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Mathematical Methods in the Applied Sciences

Math Methods Appl Sci

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.5135

High-order numerical solution of viscous Burgers’ equation using a

Cole-Hopf barycentric Gegenbauer integral pseudospectral method

Kareem T. Elgindy1,2∗Sayed A. Dahy2

1Mathematics & Statistics Department, The College of Sciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Kingdom

of Saudi Arabia

2Mathematics Department, Faculty of Science, Assiut University, Assiut 71516, Egypt

SUMMARY

We present a novel, high-order numerical method to solve viscous Burger’s equation with smooth initial and boundary data.

The proposed method combines Cole-Hopf transformation with well conditioned integral reformulations to reduce the problem

into either a single easy-to-solve integral equation with no constraints, or an integral equation provided by a single integral

boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing

a full Gegenbauer collocation scheme based on Gegenbauer-Gauss (GG) mesh grids using apt Gegenbauer parameter values

and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices

of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error

and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational

algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior

accuracy of the method even when the viscosity paramter ν→0, in the absence of any adaptive strategies typically required

for adaptive reﬁnements.

KEY WORDS: Barycentric; Burgers’ equation; Cole-Hopf; Gegenbauer-Gauss; Gegenbauer quadrature; Pseudospectral.

1. INTRODUCTION

Burgers’ equation is a non-linear parabolic partial differential equation that gained much attention in the literature

after it was ﬁrst introduced by Bateman [1] who gave its steady solutions. It was later proposed as a model of

ﬂuid motion in studies on turbulence by Burgers [2] whose name was coined with the equation due to his relevant

remarkable contributions. The breadth and the range of the applications of Burgers’ equation are certainly part of its

establishment as an important nonlinear mathematical model in applied mathematics; cf. [3–9].

The development of both analytical and numerical methods for solving Burgers’ equation provided with various

types of constraints continues to be an area of interest to scholars and researchers who endeavor to enrich profound

understanding of such important nonlinear problems and to scrutinize the quality of diverse numerical methods as a

natural ﬁrst step towards developing methods for computations of complex ﬂows. In fact, Burgers’ equation is one

of few nonlinear partial differential equations that can be solved exactly for a restricted set of initial and boundary

functions; cf. [10]. Some of the works presented in the literature to ﬁnd the analytic solution of Burgers’ equation

include the works of Hopf [11] and Cole [12] who pioneered independently a magic, nonlinear transformation,

widely known as Cole-Hopf transformation, converting Burgers’ equation into a heat equation. Benton and Platzman

[13] used the transformation to classify several distinct exact solutions of Burgers’ equation in tabular forms.

Odai et al. [14] solved the reduced linear diffusion equation produced by Cole-Hopf transformation via means of

Green’s function assuming an inﬁnite domain. Schiffner et al. [15] derived an explicit analytical solution to Burgers’

∗Correspondence to: Mathematics & Statistics Department, The College of Sciences, King Fahd University of Petroleum & Minerals, Dhahran

31261, Kingdom of Saudi Arabia. E-mail: kareem.elgindy@(kfupm.edu.sa; aun.edu.eg; gmail.com).

Prepared using speauth.cls [Version: 2010/05/13 v3.00]

2KAREEM T. ELGINDY AND SAYED A. DAHY

equation based on Volterra series. A closed form solution of the (1 + n)-dimensional Burgers’ equation was derived

by Srivastava and Awasthi [16] using homotopy perturbation, Adomian decomposition, and differential transform

methods.

It is widely known that the analytical solutions of Burgers’ equation in terms of inﬁnite series exhibit slow

convergence for certain values of the viscosity parameter ν, and may even fail for small viscosity values; cf. [17–

19]. Numerical techniques are therefore of much interest to meet the requirement of the wide range of solutions of

the Burgers’ equation. Among the common numerical techniques used are ﬁnite difference schemes, differential

quadrature methods, ﬁnite volume methods, ﬁnite element methods, Haar wavelet methods, spectral methods,

pseudospectral methods, method of lines, meshless methods, Adomian decomposition methods, B-spline methods,

discontinuous Galerkin methods, reproducing kernel functions, etc. [10,20–35]. However, such methods either lack

the exponential convergence enjoyed by spectral and pseudospectral methods, or they enjoy exponential convergence

rate in the spatial direction, but suffer from low-order convergence rate in the temporal direction, or suffer from

degradation of the observed precision due to the ill-conditioning of the employed numerical differential operators to

the extent that the development of efﬁcient preconditioners becomes extremely crucial, or subject to serious time step

restrictions that could be more severe than those predicted by the standard stability theory; cf. [36–48]. To avoid the ill-

conditioning of differential operators and the reduction in convergence rate for derivatives, an alternative direction to

the aforementioned methods is to recast the partial differential equation into its integral formulation to take advantage

of the well-conditioning of integral operators, and then discretize the latter using various discretization techniques.

This useful strategy was applied successfully in the recent papers of Elgindy [48,49] in which highly accurate

numerical schemes were established to solve the second-order, one-dimensional hyperbolic telegraph equation and

parabolic distributed parameter system-based optimal control problems.

The current research paper objectives are highly motivated by the desire to extend the recent works of Elgindy

[48,49] to solve Burgers’ equation as one of the fundamental nonlinear partial differential equations that drew

much attention over decades. In particular, we shall endeavor through this research paper to establish some novel

computational algorithms suited to provide highly accurate, fully exponentially convergent, fast, and stable numerical

solutions. The proposed computational algorithms enjoy the luxury of integrating four useful tools: (i) the superior

advantages possessed by the family of pseudospectral methods, (ii) the well-conditioning of integral operators

furnished through the use of the integral form of the dynamical system equation, (iii) the spectral accuracy provided

by the latest technology of optimal Gegenbauer barycentric quadratures, and (iv) the useful linearization strategy

provided by the powerful Cole-Hopf transformation. We show that Burgers’ equation provided by initial and Dirichlet

boundary conditions can be transformed into a single easy-to-solve integral equation with no constraints, or an integral

equation provided by a single integral boundary condition. To the best of our knowledge, the current paper presents

the ﬁrst fully exponentially convergent integral pseudospectral method in the literature for solving Burgers’ equation.

The rest of the article is organized as follows: In Section 2, we state the problem under study and describe the

assumptions on the problem data. In Section 3, we give some basic preliminaries relevant to Gegenbauer polynomials

in addition to some useful formulas for the construction of their Gegenbauer-Gauss (GG)-based differentiation

matrices in barycentric form. The proposed method is presented in Section 4followed by rigorous convergence

and error analyses conducted in Section 5. Three numerical test examples were studied in Section 6to assess

the efﬁciency and accuracy of the proposed numerical schemes. We provide some concluding remarks in Section

7. Finally, Appendix Aestablishes two efﬁcient computational algorithms for the implementation of the proposed

method.

