# Rational Spectral Collocation Combined with the Singularity Separated Method for a System of Singularly Perturbed Boundary Value Problems.

1. Introduction

The singular perturbation problems (SPPs) such as fluid mechanics boundary layers, quantum mechanics turning points, and flow of large Reynolds numbers arise in the mathematical modeling physical and engineering problems. In the past decades, singular perturbation has received extensive attention. Singular perturbation boundary value problems have steep gradients in the narrow layers, which was a serious obstacle in the calculation of classical numerical methods. There is a vast literature dealing with SPPs with smooth coefficients and source term for single equation [1-3]. Compared with single equation singularly perturbed problems, coupled systems can simulate more complicated physical phenomena. Only a few authors have developed numerical methods to deal with the problem of coupled singular perturbation system on smooth data. A finite difference method on a piecewise uniform mesh for the reaction-diffusion type was proposed . The result of Shishkin's method in terms of convergence, stability, and error estimation can be found in Madden and Stynes , Linss and Madden [6, 7], and Matthews et al. . An upwinding finite difference scheme on piecewise-uniform Shishkin meshes for the convection-diffusion type was presented, and it was solved by Jacobi iteration to compute the solution .

In this paper, a coupled system of m [greater than or equal to] 2 singularly perturbed linear equations are considered in the unknown vector function u = [([u.sub.1], ..., [u.sub.m]).sup.T]:

Lu := -[epsilon]u" - Bu' + Au = f, x [member of] [OMEGA] = (0, 1), (1)

and it yields

u(0) = [b.sub.0], u(1) = [b.sub.1], (2)

where A = ([a.sub.ij]) and B = ([b.sub.ij]) are m x m matrices and 0 < [epsilon] [much less than] 1 is a small parameter whose presence makes the problem singularly perturbed. We assumed that f = [([f.sub.1], ..., [f.sub.m]).sup.T] [member of] [C.sup.2][[0, 1].sup.m] and both [b.sub.0] = [([b.sub.01], ..., [b.sub.0m]).sup.T] and [b.sub.1] = [([b.sub.11], ..., [b.sub.1m]).sup.T] are constant vectors. These layer behaviors can be examined in two different cases:

Case 1. B [equivalent to] 0, the system is called reaction-diffusion type. Generally speaking, the solution of this type has two boundary layers with width O([square root of ([epsilon])]) at x = 0 and x = 1 under some assumptions.

Case 2. B [not equal to] 0, the system is called convection-diffusion type, the solution of this type has a single boundary layer with width O([epsilon]) at x = 0(or x = 1) under proper assumptions.

If (1) is coupled through its convective terms, we say it is strongly coupled; otherwise, if B [equivalent to] 0 or B is just a nonzero diagonal matrix, it is said to be weakly coupled.

The rational spectral collocation method was proposed in the literature . A conformal map which maps the collocation points clustered near the poles of [-1, 1] into a new set of collocation points. The parameters of the mapping are determined by the position and width of the boundary layer. Chen and Wang [11, 12] applied a rational spectral collocation in barycentric form with the sinh transform (RSCAT) method to solve a coupled system of singularly perturbed problems and third-order singularly perturbed problems.

To weaken the singularity and improve the accuracy of numerical simulation, the singularity-separated technique (SST) for singular perturbation problem with constant coefficients was proposed by Chen and Yang , and finite element methods with SST were used to solve a singular perturbation problem with a single boundary layer.

Here, we present a novel numerical method based on rational spectral collocation in barycentric form with the singularly-separated method (RSC-SSM) to solve singularly perturbed boundary value problem in various types, both weakly coupled and strongly coupled.

This paper is organized as follows. The asymptotic analysis of coupled system is outlined in Section 2. The algorithmic details of the RSC-SSM for a coupled system of singularly perturbed problems are provided and the error estimates for the method are discussed in Section 3. The coupled system of singularly perturbed problems is solved in Section 4, which supports theoretical results and provides a favorable comparison with existing methods. Finally, we present some concluding remarks in Section 5.

