International Journal of Analysis and Applications

Volume 16, Number 1 (2018), 1-15

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-1

CHEBYSHEV RATIONAL APPROXIMATIONS FOR THE ROSENAU-KDV-RLW

EQUATION ON THE WHOLE LINE

MOHAMMADREZA FOROUTAN∗ AND ALI EBADIAN

Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran

∗Corresponding author: foroutan−mohammadreza@yahoo.com

Abstract. In this paper, we consider the use of a modified Chebyshev rational approximations for the

Rosenau-KdV-RLW equation on the whole line with initial-boundary values. It is shown that the proposed

scheme leads to optimal error estimates. Furthermore, the stability and convergence of the proposed schemes

are proved. The fully discrete Chebyshev pseudo-spectral scheme is constructed. Numerical results confirm

well with the theoretical results. The idea and techniques presented in this paper will be useful to solve

many other problems.

1. Introduction

The application of spectral methods for approximating solutions of partial differential equations in un-

bounded domains has achieved great success and popularity in recent years. As a case in point, we can refer

to the book by Shen et al. [32] and a more recent research paper by Foroutan et al. [18]. In general, spectral

methods used for solving partial differential equations on unbounded domains can be classified into three

families.

The first family is to use spectral methods associated with some orthogonal systems such as the Hermite

spectral method and Laguerre spectral method (see e.g. Parand and Taghavi [22], Guo [12] and Parand et

al. [21]).

The second family replaces infinite domain with [-L,L] and semi-infinite interval with [0,L] by choosing L,

2010 Mathematics Subject Classification. 65M12, 41A20, 35Q53.

Key words and phrases. Error estimate; modified Chebyshev rational approximation; spectral method; stability of the

scheme.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

1

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-1


Int. J. Anal. Appl. 16 (1) (2018) 2

sufficiently large. This method is named the domain truncation [5].

The third family, that is used in this paper too, is based on rational approximations. For example, Boyd [6,7]

and Christov [8] developed some spectral methods on infinite intervals by using mutually orthogonal sys-

tems of rational functions. This family of spectral schemes is efficient specially for solving boundary value

problems; see also [14–17, 34]. Overall, rational spectral methods are highly flexible, but it is hard to obtain

convergence results and error estimates for those rational spectral methods. To this end, we apply conver-

gence and error estimates in the sense of Guo [11, 13]. The purpose of this paper is to develop and analyze

the modified Chebyshev rational spectral methods for Rosenau-KdV-RLW equation

vt −vxxt + vxxxxt + vxxx + vx + vvx = 0, (x,t) ∈ Ω × (0,T], (1.1)

with the initial condition:

v(x, 0) = v0(x), x ∈ Ω, (1.2)

and the boundary conditions:

lim
|x|→∞

v(x,t) = lim
|x|→∞

vx(x,t) = lim
|x|→∞

vxx(x,t) = 0, t ∈ [0,T], (1.3)

where v0(x) is a known smooth function and Ω = (−∞,∞).

The nonlinear wave is one of the most widely researched areas. The dynamics of wave behaviors can be

described by several models. Some of these well-known models are Korteweg-de Vries (KdV) equation,

regularized long wave (RLW) equation, and Rosenau equation. In the following section, we address a short

review of these important wave models.

Korteweg-de Vries (KdV) equation as one of the well-known equations in mathematics and physics:

vt + vxxx + 6vvx = 0.

This equation has been applied in many various fields and its application for describing wave propagation

and interaction has been studied widely. There are many numerical methods that can be used to solve KdV

equation such as the modified Legendre rational spectral methods applied on the semi-infinite interval [14],

explicit scheme [20], Petrov-Galerkin method on the half line [36], finite-difference method [9], and solitary

wave solution [1, 3].

The regularized long-wave (RLW) equation (also known as Benjamin-Bona-Mahony equation) which was

first introduced by Peregrine [23] to describe the development of an undular bore is presented as follows:

vt −vxxt + vx + vvx = 0.