2. PROBLEM STATEMENT

Consider the following one-dimensional, quasilinear, parabolic Burger’s equation

∂u

∂t +u∂u

∂x =ν∂2u

∂x2,(x, t)∈ D2

a,b,T ,(2.1)

subject to the initial condition

u(x, 0) = f(x), a ≤x≤b,

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 3

together with Dirichlet boundary conditions

u(a, t) = h(t), u(b, t) = k(t),0≤t≤T,

where u=u(x, t), ν > 0is the kinematic viscosity parameter, and D2

a,b,T = [a, b]×[0, T ], a, b, ∈R:b > a, T ∈R+.

We shall refer to this problem by Problem P1, and assume that it exhibits global smooth solutions. It is also assumed

that supp(k)is either [0, T ], or the empty set ∅. We refer to those two assumptions by Assumptions 1 and 2,

respectively.

3. GEGENBAUER POLYNOMIALS AND THEIR BARYCENTRIC OPERATIONAL MATRICES OF

DIFFERENTIATION

In this section we present some properties of Gegenbauer polynomials, and introduce some useful formulas for the

construction of their GG-based differentiation matrices in barycentric form. A comprehensive survey on Gegenbauer

integration matrices and their associated optimal variants can be found in [37,50,51].

The Gegenbauer polynomial G(α)

n(x), of degree n∈Z+, and associated with the parameter α > −1/2, is a real-

valued function, which appears as an eigensolution to a singular Sturm-Liouville problem in the ﬁnite domain [1,1]

[52]. It is a Jacobi polynomial, P(α,β)

n, with α=β, and can be explicitly written in standardized form as follows

G(α)

n(x) = n!Γ(α+1

2)

Γ(n+α+1

2)P(α−1/2,α−1/2)

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

Gegenbauer polynomials can be generated by the following useful three-term recurrence equation

G(α)

0(x)=1, G(α)

1(x) = x,

(n+ 2α)G(α)

n+1(x) = 2(n+α)G(α)

n(x)−nG(α)

n−1(x), n = 1,2,3, ...,

or in terms of the hypergeometric functions [49]:

G(α)

n(x) = 2F1−n, 2α+n;α+1

2;1−x

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

where 2F1(a, b;c;x)is the Gauss hypergeometric function. We denote the zeroes of the Gegenbauer polynomial

G(α)

n+1(x)(aka GG nodes) by x(α)

n,k,k= 0, ..., n, and denote their set by S(α)

n. The leading coefﬁcient of the Gegenbauer

polynomials G(α)

n(x)is denoted by K(α)

n, and deﬁned by

K(α)

n= 2n−1Γ(n+α)Γ(2α+ 1)

Γ(n+ 2α)Γ(α+ 1), n = 0,1,2, ....

The weight function for the Gegenbauer polynomials is the even function w(α)(x) = (1 −x2)α−1/2. Gegenbauer

polynomials form a complete orthogonal basis polynomials in L2

w(α)[−1,1], and their orthogonality relation is given

by the following weighted inner product

G(α)

m, G(α)

nw(α)=Z1

−1

G(α)

m(x)G(α)

n(x)w(α)(x)dx =

G(α)

n

2

w(α)δm,n =λ(α)

nδm,n,

where

λ(α)

n=

G(α)

n

2

w(α)= 21−2απΓ(n+ 2α)

n!(n+α)Γ2(α),

is the normalization factor, and δm,n is the Kronecker delta function. In two dimensions, the bivariate Gegenbauer

polynomial, G(α)

m,n(x, t), for some m, n ∈Z+

0=Z+∪ {0}is deﬁned in terms of the univariate Gegenbauer polynomial

by

G(α)

m,n(x, t) = G(α)

m(x)G(α)

n(t)∀(x, t)∈ D2= [−1,1] ×[−1,1].

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

4KAREEM T. ELGINDY AND SAYED A. DAHY

The ﬁrst-order derivative and indeﬁnite integral of G(α)

nare deﬁned by

DxG(α)

n(x) =

0, n = 0,

n(n+ 2α)

(2α+ 1) G(α+1)

n−1(x), n ≥1,(3.1)

ZG(α)

n(x)dx =(2α−1)

(n+ 1)(n+ 2α−1)G(α−1)

n+1 (x) + constant.

The deﬁnite integrals on [−1, x],for any x∈[−1,1], can also be calculated using the following closed form formulas

Zx

−1

G(α)

0(z)dz =G(α)

0(x) + G(α)

1(x),

Zx

−1

G(α)

1(z)dz =a1G(α)

2(x)−G(α)

0(x),

Zx

−1

G(α)

i(z)dz =1

2(i+α)a2,iG(α)

i+1(x) + a3,i G(α)

i−1(x)+(−1)i(a2,i +a3,i ), i ≥2,

where

a1=1+2α

4(1 + α), a2,i =i+ 2α

i+ 1 , a3,i =−i

i+ 2α−1∀i.

The GG-based linear barycentric rational Lagrange interpolant of a real function fdeﬁned on [−1,1] is given by

PB,nf(x) =

n

X

j=0

f(α)

n,j L(α)

B,n,j (x),(3.2)

where f(α)

n,j =fx(α)

n,j ∀j,

L(α)

B,n,j (x) = ξ(α)

n,j

x−x(α)

n,j ,n

X

i=0

ξ(α)

n,i

x−x(α)

n,i

, j = 0, . . . , n, (3.3)

are the Lagrange interpolating polynomials deﬁned in barycentric form, ξ(α)

n,j , j = 0, ..., n are the barycentric weights

as deﬁned by Elgindy [51]:

ξ(α)

n,j = (−1)jsin cos−1x(α)

n,j q$(α)

n,j , j = 0, . . . , n,

and $(α)

n,j , j = 0, . . . , n are the GG quadrature weights. To construct the ﬁrst-order barycentric Gegenbauer

differentiation matrix (BGDM) for the the GG points, D(1),x

B=d(1),x

B,i,j ,0≤i, j ≤n, we follow the approach

presented in Berrut et al. [53]. In particular, multiplying through, and then multiplying both sides of Eq. (3.3) by

x−x(α)

n,i to render them differentiable at x=x(α)

n,i yields

L(α)

B,n,j (x)

n

X

k=0

ξ(α)

n,k

x−x(α)

n,i

x−x(α)

n,k

=ξ(α)

n,j

x−x(α)

n,i

x−x(α)

n,j

.

