ε | N=128([1]) | N=128([9]) | N=128(Our method) |
2−8 | 3.68×10−3 | 3.33×10−4 | 3.88×10−5 |
2−10 | 8.26×10−3 | 4.70×10−3 | 3.86×10−5 |
2−12 | 8.01×10−3 | 8.20×10−2 | 3.85×10−5 |
2−16 | – | – | 3.85×10−5 |
2−20 | – | – | 3.85×10−5 |
2−24 | – | – | 3.85×10−5 |
Citation: F. Z. Geng. Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems[J]. AIMS Mathematics, 2020, 5(6): 6020-6029. doi: 10.3934/math.2020385
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In this study, we focus on the singular perturbation problems (SPPs) with turning points
{εv′′(x)+a0(x)v′(x)+a1(x)v(x)=g(x),v(0)=τ0,v(1)=τ1, | (1.1) |
where 0≤x≤1, 0<ε≪1, a0(γ)=0 for a point in interval (0,1).
The points such that a0(x)=0 are called turning points. Usually, turning points problems lead to boundary layers or interior layers. And therefore, for SPPs with turning points, the numerical solutions is a more challenging job. Under certain assumptions in [1], problem (1) has a unique solution with two boundary layers at left and right end points of the interval [0,1]. The goal of the work is to present an accurate symmetric collocation scheme for this class of SPPs.
SPPs arise in the physical theory of semiconductors, chemical reactions, biological sciences, etc. Solutions of SPPs, unlike regular problems, have boundary or interior layers. The derivatives of the solution in these subintervals grow without bound as ε→0. For SPPs, the use of classical numerical techniques for regular problems can not lead to accurate numerical solutions. In the last decade, there have been many effective numerical methods for SPPs [1,2,3,4,5,6,7,8,9,10,11,12]. However, most of the existing work has considered SPPs without tuning points, while much less is known on SPPs with tuning points. In Ref. [1,4,5,6,7,8], some existing numerical techniques have been proposed for SPPs with turning points.
Reproducing kernel method (RKM) is a powerful technique for operator equations. Over the last decades, the method has achieved important developments in fractional differential equations, singular integral equations, fuzzy differential equations, and so on [8,9,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Unfortunately, the direct employ of the method can not produce ideal approximate solutions to SPPs with turning points. In [8], Geng and Qian based on the RKM and the idea of domain decomposition, present an effective method for SPPs with turning points and having two boundary layers. However, the implement of the method is complicated. In this paper, we shall construct a piecewise reproducing kernel function (RKF) based on the inverse multiquadrics kernel functions and the exponential kernel functions. Then we present a symmetric collocation method for SPPs with turning points.
In this section, following the idea in [20], we will introduce a piecewise reproducing kernel function which is used to deal with the character of solutions to SPPs (1).
Definition 2.1. A Hilbert space H consists of complex value functions defined on E, is named a reproducing kernel Hilbert space (RKHS) if for each x∈E, the evaluation functional ex(f)=f(x) is continuous.
The application of Riesz Representation Theorem for Hilbert space yields the following theorem.
Theorem 2.1. For the above space H and each x∈E, there exists a unique Kx∈H such that (f,Kx)=f(x) for all f∈H.
Definition 2.2. Let H be a Hilbert function space defined in E. The function K:E×E→R defined by K(x,y)=Kx(y) is known as a RKF of space H.
Definition 2.3. For asymmetric function K:E×E→R, it is known as a positive definite kernel (PDK) if for any n∈N, x1,x2,…,xn∈E,c1,c2,…,cn∈R, n∑i,j=1cicjK(xi,xj)≥0.
Theorem 2.2. The RKF of a RKHS is positive definite. Also, every PDK can define a unique RKHS, of which it is the RKF.
Theorem 2.3. Both inverse multiquadrics kernel function IMQ(x,y)=1√(x−y)2+γ2 and exponential kernel function G(x,y)=eαxy(α>0) are strictly positive definite.
By Theorem 2.2, IMQ1(x,y)=IMQ(x,y),x,y∈[0,δ1] is the RKF of a RKHS H1[0,δ1], G(x,y),x,y∈[δ1,δ2] is the RKF of a RKHS H2[δ1,δ2] and IMQ2(x,y)=IMQ(x,y),x,y∈[δ2,1] is the RKF of a RKHS H3[δ2,1].
Define piecewise kernel function
K(x,y)={IMQ1(x,y),x,y∈[0,δ1),G(x,y),x,y∈[δ1,δ2],IMQ2(x,y),x,y∈(δ2,1],0,others. | (2.1) |
and Hilbert functions space
H[0,1]={w(x)|w(x)={w1(x),x∈[0,δ1),w2(x),x,y∈[δ1,δ2],w3(x),x,y∈(δ2,1].w1∈H1[0,δ1],w2∈H2[δ1,δ2],w3∈H3[δ2,1]}. | (2.2) |
The inner product in this space is
(u,v)H=(u1,v1)H1+(u2,v2)H2+(u3,v3)H3. |
Theorem 2.4. H[0,1] is a RKHS with RKF K(x,y).
