# Convergence and Stability in Collocation Methods of Equationu′(t)=au(t)+bu([t])

Authors: Han Yan; Shufang Ma; Yanbin Liu; Hongquan Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods foru′(t)=au(t)+bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.


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## Body

## 1. Introduction

This paper deals with the convergence, superconvergence, and stability of the collocation methods of the following differential equation with piecewise continuous argument (EPCA):(1.1)u′(t)=au(t)+bu([t]),t∈[0,T],u(0)=u0,
where T is an integer, a,b∈ℝ, u0∈ℂd is a given initial value, u(t)∈ℂd is an unknown function, and [·] denotes the greatest integer function. The general form of EPCA is
(1.2)u′(t)=f(t,u(t),u(α(t))),t≥0,u(0)=u0,
where the argument α(t) has intervals of constancy. This kind of equations has been initiated by Wiener [1, 2], Cooke and Wiener [3], and Shah and Wiener [4]. The general theory and basic results for EPCA have by now been thoroughly investigated in the book of Wiener [5].There are some authors who have considered the stability of numerical solutions for this kind of equations (see [6–8]). Though (1.1) is a delay differential equation (see [9–11]), the delay function t-[t] is discontinuous. In [12], the convergence and superconvergence of collocation methods for a differential equation with piecewise linear delays is concerned.Definition 1.1 (see Wiener [5]).
A solution of (1.1) on [0,∞) is a function u(t) that satisfies the following conditions.(1)
u(t) is continuous on [0,∞).(2)
The derivativeu'(t) exists at each point t∈[0,∞), with the possible exception of the point [t]∈[0,∞), where one-sided derivatives exist.(3)
(1.1) is satisfied on each interval [k,k+1)⊂[0,∞) with integral endpoints.Theorem 1.2 (see Wiener [5]).
Equation (1.1) has on [0,∞) a unique solution
(1.3)u(t)=m0({t})b0[t]u0,
where {t} is the fractional part of t and
(1.4)m0(t):=eat+(eat-1)a-1b,b0:=m0(1).
Equation (1.1) is asymptotically stable (the solution of (1.1) tends to zero as t→∞), for all u0, if and only if the inequalities
(1.5)-aea+1ea-1<-b<-a
hold.


## 2. Existence and Uniqueness of Collocation Methods

Leth:=1/p be a given step size with integer p≥1 and let the mesh on I be defined by
(2.1)Ih:={tn:0=t0<t1<⋯<tN=T}.
Accordingly, the collocation points are chosen as
(2.2)Xh:={tn,i=tn+cih:0<c1<⋯<cm≤1(0≤n≤N-1)},
where {ci} denotes a given set of collocation parameters.We approximate the solution by collocation in the piecewise polynomial spaces(2.3)Sm(0)([0,T]):={v∈C([0,T]):v|[tn,tn+1]∈πm},
where πm denotes the set of all real polynomials of degree not exceeding m. The collocation solution uh is the element in this space that satisfies the collocation equation
(2.4)uh′(t)=auh(t)+buh([t]),t∈Xh,uh(0)=u0.
Let Yn,j:=uh′(tn+cjh). Then
(2.5)uh′(tn+vh)=∑j=1m‍Lj(v)Yn,j,v∈(0,1],
where
(2.6)Lj(v):=∏i=1,i≠jm‍v-cicj-ci.
Integrating the above equality, we can get that
(2.7)uh(tn+vh)=uh(tn)+h∑j=1m‍βj(v)Yn,j,
where βj(v):=∫0v‍Lj(s)ds. So
(2.8)Yn,i=auh(tn,i)+buh([tn,i]).
Let n=kp+l, k∈ℤ, l=0,1,2,…,p-1. We have
(2.9)Ykp+l,i=auh(tkp+l,i)+buh(tkp)=a(uh(tkp+l)+h∑j=1m‍aijYkp+l,j)+buh(tkp),
where aij:=βj(ci).DenoteA=(aij)m×m, Yn=(Yn,1,Yn,2,…,Yn,m)T,  β=(β1,β2,…,βm)T, e=(1,1,…,1)T and for any xj∈ℝ, ∑j=0-1xj=0 if k=0. We have
(2.10)(Im×m-haA)Ykp+l=uh(tkp+l)ae+uh(tkp)be.
When the solution Yn of (2.10) has been found, the collocation solution on the interval [tn,tn+1] is determined by
(2.11)uh(tn+vh)=uh(tn)+hβT(v)Yn.So we can obtain the following theorem.Theorem 2.1.
Assume that the given functions in (1.1) satisfy a,b∈ℝ,K∈C(D), where D:={(t,s):0≤s≤t≤T}. Then there exists an h->0 so that for the mesh Ih with mesh diameter h>0 satisfying h<h-, and each of the linear algebraic systems (2.10) has a unique solution Yn∈ℝm. Hence the collocation of (2.4) defines a unique collocation solution uh∈Sm(0)(Ih) for the initial-value problem (1.1), and its representation on the subinterval [tn,tn+1] is given by (2.11).