It follows with S(x) =

n

P

k=0

ξ(α)

n,k x−x(α)

n,i .x−x(α)

n,kthat

S(x)DxL(α)

B,n,j (x) + L(α)

B,n,j (x)S0(x) = ξ(α)

n,j x−x(α)

n,i

x−x(α)

n,j !0

.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 5

At x=x(α)

n,i , straightforward computations yield

Sx(α)

n,i =ξ(α)

n,i , S0x(α)

n,i =Xi6=kξ(α)

n,k.x(α)

n,i −x(α)

n,k,

from which, together with L(α)

B,n,j x(α)

n,i = 0, we get

d(1),x

B,i,j =DxL(α)

B,n,j x(α)

n,i =

ξ(α)

n,j .ξ(α)

n,i

x(α)

n,i −x(α)

n,j

∀i6=j.

For i=j, we have n

X

j=0

L(α)

B,n,j (x)=1,

so n

X

j=0

D(m)

xL(α)

B,n,j (x) = 0,

for each differentiation order m≥1; thus

d(1),x

B,j,j =DxL(α)

B,n,j x(α)

n,j =−X

i6=j

DxL(α)

B,n,j x(α)

n,i .

Hence, we can easily approximate the derivative of fat the GG points using the following useful formula

f0x(α)

n,i ≈P0

B,nfx(α)

n,i =

n

X

j=0

d(1),x

B,i,j f(α)

n,j ∀i.

If f∈Cn+1[−1,1], then it can be directly shown using Eq. (3.1) that there exist some numbers ξ(α)

n,i ∈[−1,1], i =

0, . . . , n such that the truncation error associated with the above derivative approximation is deﬁned by

E(α)

nx(α)

n,i , ξ(α)

n,i =

(n+ 2α+ 1)f(n+1) ξ(α)

n,i

(2α+ 1)n!K(α)

n+1

G(α+1)

nx(α)

n,i , i = 0, . . . , n;n≥1.

4. THE CHBGPM

We initiate our proposed method by transforming Burger’s equation (2.1) into its associated linear diffusion equation

using the following non-linear Cole-Hopf transformation

u=−2ν∂

∂x ln (φ),(4.1)

where φis a smooth function that satisﬁes the following linear initial-boundary value problem P2

∂φ

∂t =ν∂2φ

∂x2,(x, t)∈ D2

a,b,T ,

with the initial condition

φ(x, 0) = exp −1

2νZx

a

f(ξ)dξ, a ≤x≤b,

and the mixed Neumann and Robin boundary conditions

φx(a, t) = −h(t)

2ν, φ(b, t)k(t)+2νφx(b, t)=0,0≤t≤T.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

6KAREEM T. ELGINDY AND SAYED A. DAHY

Using the following change of variables

y= (2x−a−b)/(b−a), τ =2

Tt−1,

we can transform the domain D2

a,b,T into D2, and restate Problem P2as the following associated problem

∂φs

∂τ =µν

∂2φs

∂y2,(y, τ )∈ D2,(4.2a)

with the initial condition

φS(y, −1) = exp λνZy

−1

f1

2((b−a)ξ+a+b)dξ,−1≤y≤1,(4.2b)

and the boundary conditions

∂φS(−1, τ )

∂y =λνh(T(τ+ 1) /2) ,−1≤τ≤1,(4.2c)

φS(1, τ )k(T(τ+ 1) /2) −1

λν

∂φS(1, τ )

∂y = 0,−1≤τ≤1,(4.2d)

where

µν=2νT

(b−a)2, λν=a−b

4ν,

and

φS(y, τ ) = φb−a

2y+b+a

2, T (τ+ 1) /2∀(y, τ )∈ D2.

We refer to the reduced problem deﬁned by Eqs. (4.2) by Problem P3. Partially integrating Eq. (4.2a) w.r.t. τon

[−1, τ ], and imposing Eq. (4.2b) yield the following integro-differential equation

φS(y, τ )−µνZτ

−1

∂2φS(y, σ)

∂y2dσ = exp λνZy

−1

f1

2((b−a)ξ+a+b)dξ.(4.3)

We can convert Eq. (4.3) into a more useful integral equation, as we shall see later, using the following substitution

ψ(y, τ ) = ∂2φS(y, τ)

∂y2,(y, τ )∈ D2,(4.4)

for some unknown function ψ. The function φScan be easily recovered from ψthrough successive integrations w.r.t.

y:

∂φS(y, τ )

∂y =Zy

−1

ψ(σ, τ )dσ +ϑ1(τ),(4.5)

φS(y, τ ) = Zy

−1Zσ

−1

ψ(σ1, τ )dσ1dσ + (y+ 1)ϑ1(τ) + ϑ2(τ),(4.6)

where

ϑ1(τ) = λνh(T(τ+ 1) /2) ,(4.7)

as can be veriﬁed from Eq. (4.2c), and ϑ2is some arbitrary function of τ. Now, suppose that Assumption 1 holds, i.e.,

k(T(τ+ 1) /2) 6= 0 ∀τ∈[−1,1]. Through Eqs. (4.2d), (4.5), and (4.6), we can deﬁne the function ϑ2explicitly in

terms of ψas follows

ϑ2(τ) = ϑ3(τ)1

∫

−1

ψ(σ, τ )dσ −1

∫

−1

σ

∫

−1

ψ(σ1, τ )dσ1dσ +ϑ1(τ) (ϑ3(τ)−2) ,(4.8)

where

ϑ3(τ) = 1

λνk(T(τ+ 1) /2) .(4.9)

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 7

Imposing Eqs. (4.4), (4.6), and (4.8), transforms the integro-differential equation (4.3) into the following integral

equation

y

∫

−1

σ

∫

−1

ψ(σ1, τ )dσ1dσ −1

∫

−1

σ

∫

−1

ψ(σ1, τ )dσ1dσ −µν

τ

∫

−1

ψ(y, σ)dσ +ϑ3(τ)1

∫

−1

ψ(σ, τ )dσ = Ω(y, τ),(4.10)

where

Ω(y, τ ) = exp λν

y

∫

−1

f1

2((b−a)ξ+a+b)dξ+ϑ1(τ) (1 −y−ϑ3(τ)) .