The RLW equation was well studied numerically and theoretically in the literature. For instance, Biswas [2]

has introduced an analytical solution of the RLW equation with power-law nonlinearity. On the other hand,

Islam et al. [19] investigated the meshfree method for the numerical solution of the RLW equation.



Int. J. Anal. Appl. 16 (1) (2018) 3

Since the case of wave-wave and wave-wall interactions cannot be described by the KdV equation, Rosenau

[29, 30] proposed an equation known as the Rosenau equation to over come this shortcoming of the KdV

equation:

vt + vxxxxt + vx + vvx = 0.

The Rosenau equation has been the subject of several analytical and numerical studies [4,24,27] and references

therein. Recently, the Rosenau-KdV-RLW equation was proposed in [25] as a conjunction of Rosenau-KdV

and Rosenau-RLW equations both of which are well studied and explained with regard to shallow water waves.

Also in this paper the results of Rosenau-KdV-RLW equation have been reported without considering the

effects of perturbation.

For theoretical investigations, Razborova et al. [26] explored the dynamics of perturbed soliton solutions to

the Rosenau-KdV-RLW equation with power-low nonlinearity. Solutions of the perturbated RosenauKdV-

RLW equation are obtained [31]. Soliton perturbation theory was applied to obtain the adiabatic parameter

dynamics of these solitary waves [28].

The remainder of the paper is organized as follow. In section 2, we first review some basic results on

Chebyshev rational functions. Some orthogonal projections with their properties are also given in this

section as they play an important role in the error analysis. In section 3, we will discuss some basic techniques

employed for stability of the spectral methods in infinite domains. In section 4, we use the results in the

previous sections to validate the convergence of proposed scheme and derive error estimates. In section 5,

we construct the fully-discrete Chebyshev pseudo-spectral scheme, and obtain the optimum error estimate

of approximation solutions. Numerical results are shown in section 6. Finally, the final section gives some

concluding remarks.

2. Modified Chebyshev rational functions

This section addresses the basic notions and working tools concerning orthogonal modified Chebyshev

rational functions. More specifically, it presents some properties of the modified Chebyshev rational functions.

The well-known Chebyshev polynomials are orthogonal in the interval [-1,1] with respect to the weight

function ρ(x) = 1√
1−x2

and can be calculated through the following recurrence formula:

T0(x) = 1,

T1(x) = x,

Tn+1(x) = 2xTn(x) −Tn−1(x), n = 1, 2, 3, ....

The new basis functions denoted by Rn(x), are defined by [14] in interval Ω = (−∞,∞).

Rn(x) =
1

√
x2 + 1

Tn(
x

√
x2 + 1

), n = 0, 1, 2, ... .



Int. J. Anal. Appl. 16 (1) (2018) 4

Rn(x) is the eigenfunction of the singular Sturm-Liouville problem

(x2 + 1)
1
2
d

dx

(
(x2 + 1)

d

dx
((x2 + 1)

1
2 w(x)

))
+ n2w(x) = 0, x ∈ Ω, n = 0, 1, 2, ... . (2.1)

and satisfies the following recurrence relation:

R0(x) =
1

√
x2 + 1

,

R1(x) =
x

x2 + 1
,

Rn+1(x) =
( 2x
√
x2 + 1

)Rn(x) −Rn−1(x), n = 1, 2, 3, ... .

{Rn(x)}n≥1 are orthogonal with respect to the weight function χ(x) = 1 in the interval (−∞,∞), with the

orthogonality property: ∫
Ω

Rn(x)Rm(x)χ(x)dx =
π

2
cnδn,m,

where δn,m is the Kronecker function and c0 = 2,cn = 1 for n ≥ 1. For 1 ≤ p ≤ ∞, we define the space

Lp(Ω) and its norm ‖w‖Lp(Ω) as usual. In particular ‖w‖∞ = ‖w‖L∞(Ω). For any nonnegative integer m, we

define the Sobolev space as follows:

Hm(Ω) = {w :
dkw

dxk
∈ L2(Ω), 0 ≤ k ≤ m}.