2. Preliminaries

For the construction of the RSC-SSM, it is necessary to understand the properties of exact solution, especially the position and width of the boundary layer. Some useful lemmas are listed, including the maximum principle, stability result, the error estimation of the solution, and its derivatives are established for the boundary value problem (1).

2.1. Weakly Coupled System of Reaction-Diffusion Problems. If A [not equal to] 0, B [equivalent to] 0 in (1), it is convenient to introduce the notation to denote the reaction-diffusion operator:

[L.sub.R]u := -[epsilon]u" + Au = f. (3)

Assume that A has positive diagonal entries and nonpositive off-diagonal entries, i.e.,

[mathematical expression not reproducible], (4)

and A is also strictly diagonally dominant i.e., [mathematical expression not reproducible] and [[summation].sup.m.sub.j=1][a.sub.ij] > [[alpha].sup.2] > 0, i = 1, ..., m. These hypotheses ensure that problem (1) has a unique solution.

Lemma 1. Under above assumptions, all eigenvalues of matrix A are positive.

Proof. Let A have an eigenvalue of [lambda] = 0, according to the Gerschgorin disc theorem, [mathematical expression not reproducible], which is a contradiction. So, there is no zero eigenvalue in A, that is, [absolute value of (A)] [not equal to] 0.

Set f(x) = [absolute value of (D + xB)], x [member of] [0, 1], where D = diag(A), B = A - D, then f(x) is a continuous function on [0, 1] and f(0) = [absolute value of (D)] = [[PI].sup.m.sub.i=1] [a.sub.ii] > 0; assume f(1) = [absolute value of (A)] < 0, then we can obtained [there exists][xi] [member of] (0, 1], such that f(x) = [absolute value of (D + [xi]B)] = 0. Moreover, A is strictly diagonally dominant, then [a.sub.ii] > [R.sub.i](A) [greater than or equal to] [[summation].sup.n.sub.j=1,j[not equal to]i] [absolute value of ([xi][a.sub.ij])], that is, D + [xi]B is also strictly diagonally dominant. Contradictions between f([xi]) = [absolute value of (D + [xi]B)] [not equal to] 0 and f([xi]) = 0 can be obtained. Therefore, f(1) = [absolute value of (A)] > 0, each of the main determinants of the strictly diagonally dominant matrix is the strictly diagonally dominant, then we have the A is positive definite matrix. So, all eigenvalues of matrix A are positive.

Lemma 2. Let us suppose that a function u(x) satisfies u(0) [greater than or equal to] 0, u(1) [greater than or equal to] 0, Lu(x) [less than or equal to] 0, [for all]x [member of] [OMEGA], then u(x) [greater than or equal to] 0; [for all]x [member of] [bar.[OMEGA]].

The direct application of the maximum principle is the following stability result.

Lemma 3. If u(x) is the solution of (2), then we have the stability bound inequality:

[parallel]u[parallel] [less than or equal to] M = max([parallel][b.sub.0][parallel], [parallel][b.sub.1][parallel]) + C max[parallel]f(x)[parallel], [parallel]u'(x)[parallel] [less than or equal to] CM/[epsilon]. (5)

Shishkin decomposition is used for asymptotic analysis of problem (3) , which splits the solution into regular and layer components:

u(x) = w(x) + v(x), (6)

where the regular component w(x) and the layer component v(x) are the solutions of the two problems:

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible], (8)

then w(x) and v(x) have the following estimates.

Lemma 4 (see ). Let u = w + v be the solution of problem (3) and w and v be the solutions of (7) and (8), respectively. Then, we have

[mathematical expression not reproducible], (9)

where j = 0, 1, 2.