To collocate the integral equation (4.10) at the GG mesh grid y(α)

Ny,r, τ (α)

Nτ,s∈ D(α)

Ny,Nτ=S(α)

Ny×S(α)

Nτ, r =

0, . . . , Ny;s= 0, . . . , Nτ, we closely follow the mathematical convention used by Elgindy [51], and denote the qth-

order Gegenbauer integration matrix and its associated optimal variant in respective order by P(q),θ

B=p(q),θ

B,r,sand

P(q),y

OB =p(q),y

OB,r,k , r, s = 0, . . . , Nθ;k= 0, . . . , My∈Z+;q∈Z+;θ∈ {y , τ}. Let also P(1),y

B,Ny+1 =p(1),y

B,Ny+1,s

and P(2),y

B,Ny+1 =p(2),y

B,Ny+1,sbe the the 1st- and 2nd-order barycentric Gegenbauer integration vectors required for

approximating single- and double- deﬁnite integrals over the interval [−1,1] and the region D2, respectively, as

described by Elgindy [51, Algorithms 6 and 7, and Eq. (3.9)]. Using these essential numerical quadrature tools, we

are able to write the discrete forms of Eqs. (4.8) and (4.10) in the following linear algebraic system form

˜

ϑ(α)

2,Ny,Nτ,s =ϑ3,s

Ny

X

k=0

p(1),y

B,Ny+1,k ˜

ψ(α)

Ny,Nτ,k,s −

Ny

X

k=0

p(2),y

B,Ny+1,k ˜

ψ(α)

Ny,Nτ,k,s +ϑ1,s (ϑ3,s −2) , s = 0,1, . . . , Nτ,

(4.11)

Ny

X

k=0 p(2),y

B,r,k −p(2),y

B,Ny+1,k +ϑ3,sp(1),y

B,Ny+1,k ˜

ψ(α)

Ny,Nτ,k,s −µν

Nτ

X

l=0

p(1),τ

B,s,l ˜

ψ(α)

Ny,Nτ,r,l = Ω(α)

r,s , r = 0, ..., Ny;s= 0, ..., Nτ,

(4.12)

where ϑi,s =ϑiτ(α)

Nτ,s,i= 1 and 3,˜

ϑ(α)

2,Ny,Nτ,s ≈ϑ2τ(α)

Nτ,s∀s,˜

ψ(α)

Ny,Nτ,r,s ≈ψy(α)

Ny,r, τ (α)

Nτ,s∀r, s, and

Ω(α)

r,s = exp

λν

My

X

k=0

p(1),y

OB,r,k f1

2((b−a)z(α∗

r)

Ny,r,k +a+b)

+ϑ1,s 1−y(α)

Ny,r −ϑ3,s, r = 0, ..., Ny;s= 0, ..., Nτ.

(4.13)

To rewrite the linear algebraic system (4.12) in the standard linear form

A Ψ =Ω,(4.14)

we introduce the useful mapping

n=index(r, s)≡r+s(Ny+ 1),

and rearrange the two-dimensional arrays ˜

ψNy,Nτ,r,sand Ω(α)

r,s , r = 0, . . . , Ny;s= 0, . . . , Nτin the form of

vector arrays, Ψ=˜

ψnL

n=0 and Ω={Ωn}L

n=0, respectively, where L=NyNτ+Ny+Nτ. The matrix elements

of the global collocation matrix Acan be computed by the following equations

Aindex(r,s),index(k,s)=p(2),y

B,r,k −p(2),y

B,Ny+1,k +ϑ3,sp(1),y

B,Ny+1,k , k = 0,1, ..., Ny;k6=r, (4.15a)

Aindex(r,s),index(r,k)=−µνp(1),τ

B,s,k , k = 0,1, ..., Nτ;k6=s, (4.15b)

Aindex(r,s),index(r,s)=p(2),y

B,r,r −p(2),y

B,Ny+1,r +ϑ3,sp(1),y

B,Ny+1,r −µνp(1),τ

B,s,s.(4.15c)

We can easily solve the linear system (4.14) for Ψvia a direct linear system solver.

Now, consider the second scenario when Assumption 2 holds. From Eq. (4.2d) we obtain

∂φS(1, τ )

∂y = 0,(4.16)

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

8KAREEM T. ELGINDY AND SAYED A. DAHY

since λν6= 0. Substituting Eq. (4.16) into Eq. (4.5) yields

−Z1

−1

ψ(σ, τ )dσ =ϑ1(τ).(4.17)

Through Eqs. (4.3), (4.4), and (4.6) we have

y

∫

−1

σ

∫

−1

ψ(σ1, τ )dσ1dσ +ϑ2(τ)−µν

τ

∫

−1

ψ(y, σ)dσ =η(y , τ),(4.18)

where

η(y, τ ) = exp λν

y

∫

−1

f1

2((b−a)ξ+a+b)dξ−(y+ 1)ϑ1(τ).

Collocating Eqs. (4.17) and (4.18) at the GG nodes yields the following system of (Ny+ 2) ×(Nτ+ 1) equations in

(Ny+ 2) ×(Nτ+ 1) unknowns, ˜

ψ(α)

Ny,Nτ,r,s and ˜

ϑ(α)

2,Ny,Nτ,s, r = 0, . . . , Ny;s= 0, . . . , Nτ:

−

Ny

X

k=0

p(1),y

B,Ny+1,k ˜

ψ(α)

Ny,Nτ,k,s =ϑ1,s,(4.19a)

Ny

X

k=0

p(2),y

B,r,k ˜

ψ(α)

Ny,Nτ,k,s +˜

ϑ(α)

2,Ny,Nτ,s −µν

Nτ

X

l=0

p(1),τ

B,s,l ˜

ψ(α)

Ny,Nτ,r,l =η(α)

r,s , r = 0, . . . , Ny;s= 0, . . . , Nτ,(4.19b)

where

η(α)

r,s = exp

λν

My

X

k=0

p(1),y

OB,r,k f1

2((b−a)z(α∗

r)

Ny,r,k +a+b)

−y(α)

Ny,r + 1ϑ1,s, r = 0, ..., Ny;s= 0, ..., Nτ.

(4.20)

To rewrite the augmented system of equations (4.19a) and (4.19b) in the standard linear form

B Ψaug =η,(4.21)

we introduce the mapping

n=index(r, s)≡r+s(Ny+ 2),

and rearrange the array of unknowns and η(α)

r,s , r = 0, ..., Ny;s= 0, ..., Nτas follows

Ψaug =h˜

ψNy,Nτ,0,0˜

ψNy,Nτ,1,0· · · ˜

ψNy,Nτ,Ny,0˜

ϑ(α)

2,Ny,Nτ,0˜

ψNy,Nτ,0,1˜

ψNy,Nτ,1,1· · · ˜

ψNy,Nτ,Ny,1

˜

ϑ(α)

2,Ny,Nτ,1· · · · · · ˜

ϑ(α)

2,Ny,Nτ,NτiT

,

η=η0,0η1,0· · · ηNy,0ϑ1,0η0,1η1,1· · · ηNy,1ϑ1,1· · · · · · ϑ1,NτT.(4.22)

The matrix elements of the global collocation matrix Bcan be computed by the following equations

Bindex(r,s),index(k,s)=p(2),y

B,r,k, k = 0,1, ..., Ny;k6=r, (4.23a)

Bindex(r,s),index(r,k)=−µνp(1),τ

B,s,k , k = 0,1, ..., Nτ;k6=s, (4.23b)