Equipped with the inner product, the semi-norm, and the norm are defined as follow:

(v,w)m =

m∑
k=0

(
dkv

dxk
,
dkw

dxk
),

| w |m= ‖
dmw

dxm
‖,

‖w‖m = (w,w)
1
2
m.

For any real r > 0, we define the space Hr(Ω) and its norm |w|r by space interpolation. To describe

approximation results, we introduce a sequence of Hilbert spaces {HrZs}s≥1. For simplicity, let ∂xw(x) =
∂
∂x
w(x), etc. Let A be the Sturm-Liouville operator in ( 2.1), namely

Aw(x) = −(x2 + 1)
1
2
d

dx

(
(x2 + 1)

d

dx
((x2 + 1)

1
2 w(x)

))
.

For any even integer r ≥ 0,

HrZs (Ω) = {w : w is a measurable on Ω and ‖w‖r,Zs < ∞},

where ‖w‖r,Z0 = ‖A
r
2 w‖, and for s ≥ 1,

‖w‖r,Zs = ‖(x
2 + 1)∂x((x

2 + 1)
1
2 w))‖r−1,Zs−1.



Int. J. Anal. Appl. 16 (1) (2018) 5

We define these spaces and their norms by space interpolation. Now let N be any positive integer. In this

case, we have:

<N = Span{R0,R1, ...,RN}.

The L2(Ω)-orthogonal projection

LN : L
2(Ω) −→<N

is defined in the following way:

(LNw −w,φ) = 0 , ∀φ ∈<N.

The Hm(Ω)-orthogonal projection

LmN : H
m(Ω) −→<N

is defined as:

(LmNw −w,φ)m = 0 , ∀φ ∈<N.

Let C denote a generic positive constant independent of any function and N. We have the following results:

Theorem 2.1. For any w ∈ HrZm (Ω) and 0 ≤ m ≤ r,

‖LmNw −w‖m ≤ CN
m−r‖w‖r,Zm.

Theorem 2.2. Let 0 ≤ µ ≤ m− 1
2

, with a positive integer m. Then for any w ∈ HmZm (Ω) ∩H
m
0 (Ω),

‖LmNw‖µ,∞ ≤ C‖w‖r,Zm.

The above two theorems were proved in [17].

In order to analyze the modified Chebyshev rational approximation of the equation ( 1.1), we need another

orthogonal projection. Let

Hm0 (Ω) = {w : w ∈ H
m(Ω) and ∂kxw(0) = 0, for 0 ≤ k ≤ m},

and

<m,0N = <N ∩H
m
0 (Ω).

We denote in particular <0N = <
1,0
N and define the orthogonal projection

L
m,0
N : H

m
0 (Ω) −→<

m,0
N

by

(L
m,0
N w −w,φ)m = 0 , ∀φ ∈<

m,0
N .



Int. J. Anal. Appl. 16 (1) (2018) 6

Theorem 2.3. For any w ∈ HrZm (Ω) ∩H
m
0 (Ω) and 0 ≤ m ≤ r,

‖Lm,0N w −w‖m ≤ CN
m−r‖w‖r,Zm.

Proof. Let

v(x) =

∫ x
0

(z2 + 1)∂z((z
2 + 1)

1
2 w(z)

)
dz,

and

ϕ(x) =
1

√
x2 + 1

∫ x
0

1

(z2 + 1)
L
m−1,0
N−1 ∂zv(z)dz.