Proof. We will prove K(x,y) given in (2) is a RKF. For each v(y)∈H, if x∈[0,δ1),
(v(y),K(x,y))H=(v1(y),IMQ1(x,y))H1+(v2(y),0)H2+(v3(y),0)H3=v1(x). |
Similarly, if x∈[δ1,δ2],
(v(y),K(x,y))H=v2(x). |
And if x∈(δ2,1],
(v(y),K(x,y))H=v3(x). |
This follows that (v(y),K(x,y))H=v(x), for x∈[0,1]. Therefore, H[0,1] is a RKHS with RKF K(x,y).
Setting δ1=¯Kε and δ2=1−¯Kε, the interval [0,1] is then divided into three subintervals [0,δ1], [δ1,δ2] and [δ2,1], where [0,δ1] and [δ2,1] are the boundary layers regions, and [δ1,δ2] is the regular region. Eq.(1) is equivalent to
{εv′′(x)+a0(x)v′(x)+a1(x)v(x)=g(x),v(0)=τ0,v(δ−1)−v(δ+1)=0,v′(δ−1)−v′(δ+1)=0,v(δ−2)−v(δ+2)=0,v′(δ−2)−v′(δ+2)=0,v(1)=τ1. | (3.1) |
Define Lv=εv′′+a0(x)v′+a1(x)v, B1v=v(0),B2v=v(δ−1)−v(δ+1), B3v=v′(δ−1)−v′(δ+1), B4v=v(δ−2)−v(δ+2), B5v=v′(δ−2)−v′(δ+2), B6v=v(1).
Choosing N distinct scattered points in [0,1], such as {x1,x2,…,xN}. We construct basis functions by the piecewise RKF K(x,y). Let ψi(x)=LyK(x,y)|y=xi,i=1,2,…,N, ψN+j(x)=Bj,yK(x,y),j=1,2…,6, here Bj,y denotes operator Bj acts on the function of y. The collocation solution vN(x) for Eq. (1) may be written as
vN(x)=N+6∑i=1βiψi(x), | (3.2) |
where {βi}N+6i=1 are constants to be determined.
We require vN(x) to satisfy the governing differential equation at all centers xj,j=1,2,…,N, i.e.
LvN(xk)=N+6∑i=1βiLψi(xk)=g(xk),k=1,2,…,N. | (3.3) |
In addition, let vN(x) satisfy boundary conditions
{B1vN(x)=N+6∑i=1βiB1ψi(x)=τ0,B2vN(x)=N+6∑i=1βiB2ψi(x)=0,B3vN(x)=N+6∑i=1βiB3ψi(x)=0,B4vN(x)=N+6∑i=1βiB4ψi(x)=0,B5vN(x)=N+6∑i=1βiB5ψi(x)=0,B6vN(x)=N+6∑i=1βiB6ψi(x)=τ1. | (3.4) |
System (6) and (7) of linear equations can be reduced to the matrix form:
Aβ=g, | (3.5) |
where β=(β1,β2,…,βN+6)⊤, g=(g(x1),g(x2),…,g(xN),τ0,0,0,0,0,τ1)⊤. Clearly, A is a symmetric matrix.
Define functionals λi=δxi∘L,i=1,2,…,N, λN+j=Bj,j=1,2,…,6, i.e. λi(v)=Lv(xi),i=1,2,…,N, λN+j(v)=Bj(v),j=1,2,…,6.
Theorem 3.1. If the functionals λi,i=1,2,…,N+6 are linearly independent, then the matrix A is invertible. That is, system (8) has a unique solution.
The present method is outlined as follows:
Step 1: Construct basis functions ψi(x) using the kernel function K(x,y), operator L and Bj,j=1,2,…,6.
Step 2: Set collocation approximation vN(x)=N+6∑i=1βiψi(x).
Step 3: Let vN satisfies the equation and boundary conditions in (4).
Step 4: Obtain the coefficients in vN and then get approximate solution vN.
Theorem 3.2. If a0(x),a1(x) and g(x)∈C2[0,1], then
∥v(x)−vN(x)∥∞=maxx∈[0,1]∣v(x)−vN(x)∣≤ch2, |
where c>0 is a real number and h=max1≤i≤N−1∣xi−xi+1∣.
Proof. Because vN(x)∈H[0,1] and satisfies formula (8), we have
vN(x)∈C1[0,1]. |
Although vN(x) only belongs to C1 on the entire interval [0,1], from the assumptions of the Theorem and the regularity of functions in H1[0,δ1],H2[δ1,δ2] and H3[δ2,1], we have
LvN(x)−g(x)∈C2 |
on three subintervals [0,δ1], [δ1,δ2] and [δ2,1]. respectively.