## 3. Global Convergence Results

In the following, unless otherwise specified, the derivatives ofu and uh denote the left derivatives.Theorem 3.1.
Assume the following:(1)
the given functions in (1.1) satisfy a,b∈ℝ, K∈Cm(D);(2)
uh∈Sm(0)(Ih) is the collocation solution to (1.1) defined by (2.10) and (2.11) with h∈(0,h-). Then the estimates
(3.1)∥u(ν)-uh(ν)∥∞:=maxt∈[0,T]|u(ν)(t)-uh(ν)(t)|≤Cν∥u(m+1)∥∞hm(ν=0,1)
hold for any set Xh(k=1,2,…) of collocation points with 0<c1<⋯<cm≤1. The constants Cν dependent on the collocation parameters {ci} and but not on h.Proof.
The collocation erroreh:=u-uh satisfies the equation
(3.2)eh′(t)=aeh(t)+beh([t]),t∈Xh,
with eh(0)=0. Assumption (1) implies that u∈Cm+1([tn,tn+1]) (at tn, the derivative of u denotes the right derivative and at tn+1, which denotes the left derivative) and hence u′∈Cm([tn,tn+1]). Thus we have, using Peano's Theorem for u′ on [tn,tn+1],
(3.3)u′(tn+vh)=∑j=1m‍Lj(v)u′(tn,j)+hmRm+1,n(1)(v),v∈(0,1],
with the Peano remainder term, and Peano kernel are given by
(3.4)Rm+1,n(1)(v):=∫01‍Km(v,z)u(m+1)(tn+zh)dz,Km(v,z):=1(m-1)!{(v-z)+m-1-∑j=1m‍Lj(v)(cj-z)+m-1},v∈(0,1].
Integration of (3.3) leads to
(3.5)u(tn+vh)=u(tn)+h∑j=1m‍βj(v)u′(tn,j)+hm+1Rm+1,n(v),v∈(0,1],
where
(3.6)Rm+1,n(v):=∫0v‍Rm+1,n(1)(s)ds.
Recalling the local representation (2.5) of the collocation solution uh on (tn,tn+1] and setting εn,j:=u'(tn,j)-Yn,j, the collocation error eh:=u-uh on (tn,tn+1] may be written as
(3.7)eh(tn+vh)=eh(tn)+h∑j=1m‍βj(v)εn,j+hm+1Rm+1,n(v),v∈(0,1],
while
(3.8)eh′(tn+vh)=∑j=1m‍Lj(v)εn,j+hmRm+1,n(1)(v),v∈(0,1].
Since eh is continuous in [0,T], and hence at the mesh points, we also have the relation