Theorem 1. The solution of (3) and (2) has the following asymptotic expansion:

u(x) = w(x) + v([bar.x]) + O([square root of ([epsilon])]), (10)

where

[mathematical expression not reproducible]. (11)

The above theorem suggests that the boundary layer regions of the solution u(x) to the reaction-diffusion problems are [0, [x.sub.*]] and [1 - [x.sub.*], 1], that is, the boundary layers are located at the two endpoints of the underlying interval [0, 1] and each of their width is [x.sub.*] = -(ln [square root of ([epsilon])]/[alpha])[square root of ([epsilon])].

2.2. Strongly Coupled System of Convection-Diffusion Problem. If B [not equal to] 0, the notation [L.sub.C] is introduced to denote the convection-diffusion operator:

[L.sub.C]u := - [epsilon]u" - Bu' + Au = f. (12)

To ensure that there is a unique solution to this problem, the following assumptions need to be satisfied: B [not equal to] 0 is strictly diagonally dominant and hence invertible and define [beta] = min([b.sub.ii]), i = 1, ..., m.

The above assumption enables us to predict that the layer in u(x) will be at x = 0 .

Theorem 2. Under above assumptions, the solution of (12) and (2) has the following asymptotic expansion:

u(x) = w(x) + v([bar.x]) + O([epsilon]), (13)

where w([bar.x]) is the layer correction function, i.e.,

[mathematical expression not reproducible]. (14)

For the convection-diffusion problems, there is a boundary layer region [0, [x.sub.*]] in the solution u of (12) and (2). That is, the boundary layers are located at the left endpoint, and its width is [x.sub.*] = - (ln [epsilon]/[beta])[epsilon].

3. The Singularity-Separated Technology

Note that if [epsilon] = [10.sup.-k,] then [mathematical expression not reproducible] can be neglected.

3.1. Weakly Coupled System of Reaction-Diffusion Problem Theorem 3. The general solution of the homogenous equation [L.sub.R]u = 0 can be expressed in the form

[mathematical expression not reproducible], (15)

where [[lambda].sub.j] is an eigenvalue of the A, the vector [V.sub.j] is the associated eigenvector of A, and [C.sub.j] and [C'.sub.j] are the constant vectors, j = 1, 2, ..., m.

Proof. We can write higher order differential equations as a system with a very simple change of variable:

[mathematical expression not reproducible]. (16)

System (16) can be written in the following matrix form:

[mathematical expression not reproducible]. (17)

Thus,

[mathematical expression not reproducible]. (18)

The eigenvalue [lambda] of D can be found from the auxiliary equation:

[mathematical expression not reproducible]. (19)

If [lambda] = 0, we have [mathematical expression not reproducible]. So, [lambda] = 0 is not the eigenvalue of D, and using the general transform, we can obtain

[mathematical expression not reproducible], (20)

therefore, we can get

[[lambda].sub.2] = [[lambda].sub.j]/[epsilon], j = 1, 2, ..., m. (21)

where [[lambda].sub.j] is an eigenvalue of A. The eigenvalue of matrix D is the square root of [[lambda].sub.j], so we can obtain all the eigenvalues of matrix D:

[s.sub.j] = [square root of ([[lambda].sub.j])]/[summation over ([epsilon])], [s'.sub.j] = -[square root of ([[lambda].sub.j])]/[square root of ([epsilon])], j = 1, 2, ..., m. (22)

Next, we need to show that [mathematical expression not reproducible] and [mathematical expression not reproducible] are the solutions of equation [L.sub.R]u = 0:

[mathematical expression not reproducible], (23)

that is, [[phi].sub.j](x) is the solution of equation [L.sub.R]u = 0. Similarly, it can be proved that [[psi].sub.j](x) is the solution of equation [L.sub.R]u = 0.

Therefore, the general solution of the homogenous equation [L.sub.R]u = 0 can be expressed in the form (15).

Corollary 1. The general solution of (3) can be expressed as follows:

[mathematical expression not reproducible], (24)

where w is a special solution, [[lambda].sub.j] is an eigenvalue of A, the vector [V.sub.j] is the associated eigenvector of A, and [C.sub.j] and [C'.sub.j] are the constant vectors, j = 1, 2, ..., m.