Bindex(r,s),index(r,s)=p(2),y

B,r,r −µνp(1),τ

B,s,s,(4.23c)

Bindex(r,s),index(Ny+1,s)= 1, s = 0,1, ..., Nτ,(4.23d)

Bindex(Ny+1,s),index(k,s)=−p(1),y

B,Ny+1,k , k = 0,1, ..., Ny.(4.23e)

Once again we can easily solve the linear system of equations (4.21) via a direct linear system solver.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 9

For both assumptions, Assumptions 1 and 2, we can estimate an approximation ˜

φ(α)

S,Ny,Nτ,r,s to the shifted function

value φS(y, τ )at any collocation point y(α)

Ny,r, τ (α)

Nτ,s∈ D(α)

Ny,Nτthrough Eq. (4.6):

˜

φ(α)

S,Ny,Nτ,r,s =

Ny

X

k=0

p(2),y

B,r,k ˜

ψ(α)

Ny,Nτ,k,s +y(α)

Ny,r + 1ϑ1,s +˜

ϑ(α)

2,Ny,Nτ,s, r = 0, . . . , Ny;s= 0, . . . , Nτ.(4.24)

Let ˜

φ(α)

Ny,Nτ,r,s and ˜

φx(α)

Ny,Nτ,r,s denote the approximations to the function value φ(x, t)and its partial derivative

w.r.t. xat any shifted GG (SGG) collocation point x(α)

Ny,r, t(α)

Nτ,s∈ˆ

D(α)

Ny,Nτ=a,bˆ

S(α)

Ny×0,T ˆ

S(α)

Nτ, where

a,bˆ

S(α)

Ny=x(α)

Ny,k :x(α)

Ny,k =1

2(b−a)y(α)

Ny,k +a+b, k = 0,...Ny,

0,T ˆ

S(α)

Nτ=t(α)

Nτ,k :t(α)

Nτ,k =T

2τ(α)

Nτ,k + 1, k = 0,...Nτ.

Then, clearly

˜

φ(α)

Ny,Nτ,r,s =˜

φ(α)

S,Ny,Nτ,r,s ∀r, s,

and we can directly estimate ˜

φx(α)

Ny,Nτ,r,s using the ﬁrst-order BGDM:

˜

φx(α)

Ny,Nτ,r,s =2

b−a

Ny

X

i=0

d(1),x

B,r,i ˜

φ(α)

Ny,Nτ,i,s, r = 0, . . . , Ny;s= 0, . . . , Nτ.(4.25)

These useful estimates allow us to easily determine approximations P(α)

Ny,Nτux(α)

Ny,r, t(α)

Nτ,sto the solution values

ux(α)

Ny,r, t(α)

Nτ,sthrough Eq. (4.1):

P(α)

Ny,Nτux(α)

Ny,r, t(α)

Nτ,s=1

λν

Ny

X

i=0

d(1),x

B,r,i ln ˜

φ(α)

Ny,Nτ,i,s, r = 0, . . . , Ny;s= 0, . . . , Nτ,

or, equivalently,

P(α)

Ny,Nτux(α)

Ny,r, t(α)

Nτ,s=−2ν˜

φx(α)

Ny,Nτ,r,s

˜

φ(α)

Ny,Nτ,r,s

, r = 0, . . . , Ny;s= 0, . . . , Nτ.(4.26)

We can further estimate the approximate solution function P(α)

Ny,Nτu(x, t)at any point (x, t)∈ D2

a,b,T using the

orthogonal shifted Gegenbauer interpolation written in Lagrange form:

P(α)

Ny,Nτu(x, t) =

Ny

X

i=0

Nτ

X

j=0

P(α)

Ny,Nτux(α)

Ny,i, t(α)

Nτ,j L(α),x

S,Ny,i(x)L(α),t

S,Nτ,j (t),(4.27)

where

L(α),x

S,Ny,i(x) = L(α)

Ny,i 2x−a−b

b−a, i = 0,...Ny;x∈[a, b],

L(α),t

S,Nτ,j (t) = L(α)

Nτ,j 2

Tt−1, j = 0,...Nτ;t∈[0, T ].

The CHBGPM can be carried out easily using Algorithms 1and 2; cf. Appendix A.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

10 KAREEM T. ELGINDY AND SAYED A. DAHY

Remark 4.1

In the cases when supp(k) = [0, T ], except for some ﬁnitely many real numbers, the CHBGPM can be carried out

straightforwardly by considering only temporal collocation points that are non-zeros of k. In particular, suppose

that tk, k = 0, . . . , Ntare the zeros of kin (0, T ), for some Nt∈Z+. We can directly implement the CHBGPM

exactly as described under Assumption 1, but using the “feasible” set of GG collocation mesh grid D(α)

Ny,Nτ:τ(α)

Nτ,i 6=

2

Ttk−1∀i, k instead.

5. CONVERGENCE AND ERROR ANALYSIS

We start this section by presenting three technical lemmas essential for the study of the convergence and truncation

error of the CHBGPM.

Lemma 5.1

Let f∈Cn+1 [−1,1] be approximated by the barycentric Gegenbauer expansion series (3.2). Then there exist some

numbers ξ(α)

n,j ∈[−1,1] , j = 0, ..., n such that

Zx(α)

n,j

−1Zx

−1

f(σ)dσdx =

n

X

i=0

p(2),x

B,j,if(α)

n,i +E(α)

2,n x(α)

n,j , ξ(α)

n,j ∀x(α)

n,j ∈S(α)

n,(5.1)

where

E(α)

2,n x(α)

n,j , ξ(α)

n,j =1

(n+ 1)! K(α)

n+1 dn+1

dxn+1 x(α)

n,j −xf(x)x=ξ(α)

n,j Zx(α)

n,j

−1

G(α)

n+1 (x)dx,

is the Gegenbauer quadrature error term.

Proof

The proof is established by combining Cauchy’s formula for repeated integration and [51, Theorem 3.1].

The following lemma gives the truncation error upper bounds associated with Formula (5.1).