Clearly ϕ ∈<m,0N . The desired result follows from the same argument as in the proof of Theorem 4 [17]. �

3. Stability of the Rosenau-KdV-RLW equation

This section discusses the application of modified Chebyshev rational approximations for the Rosenau-

KdV-RLW equation ( 1.1) and presents the stability results of the proposed approach. A weak form of

equation ( 1.1) is to find v ∈ H20 (Ω) such that for any w ∈ H20 (Ω),
 (vt,w) + (vxt,wx) + (vxxt,wxx) − (vxx,wx) + (vx,w) −

1
2
(v2,wx) = 0

v(x, 0) = vo(x), x ∈ Ω.
(3.1)

The Chebyshev rational spectral scheme of ( 3.1) is to find vN (t) ∈<0N such that for any φ ∈<
0
N ,

 (vNt,φ) + (vNxt,φx) + (vNxxt,φxx) − (vNxx,φx) + (vNx,φ) −
1
2
(v2N,φx) = 0

vN (x, 0) = L
2
Nvo(x), x ∈ Ω.

(3.2)

Since the initial value term cannot be exactly evaluated, here we consider how stable the numerical solution of

( 3.2) depending on the initial value term. We now analyze the stability of ( 3.2) in the sense of Guo [11, 13].

To this end, we suppose that vN has the error ṽN . Then for any φ ∈ <0N and t ∈ [0,T], we derive the

following equation from ( 3.2) as follows:


(vNt(t) + ṽNt(t),φ) + (vNxt(t) + ṽNxt(t),φx) + (vNxxt(t) + ṽNxxt(t),φxx)

−(vNxx(t) + ṽNxx(t),φx) + (vNx(t) + ṽNx(t),φ) − 12 ((vN (t) + ṽN (t))
2,φx) = 0

ṽN (0) = ṽN,0.

(3.3)

It can be demonstrated that ṽN must satisfy the following relation:


(ṽNt(t),φ) + (ṽNxt(t),φx) + (ṽNxxt(t),φxx) − (ṽNxx(t),φx) + (ṽNx(t),φ)

−1
2
(ṽN (t) + 2vN (t)ṽN (t),φx) = 0, ∀φ ∈<0N, 0 ≤ t ≤ T

ṽN (0) = ṽN,0.

By taking φ = 2ṽN in ( 3.4), it follows that

d

dt
‖ṽN (t)‖22 = 2A(t), (3.4)



Int. J. Anal. Appl. 16 (1) (2018) 7

where A(t) = (vN (t)ṽN (t), ṽNx(t)). Then we have:

|A(t)| ≤ ‖vN (t)‖∞‖ṽN (t)‖|ṽN (t)|1 ≤
1

2
‖vN (t)‖2∞‖ṽN (t)‖

2 +
1

2
|ṽN (t)|21. (3.5)

To describe the error, let ρ(ϑ,t) = ‖ϑN (t)‖22. Also let

|‖w|‖∞ = sup0≤t≤T‖w(t)‖∞,

and

M(w) = (1 + |‖w|‖∞).

By substituting ( 3.5) into ( 3.4) and integrating the resulting inequality for t, we find that

‖ṽN (t)‖22 ≤ M(vN )
∫ t

0

‖ṽN (τ)‖22dτ + ρ(ṽN,0, t). (3.6)

Applying the Gronwall Lemma to the above inequality, we draw the following conclusion.

Theorem 3.1. Let vN be the solution of ( 3.2), and ṽN be its error induced by ṽN,0. Then for all 0 ≤ t ≤ T ,

‖ṽN (t)‖22 ≤ ρ(ṽN,0, t)e
M(vN )t. (3.7)

4. Convergence analysis and error estimate for the Rosenau-KdV-RLW equation

In this section, we will discuss some basic techniques use to estimate error bounds for spectral methods in

infinite domains. Furthermore, to show that the convergence results of the spectral methods for the Rosenau-

KdV-RLW equation ( 1.1), we use the modified Chebyshev rational approximations. For this purpose, we

first present an a priori estimate:

Lemma 4.1. If for r ≥ 2, v0 ∈ HrZ2 (Ω) ∩H
2
0 (Ω) then for any t ∈ [0,T],

‖vN (t)‖22 ≤ 2‖v0‖
2
2 + C0N

4−2r‖v0‖2r,Z2. (4.1)

Proof. Taking φ = vN in ( 3.2), we have

1

2
∂t(‖vN‖2 + ‖vNx‖2 + ‖vNxx‖2) − (vNxx,vNx) + (vNx,vN ) −

1

2
(v2N,vNx) = 0. (4.2)

Clearly

(vNxx,vNx) =
1

2

∫
Ω

∂xv
2
Nx(x,t)dx = 0, (4.3)

(vNx,vN ) =
1

2

∫
Ω

∂xvN (x,t)dx = 0, (4.4)

(v2N,vNx) =
1

3

∫
Ω

∂xv
3
N (x,t)dx = 0. (4.5)

Hence

∂t(‖vN‖2 + ‖vNx‖2 + ‖vNxx‖2) = 0, (4.6)



Int. J. Anal. Appl. 16 (1) (2018) 8

which implies

‖vN (t)‖2 + ‖vNx(t)‖2 + ‖vNxx(t)‖2 = ‖vN (0)‖2

+‖vNx(0)‖2 + ‖vNxx(0)‖2 = ‖L2Nv0‖
2
2. (4.7)

Next, according to Theorem 2.2 for v0 ∈ H20 (Ω), we have

‖L2,0N v0‖
2
2 ≤ 2‖v0‖

2
2 + 2‖v0 −L

2,0
N v0‖

2
2 ≤ 2‖v0‖

2
2 + C0N

4−2r‖v0‖2r,Z2. (4.8)

Finally, a combination of ( 4.7) and ( 4.8) leads to the desired result. �

Therefore, the modified Chebyshev rational approximations are well suited for numerical approximation

of the Rosenau-KdV-RLW equation. Indeed, we have the following result concerning the convergence and

error estimate for ( 3.2).

Theorem 4.1. If for r ≥ 2,

v ∈ L∞(0,T; W 1,∞(Ω)) ∩H2(0,T; HrZ2 (Ω)),

and HrZ2 (Ω), then we have

‖v −vN‖22 ≤ C2(v)N
2−r, ∀t ∈ (0,T], (4.9)

where C2(v) is positive constant depending only on the norms of v and v0 in the spaces mentioned above.

Proof. For simplicity, let

ξ = v −L2,0N v and η = vN −L
2,0
N v. (4.10)

Obviously, vN −v = η − ξ. Hence, by ( 3.1) and ( 3.2) we obtain that for any φ ∈<0N ,

(ηt,φ) + (ηxt,φx) + (ηxxt,φxx) − (ηxx,φx) + (ηx,φ) + (vNvNx −vvx,φ) (4.11)

= (ξt,φ) + (ξxt,φx) + (ξxxt,φxx) − (ξx,φxx) − (ξ,φx).

Take φ = η in ( 4.11). It can be shown that

(ηxx,ηx) = 0 , (ηx,η) = 0, (4.12)

and it is clear that

(ηxxt,ηx) + (ηxt,ηx) + (ηt,η) =
1

2
∂t‖η‖22. (4.13)



Int. J. Anal. Appl. 16 (1) (2018) 9

Next, by Theorems 2.2 and 2.3 we obtain

(vNvNx −vvx,η) = (vN (ηx − ξx) + vx(η − ξ),η)

≤‖η‖2 + ‖vN (ηx − ξx)‖2 + ‖vx(η − ξ)‖2

≤‖η‖2 + C‖vN‖2∞‖ηx − ξx‖
2 + C‖vx‖2∞‖η − ξ‖

2

≤‖η‖2 + CN4−2r(‖v‖21,∞ + ‖v‖
2
r,Z2

). (4.14)