(3.9)eh(tn)=eh(tn-1+h)=eh(tn-1)+h∑j=1m‍bjεn-1,j+hm+1Rm+1,n-1(1),n=1,…,N-1,
with bj:=βj(1). The fact that eh(0)=0 yields
(3.10)eh(tn)=h∑j=1m‍bj∑r=0n-1‍εr,j+hm+1∑r=0n-1‍Rm+1,r(1),n=1,…,N-1.
We are now ready to establish the estimates in Theorem 3.1. Let n=kp+l(l=0,1,…,p-1); since the collocation error satisfies
(3.11)eh′(tkp+l,i)=aeh(tkp+l,i)+beh(tkp),
it follows from (3.7) and (3.8) that
(3.12)εkp+l,i=eh′(tkp+l,i)=aeh(tkp+l,i)+beh(tkp)=a(eh(tkp+l)+h∑j=1m‍aijεkp+l,j+hm+1Rm+1,kp+l(ci))+beh(tkp).
Denote(3.13)εn:=(εn,1,εn,2,…,εn,m)T,Rm+1,n:=(Rm+1,n(c1),Rm+1,n(c2),…,Rm+1,n(cm))T,
we can get that
(3.14)(Im×m-haA)εkp+l=eh(tkp+l)ae+eh(tkp)be+ahm+1Rm+1,kp+l.
According to Theorem 2.1, this linear system has a unique solution whenever h∈(0,h-), and hence there exists a constant D0<∞ so that ∥(Im×m-hAn)-1∥1≤D0 uniformly for 0≤n≤N-1. Here, for B∈L(ℝm),∥B∥1 denotes the matrix (operator) norm induced by the l1-norm in ℝm. Denote Mm+1:=∥u(m+1)∥∞, Km:=maxv∈[0,1]∫01‍|Km(v,z)|dz,b-:=max1≤j≤m|bj|, and β-:=max1≤i≤m,v∈[0,1]βi(v). So
(3.15)|eh(tn)|≤hb-∑r=0n-1‍∥εr∥1+hmTKmMm+1.
Equation (3.14) now leads to the estimate
(3.16)∥εkp+l∥1≤D0{m|a|hb-∑r=0kp+l-1‍∥εr∥1+|a|mhmTKmMm+1+|b|mhb-∑r=0kp-1‍∥εr∥1+|b|mhmTKmMm+1+|a|hm+1mMm+1Km}≤γ0h∑r=0kp+l-1‍∥εr∥1+γ1Mm+1hm,
with obvious meanings of γ0 and γ1. By using the discrete Gronwall inequality, its solution is bounded by
(3.17)∥εn∥1≤γ1Mm+1hmexp(γ0T)=:BMm+1hm,
and so (3.15) yields
(3.18)|eh(tn)|≤(b-B+KmT)Mm+1hm.
Denote
(3.19)Λm:=max1≤j≤m,v∈[0,1]Lj(v),
we have
(3.20)|eh(tn+vh)|≤(b-B+KmT)Mm+1hm+hβ-BMm+1hm+hm+1Mm+1Km=:C0Mm+1hm,|eh′(tn+vh)|≤Λm∥εn∥1+hmKmMm+1≤ΛmBMm+1hm+hmKmMm+1=:C1Mm+1hm.
This concludes the proof of Theorem 3.1.


