 ,

where g is expressed by (1.3) in which λ = −1, µ = 2βε.Evaluating the equations in (3.6) at points {yk}Nk=0 yields

2D(1)Û1 − Û2 = 0,4εD(2)Û2 + 2diag(â)D(1)Û2 + diag(b̂)Û2 + diag(ĉ)Û1 = f̂ ,

(3.7)

where

Û1 = [u10, . . . , u1N ]T with u1k evaluating u1(yk),

Û2 = [u20, . . . , u2N ]T with u2k evaluating u2(yk),â = [â0, . . . , âN ]T with âk evaluating â(yk),

and similar definitions for b̂, ĉ and f̂ . The diag(â) is a diagonalmatrixwith the elements of â on themain diagonal and similardefinitions are for diag(b̂), diag(ĉ).

By block matrices technique, Eqs. (3.7) can be rewritten as:2D(1) −Ediag(ĉ) 4εD(2) + 2diag(â)D(1) + diag(b̂)

Û1Û2

=

0f̂

. (3.8)

By defining

A =

2D(1) −Ediag(ĉ) 4εD(2) + 2diag(â)D(1) + diag(b̂)

, U =

Û1Û2

, d =

0f̂

,

Eq. (3.8) can be rewritten as

AU = d. (3.9)

The boundary conditions in (3.6) suggest that

Û1(1) = p, Û2(1) = q, Û2(N + 1) = r,

i.e.

U(1) = p, U(N + 2) = q, U(2N + 2) = r.

Let us consider three row vectors with length 2N + 2 : E1 = (1, 0, . . . , 0), EN+2 = (0, . . . , 1, . . . , 0), E2N+2 =(0, . . . , 1). Replacing the 1st, (N + 2)th and (2N + 2)th rows of the matrix A with E1, EN+2 and E2N+2 respectively gives usa new matrix Ã. Correspondingly, the d(1), d(N + 2) and d(2N + 2) should be replaced by p, q and r to get the new righthand side d̃. Finally, the numerical solution of (3.6) can be solved from the following linear system

ÃU = d̃. (3.10)

Remark 3.1. The above numerical method could also be applied to solve the TSPBVP with boundary conditions like u′(0) =p̂, u(1) = q̂, u′(1) = r̂ , which will be illustrated in Example 2 in Section 5.

3.3. Fast solver for the linear system (3.10)

The rational spectral collocation method we have described requires the solution of the linear system (3.10), where thematrix Ã, solution U and right hand side d̃ are respectively

Ã =D1 BC D2

, U =

U1U2

, d̃ =

d1d2

. (3.11)

Here, the sub matrices D1 and D2 are slightly modification of the matrices 2D(1) and 4εD(2) + 2diag(â)D(1) + diag(b̂),respectively; B and C are two diagonal matrices; d1 and d2 are two column vectors.

The linear system (3.10) could be solved efficiently via Schur complement. For instance, in order to solve the functionvalues U1, we merely need to solve the following:

(D1 − BD−12 C)U1 = d1 − BD−12 d2. (3.12)

Note that the main computational cost for solving (3.12) has three parts: (i) solving V1 from D2V1 = C; (ii) solving V2 fromD2V2 = d2; (iii) solving U1 from (D1 − BV1)U1 = d1 − BV2. And the first derivatives U2 can be solved in a similar way.


Here are some remarks on the computational cost.

• In the method proposed here, the original TSPBVP (1.1)–(1.2) has been rewritten into a system of ODEs (2.1), whichobviously increases the complexity of the computation. A natural question is that how much complexity has beenincreased. Recall the standard spectral collocation method for directly solving the original TSPBVP (1.1)–(1.2) leads tothe linear system

D3U1 = f1, (3.13)

where the matrix D3 arises from the discretization of third-order differential operator. Recall that the conditionnumbers of D1,D2 and D3 are roughly O(N2), O(N4) and O(N6). See Fig. 4 for numerical illustration. It follows thatthe computational cost for solving (3.12) is at most three times over the one for solving (3.13).

• It is well known that the differential matrices D(1) and D(2) defined in (3.3) and (3.4) are totally dense, and so do thematrices D1,D2 in (3.11) and D3 in (3.13). Recently, we found that these matrices enjoy a low-rank property, i.e., theiroff-diagonal blocks have small and nearly bounded (numerical) ranks. Then we can design a fast structured solver tosolve these linear systems [38].