Using Theorem 2.3 again yields

(ξt,η) + (ξxt,ηx) + (ξxxt,ηxx) ≤‖η‖22 + ‖ξt‖
2
2 ≤‖η‖

2
2 + CN

4−2r‖∂tv‖2r,Z2, (4.15)

(ξx,ηxx) ≤‖ηxx‖2 + ‖ξx‖2 ≤‖ηxx‖2 + CN4−2r‖v‖2r,Z2, (4.16)

(ξ,ηx) ≤‖ηx‖2 + ‖ξ‖2 ≤‖ηx‖2 + CN4−2r‖v‖2r,Z2. (4.17)

In addition, Theorems 2.1 and 2.3 lead to

‖η(0)‖22 ≤‖v0 −L
2
Nv0‖

2
2 + ‖v0 −L

2,0
N v0‖

2
2 ≤ CN

4−2r‖v0‖2r,Z2. (4.18)

Hence, by inserting ( 4.12)–( 4.17) in to ( 4.11), we obtain

∂t‖η‖22 ≤‖η‖
2
2 + C1(v)N

4−2r(‖v‖2r,Z2 + ‖∂tv‖
2
r,Z2

), (4.19)

where C1(v) is a positive constant depending only on

‖v‖L∞(0,T,H2
Z2

(Ω)∩W1,∞(Ω)). Substituting ( 4.18) into ( 4.20) and integrating the result with respect to t, we

obtain:

‖η‖22 ≤
∫ t

0

‖η(s)‖22ds + C2(v)N
4−2r, (4.20)

where C2(v) is a positive constant depending only on C1(v),

‖v‖H2(0,T,Hr
Z2

(Ω)) and ‖v0‖Hr
Z2

(Ω)). The desired result follows from the Gronwall inequality and Theorem

2.3. �

5. Fully-discretization scheme

In this section, we describe the numerical implementation and present some numerical results. At first,

we introduce the operator of interpolation at the Chebyshev- Gauss rational nodes

{xN,j = cot(
π(2j + 1)

2N + 2
)}0≤j≤N,

denoted by INv ∈<0N and

INv(xN,j) = v(xN,j), 0 ≤ j ≤ N.

The following Theorem can be found in [17].



Int. J. Anal. Appl. 16 (1) (2018) 10

Theorem 5.1. For any v ∈ HrZ1 (Ω) and 0 ≤ µ ≤ 1 ≤ r,

‖INv −v‖µ ≤ CNµ−r+1‖w‖r,Z1.

Let τ be the mesh size in t and set tk = kτ (k = 0, 1, ...,n + 1 = [
T
τ

]). For simplicity, we denote

vk(x) := v(x,tk) by v
k and

vk
t̂

=
vk+1 −vk−1

2τ
, v̂kt =

vk+1 + vk−1

2
. (5.1)

The fully-discretization Chebyshev pseudo-spectral method for ( 1.1)-( 1.3) is to find vkN ∈<
0
N , such that

vk
Nt̂
−

∂2

∂x2
vk
Nt̂

+
∂4

∂x4
vk
Nt̂

+
∂3

∂x3
v̂kN +

∂

∂x
v̂kN + IN (v

k
N

∂

∂x
v̂kN ) = 0. (5.2)

v0N (x) = INv0(x). (5.3)

The equations ( 5.2) and ( 5.3) are satisfied at Chebyshev-Gauss rational nodes x`,` = 0, 1, ...,N. Let

vk −vkN = (v
k −L2,0N v

k) + (L
2,0
N v

k −vkN ) = ξ
k + ηk.