Lemma 5.2

Suppose that f∈Cn+1 [−1,1], and

dn+1

dxn+1 x(α)

n,j −xf

L∞[−1,1]

≤A∈R+

0=R∪ {0},

for some number n∈Z+

0, where the constant Ais independent of n. Moreover, let Rx(α)

n,j

−1Rσ2

−1f(σ1)dσ1dσ2be

approximated by the numerical quadrature

n

P

i=0

p(2),x

B,j,if(α)

n,i , j = 0, . . . , n. Then there exist some positive constants D(α)

1

and D(α)

2independent of nsuch that the truncation error of the barycentric Gegenbauer quadrature, E(α)

2,n x(α)

n,j , ξ(α)

n,j ,

is bounded by the following inequalities

E(α)

2,n x(α)

n,j , ξ(α)

n,j ≤

A2−nx(α)

n,j + 1Γ (n+ 2α+ 1) Γ (α+ 1)

(n+ 1)! Γ (n+α+ 1) Γ (2α+ 1) , n ≥0∧α≥0,

A2−n−1x(α)

n,j + 1Γn+1

2+α

Γn+3

2Γ (n+α+ 1) ,n+ 1

2∈Z+∧ −1

2≤α < 0,

E(α)

2,n x(α)

n,j , ξ(α)

n,j <

∼A2−nx(α)

n,j + 1Γn

2+α+ 1

n

2!p(n+ 1) (n+ 2α+ 1) Γ (n+α+ 1) ,n

2∈Z+

0∧ −1

2< α < 0.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 11

Moreover, as n→ ∞, we have

E(α)

2,n x(α)

n,j , ξ(α)

n,j

∼

≤

B(α)

1e

2nx(α)

n,j + 1nα−n−3

2, α ≥0,

B(α)

2e

2nx(α)

n,j + 1n−n−3

2,−1

2< α < 0,

for all j= 0, . . . , n where B(α)

1=AD(α)

1and B(α)

2=AD(α)

2.

Proof

The proof follows readily using Cauchy’s formula for repeated integration and [51, Theorem 3.2].

Let z(α∗

r)

Ny,r,k, r = 0, . . . , Ny;k= 0, . . . , Mybe the adjoint GG nodes as deﬁned by Elgindy and Smith-Miles [50].

The following lemma estimates the asymptotic truncation error upper bound when approximating the natural

exponential function of the following deﬁnite integral

Zy(α)

Ny,r

−1

f1

2((b−a)ξ+a+b)dξ ∀r,

using the ﬁrst-order optimal Gegenbauer matrix, as the number of its columns grows large.

Lemma 5.3

Assume that

f(k)

L∞≤Ak/(My+1) ∈R+

0, k = 0, My+ 1. Then exp My

P

k=0

p(1),y

OB,r,k f1

2(b−a)z(α∗

r)

Ny,r,k +a+b!

is an approximation to exp Ry(α)

Ny,r

−1f1

2((b−a)ξ+a+b)dξ∀rwith an approximate truncation error upper

bound given by

c(α∗

r)

12−MyMyα∗

r−My−3

2eMy

1, α∗

r≥0,

c(α∗

r)

2My−α∗

r,−1/2< α∗

r<0

,

as My→ ∞, where c(α∗

r)

1and c(α∗

r)

2are positive constants dependent on α∗

r, but independent of My.

Proof. Let g(y) = Ry

−1f1

2((b−a)ξ+a+b)dξ and ˜gy(α)

Ny,r=

My

P

k=0

p(1),y

OB,r,k f1

2((b−a)z(α∗

r)

Ny,r,k +a+b)∀r.

Then it follows from [51, Theorem 4.2] that

gy(α)

Ny,r= ˜gy(α)

Ny,r+ ∆gy(α)

Ny,r, ζ (α∗

r)

My,r,

where

∆gy(α)

Ny,r, ζ (α∗

r)

My,r=

f(My+1) 1

2(b−a)ζ(α∗

r)

My,r +a+b

(My+ 1)! K(α∗

r)

My+1 Zy(α)

Ny,r

−1

G(α∗

r)

My+1 (σ)dσ,

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

12 KAREEM T. ELGINDY AND SAYED A. DAHY

and ζ(α∗

r)

My,r, r = 0, . . . , Ny, are some numbers in [−1,1]. By Taylor’s Theorem and [50, Lemmas 2.1 and 2.2], there

exists a positive real number ρsuch that

exp gy(α)

Ny,r−exp ˜gy(α)

Ny,r= exp ˜gy(α)

Ny,r

∞

X

k=1

1

k!∆gy(α)

Ny,r, ζ (α∗

r)

My,rk

< ρ exp

A0

My

X

k=0

p(1),y

OB,r,k

∆gy(α)

Ny,r, ζ (α∗

r)

My,r

≤

∼ρ A11 + y(α)

Ny,r

(My+ 1)! K(α∗

r)

My+1

exp A01 + y(α)

Ny,r

G(α∗

r)

My+1

∞

< ρ A1c(α∗

r)

32−My1 + y(α)

Ny,rMyα∗

r−My−3

2eA01+y(α)

Ny,r+My (1, α∗

r≥0,

c(α∗

r)

2My−α∗

r,−1/2< α∗

r<0!,

as My→ ∞, where c(α∗

r)

3is a positive constant dependent on α∗

r, but independent of My. This completes the proof of

the lemma.

One can easily show by a direct corollary of Lemma 5.3 that when f←λνf, the constant c(α∗

r)

1becomes

proportional with |λν|, which is inversely proportional to the viscosity parameter ν. Therefore, we expect the

truncation error upper bound to roughly increase for decreasing values of νwhile holding the value of Myﬁxed.

However, this rise of error is insigniﬁcant as the error term decays exponentially fast as My→ ∞. In particular, for

such cases, we can easily keep the error as small as desired by increasing the value of Mywithout having to increase

the number of collocation points; thus preserving the dimensionality of the produced algebraic linear system.

5.1. Convergence and error analysis of the CHBGPM under Assumption 1

The following two theorems give in respective order some upper bounds for the truncation error and the asymptotic

truncation error associated with the orthogonal collocation discretization of the integral formulation of Burger’s

equation deﬁned by Eq. (4.10) at the interior GG nodes using the CHBGPM and under Assumption 1.

Theorem 5.1 (Truncation error upper bound under Assumption 1)

Consider the integral formulation of Burger’s equation, Eq. (4.10), and its orthogonal collocation discretization given

by Eq. (4.12) at the interior GG nodes y(α)

Ny,r, τ (α)

Nτ,s,r= 0, . . . , Ny, and s= 0, . . . , Nτ, using the CHBGPM.