## 4. Global Superconvergence Results

Theorem 4.1.
Assume that the assumptions(2) of Theorem 3.1 hold, and let (1) be replaced by a,b∈Cd(I) and K∈Cd(D), with d≥m+1. If the m collocation parameters {ci} are subject to the orthogonality condition
(4.1)J0:=∫01‍∏i=1m‍(s-ci)ds=0,
then the corresponding collocation solution uh∈Sm(0)(Ih) satisfies, for h∈(0,h-),
(4.2)∥u-uh∥∞≤C2hm+1,
with C2 depending on the collocation parameters and on ∥u(m+2)∥∞ but not on h. The exponent m+1 cannot, in general, be replaced by m+2. For the derivative uh′, we attain only ∥u'-uh′∥∞=O(hm).Proof.
Let(4.3)δh(t):=-uh′(t)+auh(t)+buh([t]),t∈I,
denote the defect (or: residual) associated with the collocation solution uh∈Sm(0)(Ih) to the initial-value problem (1.1). by definition of the collocation solution the defect δh vanishes on the set Xh as follows:
(4.4)δh(t)=0∀t∈Xh.
Moreover, the uniform convergence of uh and uh′ established in Theorem 3.1 implies the uniform boundedness (as h→0) of δh on I, as well as that of its derivatives of order not exceeding d (here the derivatives refer to the left derivatives).
It following from (4.3) that the collocation error eh=u-uh satisfies the equation
(4.5)δh(t)=eh′(t)-aeh(t)-beh([t]),t∈I.
By Theorem 3.1, there exists a constant D, such that
(4.6)∥δh(t)∥∞≤DhmMm+1,
and this holds for any choice of the {ci}. On the other hand, the collocation error eh solves the initial-value problem
(4.7)eh′(t)=aeh(t)+beh([t])+δh(t),t∈I,eh(0)=0.
For t∈[k,k+1], whose solution is given by
(4.8)eh(t)=[r(t,k)+∫kt‍br(t,s)ds]eh(k)+∫kt‍r(t,s)δh(s)ds,t∈I.
The function r=r(t,s) denotes the “resolvent” (or: resolvent kernel) of (1.1) as follows:
(4.9)r(t,s):=ea(t-s),withr∈Cm+1(D).
If k=0, let t=tl+vh, v∈[0,1], and 0≤l≤p-1; we have
(4.10)eh(tl+vh)=∫0tl+vh‍r(tl+vh,s)δh(s)ds=∑j=0l-1‍∫tjtj+1‍r(tl+vh,s)δh(s)ds+∫tltl+hv‍r(tl+vh,s)δh(s)ds=h∑j=0l-1‍∫01‍r(tl+vh,tj+hs)δh(tj+hs)ds+h∫0v‍r(tl+vh,tl+hs)δh(tl+hs)ds.
Suppose now that each of the integrals over [0,1] is approximated by the interpolatory m-point quadrature formula with abscissas {ci}, then
(4.11)∫01‍r(tl+vh,tj+hs)δh(tj+hs)ds=∑i=1m‍bjr(tl+vh,tj+hci)δh(tj+hci)+Ej(v),v∈[0,1].
Here, terms Ej(v) denote the quadrature errors induced by these quadrature approximations. By assumption (4.1) each of these quadrature formulas has degree of precision m, and thus the Peano Theorem for quadrature implies that the quadrature errors can be bounded by
(4.12)|Ej(v)|≤Qhm+1,v∈[0,1],
because the defect δh is in Cm+1 on each subinterval [tn,tn+1]. Due to the special choice of the quadrature abscissas, we have ∑i=1mbjr(tl+vh,tj+hci)δh(tj+hci)=0, because δh(t)=0 whenever t∈Xh. Hence
(4.13)eh(tl+vh)=h∑j=0l-1‍Ej(v)+h∫0v‍r(tl+vh,tl+hs)δh(tl+hs)ds,v∈[0,1].
This leads to the estimate
(4.14)|eh(tl+vh)|≤h∑j=0l-1‍Qhm+1+hr0∥δh∥∞≤QThm+1+Dr0hm+1Mm+1=:C-0hm+1,
for 0≤l≤p-1 and v∈[0,1], with r0:=maxt∈I∫0T‍|r(t,s)|ds.
We assume fort∈[k-1,k](4.15)|eh(t(k-1)p+l+vh)|≤C-k-1hm+1,v∈[0,1],0≤l≤p-1.
Then for t∈[k,k+1], let t=tkp+l+vh,v∈[0,1], and 0≤l≤p-1; we have
(4.16)eh(tkp+l+vh)=[r(tkp+l+vh,k)+∫ktkp+l+vh‍br(tkp+l+vh,s)ds]eh(k)+∫ktkp+l+vh‍r(tkp+l+vh,s)δh(s)ds=[r(tkp+l+vh,k)+∫ktkp+l+vh‍br(tkp+l+vh,s)ds]eh(k)+h∑j=kpkp+l-1‍∫01‍r(tkp+l+vh,tj+hs)δh(tj+hs)ds+h∫0v‍r(tkp+l+vh,tkp+l+hs)δh(tkp+l+hs)ds.
Similarly to the case of t∈[0,1], we have
(4.17)|eh(tkp+l+vh)|≤(r0+r0|b|)C-k-1hm+1+pQhm+2+r0DMm+1hm+1=:C-khm+1.
This completes the proof.