4. Nonlinear problems

In this section, let us focus on a class of nonlinear TSPBVP as

εu′′′(x) = F(x, u, u′, u′′), x ∈ (0, 1), (4.1)

u(0) = p, u′(0) = q, u′(1) = r, (4.2)

where F(x, u, u′, u′′) is a smooth function satisfying the following conditions

Fu′(x, u, u′, u′′) > 0, Fu′′(x, u, u′, u′′) 6 −α, α > 0.

Let us assume that the reduced problem of (4.1)–(4.2), i.e.

F(x, ũ, ũ′, ũ′′) = 0, x ∈ (0, 1), (4.3)

ũ(0) = p, ũ′(1) = r, (4.4)

has a unique solution ũ0 ∈ C (3). Then the original problem (4.1)–(4.2) has a unique solution with a boundary layer of widthO(ε) near x = 0 for ε ≪ 1 (see Ref. [8]).

Besides, the nonlinear function F(x, u, u′, u′′) could be treated via Newton’s method of quasilinearization as follows

F [m+1] ≈ F [m]u · (u[m+1]

− u[m]) + F [m]u′ · ((u′)[m+1] − (u′)[m]) + F [m]u′′ · ((u

′′)[m+1] − (u′′)[m]),

wherem is a nonnegative integer and

F [m] = F(x, u[m], (u′)[m], (u′′)[m]), F [m+1] = F(x, u[m+1], (u′)[m+1], (u′′)[m+1]).

Then for each fixedm, the u[m+1] is the solution of the following linearized problem:

εu′′′[m+1]

− a[m](x)u′′[m+1]

− b[m](x)u′[m+1]

− c[m](x)u[m+1] = F̂ [m], (4.5)

u[m+1](0) = p, (u′)[m+1](0) = q, (u′)[m+1](1) = r, (4.6)

where

a[m](x) = F [m]u′′ , b[m](x) = F [m]u′ , c

[m](x) = F [m]u ,

F̂ [m] = u[m]c[m] + (u′)[m]b[m] + (u′′)[m]a[m] − F [m].

Here are some additional remarks.

• The solution of the reduced problem (4.3)–(4.4) would be taken as the initial value u[0] in the iteration (4.5).• For each fixed m, the linearized problem (4.5)–(4.6) is the TSPBVP like (1.1)–(1.2) which can be resolved by the RSCAT

method presented in above sections.• Let us denote the iteration error atmth step as

e[m]it = maxj

u[m+1] xj− u[m] xj , xj ∈ [0, 1].Then for the above Newton’s quasilinearization process, if the iteration error is less than the tolerated error, i.e. e[m]it 6 τ ,the iteration process could be stopped.


Table 1Comparison of the relative maximum errors of Example 1.

ε RSCAT method Method in [20]N e1 e2 e1

1e−2 45 2.7465e−14 2.5628e−13 a1e−4 59 2.6376e−11 8.0553e−11 1.0494e−31e−6 73 1.3614e−10 4.0762e−10 a1e−8 82 1.7240e−08 5.1601e−08 a1e−10 88 4.5012e−06 1.3470e−05 a

a Means this case is not considered in [20].

5. Numerical results

We now present several numerical experiments of TSPBVP with emphasis on the comparison between commonChebyshev collocation method (CCC) and our RSCAT method. The relative maximum errors of the solution and its firstderivative are given by

e1 = ∥û1 − uE∥∞/∥uE∥∞, e2 = ∥û2 − u′E∥∞/∥u′

E∥∞,

where û1, û2 are the numerical solution and its first derivative, i.e. the first and the second half of vector U in (3.10), anduE, u′E are the exact solution and its first derivative respectively.

Example 1. Consider the following problem with constant coefficients [20]:

εu′′′(x) + 2u′′(x) = −1, x ∈ (0, 1), (5.1)

u(0) = 1, u′(0) = 1, u′(1) = 1. (5.2)

The exact solution and its first order derivative are respectively

u(x) = 1 −ε1 − e−2x/ε

41 − e−2/ε

− x24

+ x

1 +

121 − e−2/ε

, (5.3)u′(x) =

−e−2x/ε

21 − e−2/ε

− x2

+ 1 +1

21 − e−2/ε

. (5.4)The equivalent system of ODEs is as follows:u′1(x) − u2(x) = 0,

εu′′2(x) + 2u′

2(x) = −1, x ∈ (0, 1),u1(0) = 1, u2(0) = 1, u2(1) = 1.