So, the solution u(x) is composed of two parts, u [approximately equal to] [u.sub.ss] = w + v, in which w(x) is the regular term and v(x) is the singular term, and it has the following explicit expression:

v = [m.summation over (j=1)] ([C.sub.j][[phi].sub.j](x) + [C'.sub.j][[psi].sub.j] (x)). (25)

The parameters [C.sub.j] and [C'.sub.j], j = 1, 2, ..., m, in (25) can be defined by the boundary conditions. For any f(x), how to construct the regular solution w is key.

We construct an auxiliary third boundary value problem to solve the special solution of (3):

[mathematical expression not reproducible], (26)

then [epsilon]w"(0) = 0, [epsilon]w"(1) = 0, and the singularities of the solution are weakened. Based on the boundary condition (2), we can obtain the undetermined coefficients [C.sub.j] and [C'.sub.j] in explicit singular function (15):

[mathematical expression not reproducible]. (27)

Due to the linear independence of [{[V.sub.j]}.sup.m.sub.j=1], the matrix V = [[V.sub.1], [V.sub.2], ..., [V.sub.m]] is invertible. Equation (27) has a unique solution.

3.2. Strongly Coupled System of Convection-Diffusion Problem

Theorem 4. If A [equivalent to] 0, the general solution of (12) can be expressed as follows:

[mathematical expression not reproducible], (28)

where w is a special solution, [[lambda].sub.j] and [V.sub.j] are the distinct eigenvalues and eigenvectors of B, and [C.sub.j] is the constant which is determined by boundary condition (2).

Proof. Let u' = v, then (12) can be transformed into an equivalent problem including a system of two ODEs as follows:

[mathematical expression not reproducible]. (29)

System (29) can be written in the following matrix form:

[mathematical expression not reproducible], (30)

where [E.sub.m] and [O.sub.m] are identity matrix and all-zeros matrix, respectively. Then, we have

[mathematical expression not reproducible]. (31)

Let [lambda] is an eigenvalue of D, and

[mathematical expression not reproducible]. (32)

The eigenvalue [lambda] = [s.sub.0] = 0 has multiplicity m, [lambda] = [s.sub.j] = -([[lambda].sub.j]/[epsilon]), (j = 1, 2, ..., m).

Next, we need to show that [mathematical expression not reproducible] is the solution of equation [L.sub.C]u = 0:

[mathematical expression not reproducible]. (33)

So, the general solution of (12) can be expressed in the form (28).

Let the singular correction function be

[mathematical expression not reproducible]. (34)

In the same way, the regular term w is a solution of an auxiliary third boundary value problem:

[mathematical expression not reproducible]. (35)

According to the boundary conditions, we can obtain the undetermined coefficients in an explicit singular function:

[m.summation over (j=1)] [C'.sub.j][V.sub.j] = [b.sub.0] - w(0). (36)

4. The Rational Spectral Collocation Method Combined with Singularity-Separated Method

4.1. The Barycentric Form of Rational Interpolation. Rational function [p.sub.N](x) in barycentric form, which interpolates function u(x) at N + 1 distinct points [{[x.sub.k]}.sup.N.sub.k=0] can be expressed as 

u(x) [approximately equal to] [p.sub.N](x) = [[summation].sup.N.sub.k=0]([[omega].sub.k]/x - [x.sub.k])u([x.sub.k])/ [[summation].sub.N.sub.k=0][[omega].sub.k]/x - [x.sub.k]], (37)

where [{[[omega].sub.k]}.sup.N.sub.k=0] are barycentric weights. For Chebyshev-Gauss-Lobatto points [x.sub.k] = - cos(k[pi]/N), barycentric weights are chosen as 

[[omega].sub.0] = 1/2, [[omega].sub.k] = [(-1).sup.k], k = 1, 2, ...,N - 1, [[omega].sub.N] = [(-1).sup.N]/2. (38)

According to the theorem in Baltensperger et al. , we have the convergence analysis of the rational interpolation polynomial in barycentric form with transformed Chebyshev points as follows:

[parallel][p.sub.N](x) - u(x)[parallel] = O ([[rho].sup.-N]).[for all]x [member of] [-1, 1], (39)

where [p.sub.N] (x) is the rational interpolating polynomial in barycentric form of u, g is a conformal map, and [rho] > 1 is the sum of its major and minor axes of an ellipse.