## 5. The Local Superconvergence Results onIh

Theorem 5.1.
Assume the following:(a)
a,b∈Cm+κ(I) and K∈Cm+κ(D), for some κ with 1≤κ≤m and value as specified in (b) below,(b)
Them distinct collocation parameters {ci} are chosen so that the general orthogonality condition
(5.1)Jν:=∫01‍sν∏i=1m‍(s-ci)ds=0,ν=0,…,κ-1holds, with Jκ≠0.
Then, for all meshesIh with h∈(0,h-), the collocation solution uh∈Sm(0)(Ih) corresponding to the collocation points Xh based on these {ci} satisfies
(5.2)max{|u(t)-uh(t)|:t∈Ih}≤C3hm+κ,
where C3 depends on the collocation parameters and on ∥u(m+κ+1)∥∞ but not on h.Proof.
Ifk=0, for t=tl(0≤l≤p-1)(5.3)eh(tl)=∫0tl‍r(tl,s)δh(s)ds=∑j=0l-1‍∫tjtj+1‍r(tl+vh,s)δh(s)ds=h∑j=0l-1‍∫01‍r(tl+vh,tj+hs)δh(tj+hs)ds=h∑j=0l-1‍(∑i=1m‍bjr(tl+vh,tj+hci)δh(tj+hci)+Ej(v)),
with
(5.4)|Ej(v)|≤Chm+κ,
so
(5.5)|eh(tl)|≤Chm+κ.
By the induction method similarly to the proof of Theorem 4.1, the assertion of Theorem 5.1 follows.


## 6. Numerical Stability

In this section, we will discuss the stability of the collocation methods. We introduce the setH consisting of all pairs (a,b)∈ℝ2 which satisfy the condition
(6.1)H:={(a,b):-aea+1ea-1<b<-a},
and divide the region into three parts:
(6.2)H0:={(a,b):(a,b)∈H,a=0},H1:={(a,b):(a,b)∈H,a<0},H2:={(a,b):(a,b)∈H,a>0}.
By (2.9) and (2.10), we can obtain that
(6.3)u(tkp+l+1)=R(x)u(tkp+l)+α(x,y)u(tkp),l=0,1,…,p-1,
where x:=ha,y:=hb, R(x):=1+bTx(I-Ax)-1e, and α(x,y):=y(1+xbT(I-Ax)-1e)=ybT(I-Ax)-1e.LetUk:=(ukp,ukp+1,…,ukp+p)T and B:=∏i=1pBi. It is easy to see
(6.4)Uk=BUk+1,k=1,2,…,
where
(6.5)B=(0⋯0b1,p+10⋯0b2,p+1⋯⋯⋯⋯0⋯0bp+1,p+1),bi,p+1={1+(1+ab)[R(x)i-1-1],a≠0,1+(i-1)hb,a=0.i=1,2,…,p+1.Letφ(x):=bT(I-xA)-1e. Then there exists δ>0 such that
(6.6)φ(x)>0∀xwith|x|≤δ,
since φ(0)=1 and φ(x) is continuous in a neighborhood of zero. In the rest of the paper we define
(6.7)M:={1,a≤0,aδ,a>0.Definition 6.1 (see [6]).
Process (2.11) for (1.1) is called asymptotically stable at (a,b) if and only if for all m≥M and h=1/m.  (i)
(I-xA) is invertible.(ii)
for any givenui(1≤i≤m) relation (6.4) defines Uk(k=1,2,…) that satisfy Uk→0 for k→∞.Definition 6.2 (see [6]).
The set of all pairs(a,b) at which the process (2.11) for (1.1) is asymptotically stable is called asymptotical stability region denoted by S.Theorem 6.3 (see [6]).
Suppose that the collocation method isA0-stable and the stability function is given by the (r,s)-Padé approximation to the exponential ex. Then H1⊆S if and only if r is even.Theorem 6.4 (see [6]).
Suppose that the stability function of the collocation method is given by the(r,s)-Padé approximation to the exponential ez. Then H2⊆S if and only if s is even.Theorem 6.5 (see [6]).
For all the collocation methods, we haveH0⊆S.Using the above theorems we can formulate the following result.Theorem 6.6 (see [6]).
Suppose that the collocation method isA0-stable and the stability function is given by the (r,s)-Padé approximation to the exponential ex. Then H0⊆S and H⊆S if and only if both r and s are even,
(6.8)H1⊆Siffr,H2⊆Siffsis  even.Corollary 6.7.
For theA-stable higher order collocation methods, it is easy to see from Theorem 6.6.  (i)
For theν-stage Gauss-Legendre method, H⊆S if and only if ν is even.(ii)
For theν-stage Lobatto IIIA method, H⊆S if and only if ν is old.(iii)
For theν-stage Radau IIA method, H1⊆S if and only if ν is old and H2⊆S if and only if ν is even.