Note that (ξk,w) = 0, ∀w ∈ <0N . Taking the inner product of ( 1.1) with w and subtracting ( 5.2) from

( 1.1), we get

(ηk
t̂
,w) + (

∂ηk
t̂

∂x
,
∂w

∂x
) − (

∂2ηk
t̂

∂x2
,
∂2w

∂x2
) + (

∂η̂k

∂x
,
∂2w

∂x2
) − (η̂k,

∂w

∂x
) (5.4)

+(IN (v
k ∂v̂

k

∂x
−vkN

∂v̂kN
∂x

),w) = (Γk,w),

where Γk is truncation error

Γk = (vk
t̂
−
∂vk

∂t
) +

∂2

∂x2
(
∂vk

∂t
−vk

t̂
) +

∂4

∂x4
(vk
t̂
−
∂vk

∂t
) +

∂3

∂x3
(vk − v̂k) (5.5)

+
∂

∂x
(vk − v̂k) + (vk

∂v̂k

∂x
− IN (vkN

∂v̂kN
∂x

)).

By applying Taylor’s theorem, Theorems 2.3 and 5.1, we get

Γk =
τ2

12
(
∂3v

∂t3
(tk1 ) +

∂3v

∂t3
(tk2 )) +

τ2

12

∂2

∂x2
(
∂3v

∂t3
(tk3 ) +

∂3v

∂t3
(tk4 ) +

τ2

12

∂4

∂x4
(
∂3v

∂t3
(tk5 )

+
∂3v

∂t3
(tk6 ) +

τ2

4

∂3

∂x3
(
∂2v

∂t2
(tk7 ) +

∂2v

∂t2
(tk2 )) +

τ2

4

∂

∂x
(
∂2v

∂t2
(tk9 ) +

∂2v

∂t2
(tk10))

+(vk
∂v̂k

∂x
− IN (vkN

∂v̂kN
∂x

)).

where tk−1 ≤ tk` ≤ t
k+1,` = 1, ..., 10. Taking w = η̂k in equation ( 5.5), we have

(Γk, η̂k) =
‖ηk+1‖2 −‖ηk−1‖2

4τ
+
‖ηk+1x ‖2 −‖ηk−1x ‖2

4τ
+
‖ηk+1xx ‖2 −‖ηk−1xx ‖2

4τ

+(
∂η̂k

∂x
,
∂2η̂k

∂x2
) − (η̂k,

∂η̂k

∂x
) + Fk. (5.6)



Int. J. Anal. Appl. 16 (1) (2018) 11

where

Fk = (IN (v
k ∂v̂

k

∂x
−vkN

∂v̂kN
∂x

), η̂k).

Clearly,

(
∂η̂k

∂x
,
∂2η̂k

∂x2
) = 0, (η̂k,

∂η̂k

∂x
) = 0. (5.7)

In the forthcoming discussions, we denote by C a positive constant independent of τ and N, and estimate

Fk of ( 5.6). By applying Taylor’s Theorem, Cauchy-Schwartz inequality and algebraic inequality, we deduce

that

|Fk| ≤ C(N−2r + ‖ηk‖2 + ‖η̂k‖2 + ‖η̂k‖21). (5.8)

Furthermore, by Theorem 2.3 and the Cauchy-Schwartz inequality, we obtain

|(Γk, η̂k)| ≤ C(N−2r + τ4 + ‖η̂k‖2). (5.9)

Inserting ( 5.7)-( 5.9) into ( 5.6), we get

‖ηk+1‖2 −‖ηk−1‖2

4τ
+
‖ηk+1x ‖2 −‖ηk−1x ‖2

4τ
+
‖ηk+1xx ‖2 −‖ηk−1xx ‖2

4τ

≤ C(N−2r + τ4 + ‖ηk‖2 + ‖η̂k‖2 + ‖η̂k‖21). (5.10)

Obviously, we can verify from ( 5.1) that

‖v̂k‖2 ≤
1

2
(‖vk+1‖2 + ‖vk−1‖2).

By summing up ( 5.10) for k = 1, 2, ...,n, we get

‖ηn‖22 ≤ C(‖η
0‖22 + τ

4 + N−2r) + Cτ

n−1∑
k=1

‖ηn‖22.

Note that ‖η0‖22 = 0. Tanks to the discrete Gronwall inequality, we get

C(τ4 + N−2r) ≤ MeCT .