(5.5)

According to Remark 2.1, we set β = − ln(ε)a(0) = −ln(ε)2 .

In Fig. 1, we plot the relative maximum errors in semi-log scale, for both CCC method and RSCAT method. It shows thatthe convergence rates are greatly improved in our RSCAT method. Besides, Table 1 illustrates the comparison of numericalresults obtained by our RSCAT method and the method in [20]. We observe that our method gives much better results withsignificantly less number of unknowns.

Fig. 2 displays the plots of numerical and exact solutions for the case with ε = 1e−10, in which (i) and (iii) show thesolutions in the whole interval [0, 1] while (ii) and (iv) show the solutions in the boundary layer region [0, βε]. We observethat in this case the boundary layer of the solution’s first order derivative is much more narrow and steep than the solutionitself. Furthermore, Fig. 3 shows the point wise errors of the function and its derivative in the boundary layer region. We seethat the errors are almost balanced for each point, due to the fact that there are reasonably many collocation points in theboundary layer region, which verifies the correctness of our method for choosing the parameters in the mapping (1.3).

In addition, in order to verify the conclusion about the computational cost discussed in Section 3.3, by Fig. 4, we showthe condition numbers of the matrices D1,D2 and D3, which are defined in (3.11) and (3.13). It clearly illustrates that thecondition numbers of the kth order differential matrices D(k) increase like O(N2k).

Example 2. Consider the following problem with variable coefficients [21]:

εu′′′(x) + (1 − x)u′′(x) − u′(x) + xu(x) = f (x), x ∈ (0, 1), (5.6)

u′(0) =1/ε

e−1/ε − 1− 2, u(1) = 0, u′(1) = 0 (5.7)


(a) ε = 1e − 2. (b) ε = 1e − 4.

(c) ε = 1e − 6. (d) ε = 1e − 8.

Fig. 1. Relative errors in Example 1: RSCAT method (solid lines) vs. CCC method (dashed lines).

where the function f (x) is chosen so that the exact solution and its first order derivative of problem (5.6)–(5.7) arerespectively:

u(x) = 1 −1 − e−x/ε

1 − e−1/ε+ (x − 1)2, (5.8)

u′(x) =−e−x/ε

ε1 − e−1/ε

+ 2(x − 1). (5.9)The equivalent system of two ODEs is as follows:

u′1(x) − u2(x) = 0,εu′′2(x) + (1 − x)u

′

2(x) − u2(x) + xu1(x) = f (x), x ∈ (0, 1),

u2(0) =1/ε

e−1/ε − 1− 2, u1(1) = 0, u2(1) = 0.

(5.10)

According to Remark 2.1, we set β = − ln(ε)a(0) = − ln(ε).We compare the maximum relative errors of our RSCAT method with CCC method in Fig. 5 and with the method in [21]

in Table 2. Both of them verify the high accuracy and efficiency of our method.Additionally, Fig. 6 displays the plots of numerical and exact solutions for the case with ε = 1e−7, both in the whole

region and boundary layer region. We observe that unlike Example 1, in this case the boundary layers of both the solutionand its first order derivative are obvious.

Example 3. Consider the following non-linear problem [19,22]:

εu′′′(x) + 2u′′(x) − 4u′(x) +12u(x)2 = f (x), x ∈ (0, 1), (5.11)

u(0) = 1, u′(0) = 1, u′(1) = 1, (5.12)


(a) Function in overall region. (b) Function near boundary region.

(c) Derivative in overall region. (d) Derivative near boundary region.

Fig. 2. Example 1 with ε = 1e−10,N = 88: numerical solutions vs. exact solutions.

(a) Function. (b) First derivative.

Fig. 3. Example 1 with ε = 1e−10,N = 88: point wise errors near the boundary region.

where the function f (x) is chosen such that (5.3) and (5.4) are the exact solution and its first order derivative of this problemrespectively. The linearized problem of (5.11)–(5.12) is

εu′′′[m+1]

+ 2u′′[m+1]

− 4u′[m+1]

+ u[m]u[m+1] = f (x) +12

u[m]

2, (5.13)

u[m+1](0) = 1,u′[m+1]

(0) = 1,u′[m+1]

(1) = 1. (5.14)


Fig. 4. Example 1 with ε = 1e−2: condition numbers of D1,D2,D3 .