## 7.  Numerical Experiments

In order to give a numerical illustration to the conclusions in the paper, we consider the following two problems ([6]):
(7.1)u1′(t)=-20u1(t)-10.3u1([t]),u1(0)=1,(7.2)u2′(t)=10u2(t)-10.0001u2([t]),u2(0)=1.
It can be checked that (-20,-10.3)∈H1 and (10,-10.0001)∈H2.For illustrating the convergence and superconvergence orders in this paper, we choosem=2 and m=3 and use the Gauss collocation parameters: c1=(3-3)/6, c2=(3+3)/6, the Radau IIA collocation parameters: c1=1/3, c2=1, the Lobatto IIIA collocation parameters: c1=0, c2=1, and three sets of random collocation parameters: c1=1/4, c2=1/2; c1=1/3, c2=2/3; c1=2/3, c2=1, respectively, for m=2; and we use the Gauss collocation parameters: c1=(5-15)/10, c2=1/2, and c3=(5+15)/10, the Radau IIA collocation parameters: c1=(4-6)/10, c2=(4+6)/10, and c3=1, the Lobatto IIIA collocation parameters: c1=0, c2=1/2, and c3=1, and three sets of random collocation parameters: c1=1/3, c2=1/2, c3=2/3; c1=1/4, c2=1/3, c3=1/2; c1=1/2, c2=2/3, c3=1, respectively, for m=3. In Tables 1, 2, 3, and 4 we list the absolute values of the absolute errors of ut=10 for the six collocation parameters and for m=2 and m=3, respectively, and the ratios of the absolute values of the errors of N=100 over that of N=200.Table 1
The absolute values of absolute errors ofuh for example (7.1) with m=2.
N
Gauss
Radau IIA
Lobatto IIIA
(
1
/
4
,
1
/
2
)
(
1
/
3
,
2
/
3
)
(
2
/
3
,
1
)
2
1
1.2551
e
-
03
3.1476
e
-
04
1.3124
e
-
03
2.9655
e
+
02
1.3113
e
-
03
9.3953
e
-
06
2
2
4.5928
e
-
06
1.4601
e
-
06
8.5094
e
-
04
1.2874
e
-
03
6.4778
e
-
05
8.1731
e
-
10
2
3
2.4209
e
-
10
7.8934
e
-
11
8.1731
e
-
10
8.5023
e
-
07
1.0324
e
-
08
9.3978
e
-
10
2
4
6.1162
e
-
12
2.8697
e
-
11
7.7079
e
-
11
5.7012
e
-
10
1.3715
e
-
10
1.8633
e
-
10
2
5
3.4588
e
-
13
4.5808
e
-
12
3.9746
e
-
11
4.1164
e
-
11
2.0163
e
-
11
4.3362
e
-
11
2
6
2.1210
e
-
14
6.2301
e
-
13
1.2117
e
-
11
7.6587
e
-
12
4.4835
e
-
12
1.1275
e
-
11
Ratio
1.6307
e
+
01
7.3526
e
+
00
3.2802
e
+
00
5.3747
e
+
00
4.4973
e
+
00
3.8460
e
+
00Table 2
The absolute values of absolute errors ofuh for example (7.1) with m=3.
N
Gauss
Radau IIA
Lobatto IIIA
(
1
/
3
,
1
/
2
,
2
/
3
)
(
1
/
4
,
1
/
3
,
1
/
2
)
(
1
/
2
,
2
/
3
,
1
)
2
1
3.1476
e
-
04
9.9709
e
-
05
1.2551
e
-
03
1.2301
e
-
03
4.9431
e
+
04
4.8903
e
-
06
2
2
3.2248
e
-
11
1.6050
e
-
08
4.5928
e
-
06
2.6790
e
-
06
9.5944
e
-
04
3.2687
e
-
09
2
3
4.7508
e
-
12
2.0843
e
-
11
2.4209
e
-
10
6.7725
e
-
11
7.9577
e
-
11
1.1031
e
-
10
2
4
6.3943
e
-
14
5.8750
e
-
13
6.1162
e
-
12
6.7093
e
-
12
2.9944
e
-
11
1.6213
e
-
11
2
5
9.5085
e
-
16
1.9290
e
-
14
3.4588
e
-
13
4.1702
e
-
13
3.1783