Moreover, suppose that the following assumptions hold

f(k)

L∞

≤Ak/(My+1) ∈R+

0, k = 0, My+ 1,(5.2a)

∂Ny+1ψ

∂yNy+1

L∞

≤A2∈R+

0,

∂Nτ+1ψ

∂τ Nτ+1

L∞

≤A3∈R+

0,(5.2b)

∂Ny+1

∂yNy+1 y(α)

Ny,r −yψ

L∞

≤A4∈R+

0∀r, (5.2c)

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 13

where the constants Ai, i = 0,...,4, are independent of Nyand Nτ. If we denote the corresponding truncation error

at each GG node by Eψ

r,s, then an upper bound for Eψ

r,s ∀r, s is given by

2−NyΓ(α+ 1)Γ(Ny+ 2α+ 1) A4y(α)

Ny,r + 3+ 2A2|ϑ3,s|

Γ(2α+ 1)(Ny+ 1)!Γ(Ny+α+ 1) , Ny≥0∧α≥0

2−Ny−1ΓNy+1

2+αA4y(α)

Ny,r + 3+ 2A2|ϑ3,s|

ΓNy+3

2Γ(Ny+α+ 1)

,Ny+ 1

2∈Z+∧ −1

2< α < 0

2−NyΓNy

2+α+ 1A4y(α)

Ny,r + 3+ 2A2|ϑ3,s|

Ny

2!p(Ny+ 1)(Ny+ 2α+ 1) Γ(Ny+α+ 1) ,Ny

2∈Z+

0∧ −1

2< α < 0

+µν

2−NτA3τ(α)

Nτ,s + 1Γ(α+ 1)Γ(Nτ+ 2α+ 1)

(Nτ+ 1)!Γ(Nτ+α+ 1)Γ(2α+ 1) , Nτ≥0∧α≥0

2−Nτ−1A3τ(α)

Nτ,s + 1ΓNτ+1

2+α

ΓNτ+3

2Γ(Nτ+α+ 1) ,Nτ+ 1

2∈Z+∧ −1

2< α < 0

2−NτA3τ(α)

Nτ,s + 1ΓNτ

2+α+ 1

Nτ

2!p(Nτ+ 1)(Nτ+ 2α+ 1) Γ(Nτ+α+ 1) ,Nτ

2∈Z+

0∧ −1

2< α < 0

+c(α∗

r),ν

4M−My+α∗

r−3

2

ye

2My

1, α∗

r≥0

c(α∗

r)

2M−α∗

r

y,−1

2< α∗

r<0

,

(5.3)

where c(α∗

r),ν

4is a positive constant dependent on α∗

rand ν, but independent of My.

Proof

Let y(α)

Ny,Ny+1 = 1, and denote the absolute values of the truncation error in the approximations of

Zy(α)

Ny,r

−1Zσ

−1

ψ(σ1, τ )dσ1dσ, Zτ(α)

Nτ,s

−1

ψ(y, σ)dσ , Z1

−1

ψ(σ, τ )dσ, and exp λνZy(α)

Ny,r

−1

f1

2((b−a)ξ+a+b)dξ!

by E1,r,s, E2,r,s, E3,s , and ERHS,r, respectively, for each rand s. We ﬁnd through Lemma 5.2 and [51, Theorem 3.2]

that

E1,r,s ≤

A42−Nyy(α)

Ny,r + 1Γ (Ny+ 2α+ 1) Γ (α+ 1)

(Ny+ 1)! Γ (Ny+α+ 1) Γ (2α+ 1) , Ny≥0∧α≥0,

A42−Ny−1y(α)

Ny,r + 1ΓNy+1

2+α

ΓNy+3

2Γ (Ny+α+ 1)

,Ny+ 1

2∈Z+∧ −1

2< α < 0,

(5.4a)

E1,r,s <

A42−Nyy(α)

Ny,r + 1ΓNy

2+α+ 1

Ny

2!p(Ny+ 1) (Ny+ 2α+ 1) Γ (Ny+α+ 1),Ny

2∈Z+

0∧ −1

2< α < 0,(5.4b)

E2,r,s ≤

A32−Nττ(α)

Nτ,s + 1Γ (Nτ+ 2α+ 1) Γ (α+ 1)

(Nτ+ 1)! Γ (Nτ+α+ 1) Γ (2α+ 1) , Nτ≥0∧α≥0,

A32−Nτ−1τ(α)

Nτ,s + 1ΓNτ+1

2+α

ΓNτ+3

2Γ (Nτ+α+ 1) ,Nτ+ 1

2∈Z+∧ −1

2< α < 0,

(5.5a)

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

14 KAREEM T. ELGINDY AND SAYED A. DAHY

E2,r,s <

A32−Nττ(α)

Nτ,s + 1ΓNτ

2+α+ 1

Nτ

2!p(Nτ+ 1) (Nτ+ 2α+ 1) Γ (Nτ+α+ 1),Nτ

2∈Z+

0∧ −1

2< α < 0,(5.5b)

E3,s ≤

A221−NyΓ (Ny+ 2α+ 1) Γ (α+ 1)

(Ny+ 1)! Γ (Ny+α+ 1) Γ (2α+ 1) , Ny≥0∧α≥0,

A22−NyΓNy+1

2+α

ΓNy+3

2Γ (Ny+α+ 1)

,Ny+ 1

2∈Z+∧ −1

2< α < 0,

(5.6a)

E3,s <

A221−NyΓNy

2+α+ 1

Ny

2!p(Ny+ 1) (Ny+ 2α+ 1) Γ (Ny+α+ 1),Ny

2∈Z+

0∧ −1

2< α < 0.(5.6b)

Through [51, Theorem 3.2] and Lemma 5.3, we can derive the following useful inequality

ERHS,r <

∼c(α∗

r),ν

42−MyMyα∗

r−My−3

2eMy

1, α∗

r≥0,

c(α∗

r)

2My−α∗

r,−1/2< α∗

r<0

,(5.7)

as My→ ∞. The truncation error upper bound (5.3) follows directly by realizing that

Eψ

r,s ≤E1,r,s +E1,Ny+1 +µνE2,r,s +|ϑ3,s|E3,s +ERHS,r.

The following theorem highlights the asymptotic truncation error upper bound under Assumption 1, as Ny, Nτ,

and My→ ∞.

Theorem 5.2 (Asymptotic truncation error upper bound under Assumption 1)

Consider the integral formulation of Burger’s equation, Eq. (4.10), and its orthogonal collocation discretization at the

interior GG nodes y(α)

Ny,r, τ (α)

Nτ,s,r= 0, . . . , Ny, and s= 0, . . . , Nτ, Eq. (4.12), using the CHBGPM. There exist

some non-negative constants B(α)

i, i = 1,3dependent on αand B(α)

i,ν , i = 2,4dependent on αand ν, but independent

of Nyand Nτ, such that the associated asymptotic truncation error at each GG node, as Ny, Nτ, and My→ ∞, is

bounded by

B(α)

1Nα−Ny−3/2

ye

2Ny+B(α)

2,ν Nα−Nτ−3/2

τe

2Nτ, α ≥0

B(α)

3N−Ny−3/2

ye

2Ny+B(α)

4,ν N−Nτ−3/2

τe

2Nτ,−1

2< α < 0

+c(α∗

r),ν

4e

2MyMα∗

r−My−3/2

y

1, α∗

r≥0

c(α∗

r)

2M−α∗

r

y,−1

2< α∗

r<0

.

(5.8)

Proof

The proof is straightforward using the asymptotic results of Lemma 5.2 and [51, Theorem 3.2].