By applying [28], ∣v(x)−vN(x)∣≤ch2 on three subintervals. Hence,
∥v(x)−vN(x)∥∞=maxx∈[0,1]∣v(x)−vN(x)∣≤ch2. |
Define the maximum absolute errors EN=∥v−vN∥∞≜maxx∈[0,1]∣v(x)−vN(x)∣.
Example 1
We applied our method to the SPPs given by [1,9]
εv′′(x)−2(2x−1)v′(x)−4v(x)=0,x∈(0,1), | (4.1) |
with the boundary conditions v(0)=v(1)=1. Its true solution is v(x)=e−2x(x−1)ε. Choosing δ1=¯Kε,δ2=1−¯Kε,¯K=10, γ=2ε,α=1, xi={δ1∗(i−1)N1,i=1,2,…,N1,δ1+(δ2−δ1)∗(i−N1)N2−N1+1,i=N1+1,…,N2,δ2+(1−δ2)∗(i−N2)N−N2,i=N2+1,…,N,(N1=59,N2=69,N=128) in the present method, Table 1 shows the maximum absolute errors compared with the methods in [1,9]. If the zero mean normally distributed white noise with standard derivation of 0.01 is added to the right function g(x)=0, the obtained errors are depicted in Figure 1. It is found from Table and Figure 1 that our novel approach has higher accuracy and is robust.
ε | N=128([1]) | N=128([9]) | N=128(Our method) |
2−8 | 3.68×10−3 | 3.33×10−4 | 3.88×10−5 |
2−10 | 8.26×10−3 | 4.70×10−3 | 3.86×10−5 |
2−12 | 8.01×10−3 | 8.20×10−2 | 3.85×10−5 |
2−16 | – | – | 3.85×10−5 |
2−20 | – | – | 3.85×10−5 |
2−24 | – | – | 3.85×10−5 |
Example 2 Next, we consider the SPPs given by [1,9]
{εv′′(x)−2(2x−1)v′(x)−4v(x)=g(x),x∈(0,1),v(0)=v(1)=1, | (4.2) |
where g(x)=4(4x−1). Its exact solution v(x)=−2x+(erf(2x−1√2√ε)erf(1√2√ε)+2)e−2(1−x)xε. Taking δ1=¯Kε,δ2=1−¯Kε,¯K=10, γ=2ε,α=1, xi={δ1∗(i−1)N1,i=1,2,…,N1,δ1+(δ2−δ1)∗(i−N1)N2−N1+1,i=N1+1,…,N2,δ2+(1−δ2)∗(i−N2)N−N2,i=N2+1,…,N, (N1=59,N2=69,N=128) in the present method, Table 2 shows the maximum absolute errors compared with the methods in [1,9]. Figure 2 shows the absolute errors on the left boundary layer for ε=10−6. Figure 3 shows the absolute errors on the right boundary layer for ε=10−6. Figure 4 shows the absolute errors on the regular interval for ε=10−6.
ε | N=256([1]) | N=256([9]) | N=128(Our method) |
2−8 | 1.33×10−3 | 2.26×10−4 | 1.29×10−5 |
2−12 | 1.22×10−3 | 5.02×10−4 | 1.28×10−5 |
2−16 | – | – | 1.28×10−5 |
2−20 | – | – | 1.28×10−5 |
2−24 | – | – | 1.28×10−5 |
By using the inverse multiquadrics RKF and the exponential RKF, a new piecewise RKF is builded. Take advantage of the piecewise RKFs, a new symmetric collocation scheme is presented for linear stationary SPPs. The significant advantage of the present novel technique is that it can adapt to the character of boundary layers of the solution to SPPs. Our method can give accurate approximation to the solutions of the considered SPPs on both the boundary intervals and the regular interval. The results of numerical tests illustrate that the method is uniformly effective for any small ε.
This research was financially supported by the National Natural Science Foundation of China (No.11271100) and China Postdoctoral Science Foundation (2019M651765). The authors was also supported by Natural Science Foundation for the Youth of Jiangsu Province (No.BK20181030).
All authors declare no conflicts of interest in this paper.
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ε | N=128([1]) | N=128([9]) | N=128(Our method) |
2−8 | 3.68×10−3 | 3.33×10−4 | 3.88×10−5 |
2−10 | 8.26×10−3 | 4.70×10−3 | 3.86×10−5 |
2−12 | 8.01×10−3 | 8.20×10−2 | 3.85×10−5 |
2−16 | – | – | 3.85×10−5 |
2−20 | – | – | 3.85×10−5 |
2−24 | – | – | 3.85×10−5 |
ε | N=256([1]) | N=256([9]) | N=128(Our method) |
2−8 | 1.33×10−3 | 2.26×10−4 | 1.29×10−5 |
2−12 | 1.22×10−3 | 5.02×10−4 | 1.28×10−5 |
2−16 | – | – | 1.28×10−5 |
2−20 | – | – | 1.28×10−5 |
2−24 | – | – | 1.28×10−5 |