‖ηn‖22 ≤ C(τ
4 + N−2r)eC(n+1)τ.

where M is the positive constant. Therefore, we have the following result.

Theorem 5.2. Let v be the solution of ( 1.1) and v ∈ HrZm (Ω) ∩H
m
0 (Ω), 0 ≤ m ≤ r. Also, let vN be the

solution of ( 5.2) in <0N and τ is sufficiently small. Then there exist constant M, independent of τ and N

such that for k = 1, 2, ...,n− 1,

‖vk+1 −vk+1N ‖2 ≤ M(τ
2 + N−r).



Int. J. Anal. Appl. 16 (1) (2018) 12

6. Numerical experiments

In this section, we will conduct some numerical experiments to verify the theoretical results obtained in

the previous sections. We report ‖E‖∞ and the ‖E‖2 errors of the solution that are defined as

‖E‖∞ = max
0≤j≤N

|v(xj, t) −vN (xj, t)|,

and

‖E‖2 =
∑N
j=0 |v(xj, t) −vN (xj, t)|

2∑N
j=0 |v(xj, t)|2

,

Where vN (xj, t) is the solution of numerical scheme ( 5.2) and ( 5.3), while v(xj, t) is the exact solution of

( 1.1)-( 1.3). Consider the initial boundary value problem of the Rosenau-KdV-RLW equation ( 1.1)-( 1.3)

with the exact solution as follow [35]:

v(x,t) =
5

456
(25 − 13

√
457)sech4[

1
√

288

√
−13 +

√
457(x− (

241 + 13
√

457

266
)t)],

and the initial condition is set as

v0(x) =
5

456
(25 − 13

√
457)sech4[(

1
√

288

√
−13 +

√
457)x].

‖E‖∞ and ‖E‖2 errors for various of N with T = 20 and τ = 0.1, are reported in Table 1. We can observe

from the table, that the results from the present study are in good agreement with the exact solutions.

N ‖E‖∞ ‖E‖2

10 6.008 × 10−2 9.863 × 10−2

20 4.439 × 10−3 8.746 × 10−3

30 3.540 × 10−5 5.298 × 10−5

40 3.516 × 10−5 5.255 × 10−5

50 3.510 × 10−5 5.140 × 10−5

Table 1. Norm infinity and norm relative of errors for τ = 0.1,T = 20 and several values of N

Table 2 gives the errors between numerical solutions and exact solutions. We can see that when we use

smaller time and temporal mesh, numerical solutions are almost the same as the exact solutions.



Int. J. Anal. Appl. 16 (1) (2018) 13

τ ‖E‖∞ ‖E‖2

0.1 1.439 × 10−5 3.611 × 10−5

0.05 7.160 × 10−6 6.228 × 10−6

0.025 5.343 × 10−7 8.390 × 10−7

0.0125 1.905 × 10−7 3.110 × 10−7

0.00625 1.777 × 10−8 1.943 × 10−8

Table 2. Norm infinity and norm relative of errors for N = 35,T = 10 and several values of τ

7. Conclusion

In this paper, we proposed the modified Chebyshev rational approximations for the Rosenau-KdV-RLW

equation on the infinite intervals. The corresponding spectral scheme was constructed and its convergence

was proved. To show the performance of the modified Chebyshev rational approximations, an a prior

estimate was derived. We proved that the numerical solutions tend to weak solutions of ( 1.1) in a suitable

sense. The numerical results demonstrate that the suggested method possesses high-order accuracy for the

Rosenau-KdV-RLW equation on the whole line with analytical solutions.

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	1. Introduction
	2. Modified Chebyshev rational functions
	3. Stability of the Rosenau-KdV-RLW equation
	4. Convergence analysis and error estimate for the Rosenau-KdV-RLW equation
	5. Fully-discretization scheme
	6. Numerical experiments
	7. Conclusion
	References