An advantage of the rational interpolation in barycentric form is that its derivatives can be calculated directly using differentiation formulae, instead of using the differential quotient rule repeatedly. The nth order derivative of polynomial [p.sub.N](x) evaluated at the point [x.sub.j], and can be expressed in the form

[mathematical expression not reproducible], (40)

where [D.sup.(n).sub.jk] is the entry of the nth order differentiation matrix. The entries of the first- and second-order differentiation matrices, are given by 

[mathematical expression not reproducible]. (41)

As suggested in (39), the convergence rate of the rational spectral collocation method mainly depends on the analytic region of u in the complex plane. Therefore, the conformal map g could be chosen to enlarge the ellipse of analyticity of u o g. Thus, compared with the Chebyshev spectral method, a better approximation of u could be obtained.

Note that differentiation matrices (41) only rely on weights [[omega].sub.k] and new points [[??].sub.k], which is why the underlying equation does not need to be converted to new coordinates after maps.

4.2. The sinh Transform. In order to approximate the rapid changes in the boundary layer region. Tee and Trefethen have constructed the conformal map :

g(x) = [lambda] + [mu] sinh [([sinh.sup.-1] ([1 - [lambda]]/[mu]) + [sinh.sup.-1] ([1 + [lambda]]/[mu])) [x - 1]/2 + [sinh.sup.-1] ([1 - [lambda]]/[mu])], (42)

where [lambda] and [mu] are the location and width of the boundary layers, respectively. The transformed Chebyshev points [{[g.sub.[lambda],[mu]]([x.sub.k]).sup.N.sub.k=0] are clustered near the location of boundary layer x = [lambda], and the density is determined by the width of boundary layer. For the convection-diffusion type, parameters in (42) should be chosen as follows:

[lambda] = -1, [mu] = -2 [ln[epsilon]/[beta]] [epsilon]. (43)

In order to better distinguish the singular perturbation problem with two boundary layers, Tee proposed the combined sinh transform as

[mathematical expression not reproducible]. (44)

All derivatives of the piecewise map [??] at x = 0 are continuous so that the spectral accuracy could be preserved. For the reaction-difffusion type, the parameter in (44) should be chosen as [mu] = - 2(ln [square root of ([epsilon])]/[alpha]) [square root of ([epsilon])].

4.3. RSC-SSM. In this part we will show the RSC-SSM in a detailed algorithm.

For reaction-diffusion, we solve auxiliary third boundary value problem (26) using a rational spectral collocation method with the sinh transformation. First, by introducing the transformation x = (y + 1)/2 and defining [??](y) = u(x) = u((y + 1)/2), then u'(x) = 2[??]'(y) and u" (x) = 4[??]"(y). Evaluating equations in (26) at points [y.sub.k], k = 0, 1, ..., N, yields

(-4[epsilon][E.sub.m] [cross product] [D.sup.(2)] + A [cross product] [E.sub.m]) W = F, (45)

where

[mathematical expression not reproducible], (46)

for i = 1, ..., m, [cross product] is Kronecker product.

Boundary conditions in (26) suggest that

(A [cross product] [E.sub.m](1)) W = [[??].sub.i] ([y.sub.0]), (A [cross product] [E.sub.m](N + 1)) W = [[??].sub.i] ([y.sub.N]), i = 1, 2, ..., m. (47)

Solving the linear algebra system composed of (45) and (47), we can obtain the numerical solution of (26), which has a boundary layer which occurs at y = 1 with the width [mu] = 2[x.sub.*]. Let the transformed Chebyshev collocation points be

Y = [{[y.sub.k]}.sup.N.sub.k=0] = [{[[??].sub.[mu]] (-cos(k[pi]/N))}.sup.N.sub.k=0], (48)

where [[??].sub.[mu]] is expressed by (44) in which [mu] = 2[x.sub.*].