(a) ε = 1e − 4. (b) ε = 1e − 5.

(c) ε = 1e − 6. (d) ε = 1e − 7.

Fig. 5. Relative errors in Example 2: RSCAT method (solid lines) vs. CCC method (dashed lines).

We solve this problem using the procedure shown in Section 4. As shown in Fig. 7, the error tolerance is chosen asτ = 1e−12 and the scheme (5.13) converges after at most three iterations for the cases that ε varies from 1e−2 to 1e−8.Since the exact solution of this case is the same as Example 1, we do not plot the image but give the computational errorsin Table 3. The superiority of the results obtained by our RSCAT method is obvious.

6. Concluding remarks

In this paper, a new numerical method, named RSCAT, has been developed for solving a class of third-order singularlyperturbed boundary value problems. The key to the success of this method is that the results from asymptotic theory


(a) Function in overall region. (b) Function near boundary region.

(c) Derivative in overall region. (d) Derivative near boundary region.

Fig. 6. Example 2 with ε = 1e−7,N = 97: numerical solutions vs. exact solutions.


ε RSCAT method Method in [21]N e1 e2 e1 e2

1e−2 50 4.8850e−15 6.3392e−15 a a1e−3 62 9.1776e−13 4.3695e−14 a a1e−4 74 1.9931e−11 8.5406e−14 2.9000e−3 9.1000e−51e−5 75 6.2989e−09 2.5800e−12 a a1e−6 76 3.9218e−06 1.2663e−10 2.9000e−3 8.8000e−51e−7 97 3.3441e−04 1.2381e−09 a a

a Means this case is not considered in [21].


ε RSCAT method Method in [19] Method in [22] (N = 45)N e1 e2 e1 e2 e1 (CM-I) e1 (CM-II)

1e−2 45 4.1099e−14 2.2785e−13 a a 1.19e−6 4.77e−71e−4 65 6.0397e−14 4.0577e−12 a a 8.30e−4 1.95e−41e−5 73 2.9211e−13 1.9935e−11 1.1097e−3 2.2134e−07 7.36e−3 3.06e−41e−6 77 6.5289e−12 1.3035e−10 a a a a1e−8 82 7.9811e−09 4.7602e−08 a a a a

a Means this case is not considered in [19,22].

are employed to determine the parameters in the sinh transformation of the rational spectral collocation method inbarycentric form. The details of the numerical algorithms show that our method is very easy to use and ready for computer


Fig. 7. Convergence of iteration errors in Example 3 with different ε and N .

implementation. Numerical experiments illustrate that the proposed RSCAT method, which enjoys high accuracy andefficiency in both linear and nonlinear problems, is obviously a good alternative to other numerical methods available forTSPBVP in the literature.

Here, we restricted our attention to the one-dimensional third-order singularly perturbed problems with rigorousanalysis and ample numerical examples here. Actually, the theoretical and numerical frameworks presented in this paper areessential for extension to more complicated problems. Currently, we are working on the following situations: (i) Combiningthe asymptotic analysis presented here with the newmapping proposed in [39], which is a generalized version of (1.3), onecan likely construct the rational spectral collocation method for TSPBVP with several interior and/or boundary layers; (ii) Weplan to employ the strategy of adaptivity based on local error estimators [40,41] to construct the adaptive rational spectralor spectral-element schemes for the singularities with a priori unknown location or time-dependent problemswithmovinglayers.

Acknowledgments

Thework of YingweiWang is partially supported by NSF grants DMS-1217066 andDMS-1419053. The authors also thankthe anonymous referees and editors for helpful comments and suggestions which led to a significant improvement of thepresentation.

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A rational spectral collocation method for third-order singularly perturbed problemsIntroductionAnalytic analysisAsymptotic expansion approximationThe location and width of boundary layer

Rational spectral collocation method in barycentric formThe barycentric form of rational interpolationNumerical discretization of the ODE system (2.1)Fast solver for the linear system (3.10)

Nonlinear problemsNumerical resultsConcluding remarksAcknowledgmentsReferences

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