e
-
12
2.5836
e
-
12
2
6
1.0192
e
-
17
6.3187
e
-
16
2.1210
e
-
14
2.5846
e
-
14
3.3739
e
-
13
3.7792
e
-
13
Ratio
9.3298
e
+
01
3.0528
e
+
01
1.6307
e
+
01
1.6135
e
+
01
9.4201
6.8363
e
+
00Table 3
The absolute values of absolute errors ofuh for example (7.2) with m=2.
N
Gauss
Radau IIA
Lobatto IIIA
(
1
/
4
,
1
/
2
)
(
1
/
3
,
2
/
3
)
(
2
/
3
,
1
)
2
1
9.0789
e
-
01
9.1680
e
-
01
9.1647
e
-
01
9.0070
e
-
01
9.1460
e
-
01
9.1700
e
-
01
2
2
2.3033
e
-
01
8.6133
e
-
01
4.2429
e
-
01
5.4055
e
-
01
7.4023
e
-
01
9.1661
e
-
01
2
3
8.9360
e
-
03
1.2191
e
-
01
8.3085
e
-
02
1.3822
e
-
01
1.5563
e
-
01
7.6499
e
-
01
2
4
5.0981
e
-
04
9.9479
e
-
03
5.9089
e
-
02
3.4017
e
-
02
2.8752
e
-
02
1.5140
e
-
01
2
5
3.1272
e
-
05
1.0927
e
-
03
1.8134
e
-
02
8.7800
e
-
03
6.5674
e
-
03
2.5303
e
-
02
2
6
1.9458
e
-
06
1.2993
e
-
04
4.7169
e
-
03
2.2713
e
-
03
1.6041
e
-
03
5.3924
e
-
03
Ratio
1.6071
e
+
01
8.4098
e
+
00
3.8445
e
+
00
3.8656
e
+
00
4.0940
e
+
00
4.6924
e
+
00Table 4
The absolute values of absolute errors ofuh for example (7.2) with m=3.
N
Gauss
Radau IIA
Lobatto IIIA
(
1
/
3
,
1
/
2
,
2
/
3
)
(
1
/
4
,
1
/
3
,
1
/
2
)
(
1
/
2
,
2
/
3
,
1
)
2
1
4.8459
e
-
02
9.0087
e
-
01
9.0789
e
-
01
9.0508
e
-
01
7.4280
e
-
02
9.1693
e
-
01
2
2
7.0515
e
-
03
5.0901
e
-
02
2.3033
e
-
01
8.2874
e
-
02
2.5303
e
-
02
8.3086
e
-
02
2
3
9.4311
e
-
05
1.3131
e
-
03
8.9360
e
-
03
9.9825
e
-
03
1.2122
e
-
02
6.3787
e
-
02
2
4
1.4090
e
-
06
3.5203
e
-
05
5.0981
e
-
04
6.1535
e
-
04
2.2659
e
-
03
7.4707
e
-
03
2
5
2.1766
e
-
08
1.0294
e
-
06
3.1272
e
-
05
3.8124
e
-
05
3.4308
e
-
04
7.9289
e
-
04
2
6
3.3592
e
-
10
3.1217
e
-
08
1.9458
e
-
06
2.3768
e
-
06
4.7157
e
-
05
9.0681
e
-
05
Ratio
6.4794
e
+
01
3.2977
e
+
01
1.6071
e
+
01
1.6040
e
+
01
7.2752
e
+
00
8.7437
e
+
00From the above tables, we can see that the convergence orders are consistent with our theoretical analysis.In Figures1, 2, 3, 4, 5, 6, 7, and 8, we draw the absolute values of the numerical solution of collocation methods. It is easy to see that the numerical solution is asymptotically stable.Figure 1
The Gauss collocation method withm=2 and p=50 for (7.1).Figure 2
The Radau IIA collocation method withm=2 and p=100 for (7.1).Figure 3
The Gauss collocation method withm=3 and p=50 for (7.1).Figure 4
The Radau IIA collocation method withm=3 and p=1000 for (7.1).Figure 5
The Gauss collocation method withm=2 and p=50 for (7.2).Figure 6
The Radau IIA collocation method withm=2 and p=1000 for (7.2).Figure 7
The Gauss collocation method withm=3 and p=50 for (7.2).Figure 8
The Radau IIA collocation method withm=3 and p=1000 for (7.2).



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