As for the convection-diffusion type, evaluating equations in (35) at points [y.sub.k], k = 0, 1, ..., N, yields

(-4[epsilon][E.sub.m] [cross product] [D.sup.(2)] - 2B [cross product] [D.sup.(1)]) W = F. (49)

Boundary conditions in (35) suggest that

[mathematical expression not reproducible]. (50)

Solving the linear algebra system composed of (49) and (50), we can obtain the numerical solution of (35).

The numerical solution of original problem (1) and (2) can be obtained as [u.sup.N] = W + V.

4.4. Errors Analysis

Theorem 5. Let u([x.sub.i]) and [u.sup.N]([x.sub.i]) be the exact solution and numerical solution of the original problem (3) and (2), respectively. Then, we have

[mathematical expression not reproducible], (51)

where [[??].sub.A] is the spectral radius of A.

Proof. Let z = u - [u.sup.N], then the error z satisfies

[parallel]z[parallel] = [parallel]u - [u.sup.N][parallel] = [parallel]u - [u.sub.s] + [u.sub.s] - [u.sup.N][parallel] [less than or equal to] [parallel]u - [u.sub.s][parallel] + [parallel]w + v - [u.sup.N][parallel]. (52)

According to the proof of Theorem 5, the singular correct function v(x) is the solution of the homogeneous equations Lv = 0. So, set R = u - [u.sub.SS]:

L (u - [u.sub.S]) = L(u - w - v) = -[epsilon]u" + Au - (-[epsilon]w" + Au) - Lv = 0. (53)

Simultaneously, obtained from (27)

[mathematical expression not reproducible]. (54)

According to the Lemma 3, we can get

[mathematical expression not reproducible]. (55)

And, using the convergence of rational spectral collocation (39), we obtain

[parallel]w + v - [u.sup.N][parallel] [less than or equal to] [c.sub.2][[rho].sup.-N]. (56)

Joint (55) and (56), we have the error estimation

[mathematical expression not reproducible]. (57)

Similarly, for the convection-diffusion type, we have

[mathematical expression not reproducible], (58)

where [[??].sub.B] is the spectral radius of B.

So, I can get the convergence of the RSC-SSM that can almost achieve spectral accuracy.

5. Numerical Experiments

In order to demonstrate the accuracy and efficiency of the RSC-SSM, we use RSCAT and RSC-SSM to solve singularly perturbed problems with exact solutions and compare the results. We verify the theoretical results obtained in the previous section through numerical experiments. The relative maximum errors of the solution are given by

e = [[parallel][u.sub.N] - [u.sub.E][parallel].sub.[infinity]]/ [[parallel][u.sub.E][parallel].sub.[infinity]], (59)

where [u.sub.N] and [u.sub.E] are the numerical solution and the exact solution, respectively.

In our computations, all experiments are carried outperformed using MATLAB (version R2014a) on a personal computer with a 2.5 GHz central processing unit (Intel Core i5-2450M), 4.00 GB of memory, and Windows 7 operating system.

Example 1. Consider the reaction-diffusion problem with constant coefficients:

[mathematical expression not reproducible]. (60)

The exact solution of this problem is

u = 1/2 [(1, 1).sup.T] [u.sub.1] + 1/2 [(-1, 1).sup.T] [u.sub.2], (61) where

[mathematical expression not reproducible]. (62)

This problem has two boundary layer regions, i.e., the boundary layer are located at the two endpoints of the underlying interval [0, 1]. In our method, we use the transform (44), and the parameter is chosen as [alpha] = 0.99. The eigenvalues of A/[epsilon] are [[lambda].sub.1] = 1/[epsilon] and [[lambda].sub.2] = 3/[epsilon], and [V.sub.1] = [(1, 1).sup.T] and [V.sub.2] = [(1, -1).sup.T] are the corresponding eigenvectors. So, the general solution of this problem can be expressed (24), and the special solution w is a solution of an auxiliary third boundary value problem (26). The undetermined coefficients of singular term can be obtained by (27).

Figure 1 plots the relative maximum errors of both RSCAT and RSC-SSM in semilog scale for various values of [epsilon]. Compared with RSCAT, the convergence rates have considerably improved in our RSC-SSM on over trend. Furthermore, Table 1 provides the comparison of numerical results obtained by our RSC-SSM and RSCAT with several choices of [epsilon]. We perceive that our method gives much better results with significantly fewer number of unknowns. With the increase of N, the relative maximum error of RSCAT method decreases, while quite the opposite is true for the RSC-SSM. Figure 2 displays plots of numerical and exact solutions with [epsilon] = [10.sup.-2] and [epsilon] = [10.sup.-6]. There are more points located in the boundary layer region. Figure 3 shows the point-wise errors of the function in the whole region, it is observed that compared with RSCAT, the smaller the parameter [epsilon] is, the more accurate the RSC-SSM result.

Example 2. Consider the convection-diffusion problem with constant coefficients:

[mathematical expression not reproducible]. (63)

The exact solution of this problem is

u = [(1, 1, 1).sup.T] [e.sup.-x/[epsilon]] + [(-2, 1, 1).sup.T] [e.sup.-4x/[epsilon]] + [(0, 2, -2).sup.T] [e.sup.-6x/[epsilon]] + [(x, -2x, x - 1).sup.T]. (64)

For the sake of comparison, the parameters in transform (42) are chosen as [beta] = 2.55. This problem has rapid changes near the point x = 0. The eigenvalues of B/[epsilon] are [[lambda].sub.1] = 1/[epsilon], [[lambda].sub.2] = 4/[epsilon], and [[lambda].sub.3] = 6/[epsilon], and the corresponding eigenvectors are [V.sub.1] = (1, 1, 1)T, [V.sub.2] = [(2, -1, -1).sup.T], and [V.sub.3] = [(0, -1, 1).sup.T]. So, the general solution of this problem can be expressed (28), and the special solution w is a solution of auxiliary third boundary value problem (35). The undetermined coefficients of singularity term can be obtained by (36). We compare the maximum relative errors of the RSC-SSM with the RSCAT method in Figure 4 and Table 2, and both verify the high accuracy and efficiency of our method.

Figure 4 plots the relative maximum errors of both RSCAT and RSC-SSM in semilog scale for various values of [epsilon]. It shows that the convergence rates have greatly improved in our RSC-SSM. Besides, Table 2 illustrates the comparison of numerical results obtained by our RSC-SSM and the method in  with several choices of [epsilon]. We observe that our method gives much better results with significantly less number of unknowns. Figure 5 displays the plots of numerical and exact solutions for the case with [epsilon] = [10.sup.-2] and [epsilon] = [10.sup.-6]. There are more points located in the boundary layer region. Figure 6 shows the pointwise errors of the function in the whole region; it is observed that compared with RSCAT, we can conclude that with the increase of N, the error of the RSC-SSM remains stable.

6. Concluding Remarks

In this paper, a new numerical method RSC-SSM has been proposed to solve a system of singularly perturbed boundary value problems. The numerical solution consists of two parts: the solution of weaker singularly auxiliary equation and the singular correction term. Numerical experiments confirm that compared to RSCAT, the present RSC-SSM has the following advantages:

(1) The RSC-SSM is easy to implement, and it enjoys computational efficiency, accuracy, and stability over some popular numerical approaches.

(2) The accuracy of numerical approximation depends not only on the number of the grid nodes, but also on parameter [epsilon]. The smaller the [epsilon] is, the thinner the boundary layer is, and the better results obtained.

(3) In the RSC-SSM, we make good use of the eigenvalue of coefficient matrix, which plays a important role in constructing general solution of singularly perturbed problems.

The numerical results demonstrate the spectral accuracy of proposed algorithms and agree with the theoretical analysis very well. And, the theoretical and numerical frameworks presented in this paper are essential for extension to more complicated problems.

https://doi.org/10.1155/2019/9030565

Data Availability

The data used to support the findings of this study are included within the article

Conflicts of Interest

The author declares that there are conflicts of interest.

Acknowledgments

The author is grateful to Prof. Y J Wu for his valuable suggestions and help for the work related to this paper. The work is partially supported by the First-Class Disciplines Foundation of Ningxia (No. NXYLXK2017B09), Natural Science Foundation of Ningxia (No. NZ17105), and the Science Foundation of North Minzu University (No. 2019XYZSX04).

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Lufeng Yang [ID] (1,2)

(1) School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

(2) School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China

Correspondence should be addressed to Lufeng Yang; ylf-sd@163.com

Received 16 September 2019; Accepted 23 October 2019; Published 14 November 2019

Caption: Figure 1: Relative maximum errors in Example 1. (a) [epsilon] = [10.sup.-1]. (b) [epsilon] = [10.sup.-4]. (c) [epsilon] = [10.sup.-6]. (d) [epsilon] = [10.sup.-8].

Caption: Figure 2: Numerical solutions vs. exact solutions ([u.sub.1]: solid lines and [u.sub.2]: dashed lines). (a) [epsilon] = [10.sup.-2] and N = 40. (b) [epsilon] = [10.sup.-6] and N = 60.

Caption: Figure 3: Pointwise relative errors in Example 1: RSCAT (dashed lines) vs. RSC-SSM (solid lines). (a) [epsilon] = [10.sup.-2] and N = 20. (b) [epsilon] = [10.sup.-6] and N = 60.

Caption: Figure 4: Relative maximum errors in Example 2: RSCAT (dashed lines) vs. RSC-SSM (solid lines). (a) [epsilon] = [10.sup.-2]. (b) [epsilon] = [10.sup.-6].

Caption: Figure 5: Numerical solutions vs. exact solutions. RSCAT (dashes lines) vs. RSC-SSM (solid lines). (a) [epsilon] = [10.sup.-2] and N = 20. (b) [epsilon] = [10.sup.-6] and N = 60.

Caption: Figure 6: Pointwise errors. RSCAT (dashes lines) vs. RSC-SSM method (solid lines).
```Table 1: Comparison of the relative maximum errors for Example 1.

RSC-SSM                    RSCAT

[epsilon]    N        u            v            u            v

1e - 2       20   5.19e - 05   4.14e - 05   1.61e - 03   1.29e - 03
1e - 4       40   2.33e - 12   4.10e - 12   6.74e - 06   5.70e - 06
1e - 6       60   7.49e - 14   2.04e - 13   1.05e - 07   1.06e - 07
1e - 8       60   1.50e - 15   8.66e - 16   5.63e - 06   4.77e - 06

Table 2: Comparison of the relative maximum errors for Example 2.

RSC-SSM

[epsilon]   N        u            v            w

1e - 2      20   3.39e - 15   1.55e - 15   3.11e - 15
1e - 4      40   3.12e - 14   9.77e - 15   3.52e - 14
1e - 6      60   4.33e - 13   1.31e - 13   2.67e - 13
1e - 8      60   6.87e - 12   1.23e - 12   3.18e - 12

RSCAT

[epsilon]       u            v            w

1e - 2      1.21e - 03   4.70e - 03   1.76e - 02
1e - 4      1.53e - 05   2.58e - 04   1.01e - 03
1e - 6      1.38e - 06   2.05e - 05   8.35e - 05
1e - 8      3.34e - 04   9.71e - 03   3.81e - 02
```
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