Theorem 5.2 shows that the truncation error in the discretization of Eq. (4.10) converges uniformly to zero on

D(α)

Ny,Nτwith an exponential rate of convergence.

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 15

5.2. Uniform convergence of the CHBGPM under Assumption 1

Recent comprehensive theoretical and experimental studies conducted on Gegenbauer collocations based on Gauss

and ﬂipped Gauss-Radau points that can be found in [48–51,54–56] manifest that such a class of numerical schemes

behaves at its best for problems exhibiting smooth solutions when the Gegenbauer parameter αfalls within a certain

“apt” interval inside (−1/2,∞). That interval was named “the Gegenbauer collocation interval of choice” by Elgindy

[56], and was later slightly modiﬁed by Elgindy and Refat [55] into “the Gegenbauer parameter collocation interval

of choice” to distinguish that interval from the regular interval [−1,1] used in most collocation schemes applied using

the orthogonal Jacobi family of polynomials. In particular, to maintain stable computations and to avoid degradation

of accuracy, Gegenbauer collocations for small/medium numbers of collocation points and Gegenbauer expansion

terms should be performed for values of αthat falls in the following interval:

IG

ε,r ={α| − 1/2 + ε≤α≤r, 0< ε 1, r ∈[1,2]}.

In fact, the new theoretical and numerical ﬁndings that can be found in [55] reveal that the recommended interval IG

ε,r

is “largely dependent” on the number of collocation points, say n, and it was concluded that the interval IG

ε,r shrinks

as ngrows larger in the sense that there exists a positive integer n∗so that IG

ε,r ⊆[l, 0], for some negative real number

l≥ −0.5 + εif n>n∗. That is, ‘collocations for adequate non-positive values of αinside the interval [l, 0] are highly

recommended for values of n>n∗.’ For values of n>n∗, and as ncontinuously grows up, the interval IG

ε,r ⊆[l, 0]

continues to shrink from the left endpoint towards zero such that

lim

n→∞ IG

ε,r ={0},

i.e., collocations at the ﬂipped-Gauss-Chebyshev-Radau points (also Gauss-Chebyshev (GC) points) should be put

into effect for exceedingly large numbers of collocation points and Gegenbauer expansion terms if the approximations

are sought in the inﬁnity norm.

Let ˜

ψ(α)

Ny,Nτ(y, τ )be the bivariate Lagrange interpolating polynomial that ﬁts the data points set

y(α)

Ny,r, τ (α)

Nτ,s,˜

ψ(α)

Ny,Nτy(α)

Ny,r, τ (α)

Nτ,s, r = 0, . . . , Ny;s= 0, . . . , Nτ,

and denote the set of mesh grid D(α)

Ny,Nτ:α∈IG

ε,r by D(α),G

Ny,Nτ. The above argument motivates us to adhere to the

following rule of thumb.

Rule of Thumb 1. “For smooth solutions, and for collocations performed at D(α),G

Ny,Nτ, as recently prescribed

by Elgindy and Refat [55], the CHBGPM generates a sequence of approximate smooth functions, ˜

ψ(α)

Ny,Nτ, that

converge uniformly to ψon D(α),G

Ny,Nτwith the same order of exponential rate of convergence as that stated in Theorem

5.2.”

Denote the set of SGG mesh grid ˆ

D(α)

Ny,Nτ:α∈IG

ε,r by ˆ

D(α),G

Ny,Nτ. The following two theorems show that P(α)

Ny,Nτu→

uuniformly on ˆ

D(α),G

Ny,Nτif Rule of Thumb 1 holds true.

Theorem 5.3

Suppose that Rule of Thumb 1 holds true. Then ˜

φ(α)

S,Ny,Nτ→φSand ∂

∂y ˜

φ(α)

S,Ny,Nτ→∂φS/∂y uniformly on D(α),G

Ny,Nτ

with the same order of exponential convergence as that given by Theorem 5.2.

Proof

Substitute Eq. (4.8) into Eq. (4.6) to get

φS(y, τ ) = Zy

−1Zσ

−1

−Z1

−1Zσ

−1ψ(σ1, τ )dσ1dσ +ϑ3(τ)Z1

−1

ψ(σ, τ )dσ +ϑ1(τ) (y+ϑ3(τ)−1) .

Deﬁne the smooth function ¯

φSon D2

a,b,T by

¯

φS(y, τ ) = Zy

−1Zσ

−1

−Z1

−1Zσ

−1˜

ψ(α)

Ny,Nτ(σ1, τ )dσ1dσ +ϑ3(τ)Z1

−1

˜

ψ(α)

Ny,Nτ(σ, τ )dσ +ϑ1(τ) (y+ϑ3(τ)−1) .

(5.9)

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

16 KAREEM T. ELGINDY AND SAYED A. DAHY

Moreover, let ¯

φ(α)

S,Ny,Nτ,r,s denotes ¯

φSy(α)

Ny,r, τ (α)

Nτ,s∀r, s. Since integration preserves uniform convergence, and

by assumption, ˜

ψ(α)

Ny,Nτ→ψuniformly on D(α),G

Ny,Nτ, and the approximate function ˜

ψ(α)

Ny,Nτis continuous on D2

a,b,T ,

then ¯

φS→φSuniformly on D(α),G

Ny,Nτwith the referred exponential order of convergence. Now, to prove that

˜

φ(α)

S,Ny,Nτ→φSuniformly on D(α),G

Ny,Nτ, notice that the discrete form of Eq. (5.9) on D(α),G

Ny,Nτis equivalent to that

of Eq. (4.24), which yields ˜

φ(α)

S,Ny,Nτ,r,s ∀r, s. Using [51, Theorem 3.2] and Theorem 5.2 we can show that

¯

φ(α)

S,Ny,Nτ,r,s −˜

φ(α)

S,Ny,Nτ,r,s

∼

≤

B(α)

1e

2NyNα−Ny−3

2

y, α ≥0,

B(α)

2e

2NyN−Ny−3

2

y,−1/2< α < 0,

for some non-negative numbers B(α)

1and B(α)

2, as Ny, Nτ→ ∞. This shows that ˜

φ(α)

S,Ny,Nτ→φSuniformly on

D(α),G

Ny,Nτ. By a similar argument, and through Eq. (4.5), we can also show that ∂

∂y ˜

φ(α)

S,Ny,Nτ→∂φS/∂y uniformly on

D(α),G

Ny,Nτ.

Theorem 5.4

Suppose that Rule of Thumb 1 holds true. Moreover, let F(y) = φ(y, τ)∀y∈[−1,1], and for any arbitrary τ∈[−1,1];

also, denote its approximation generated by the CHBGPM by FNy(y). Then the quotient F0

Ny(y).FNy(y)→

F0(y)/F(y)uniformly on [−1,1], as Ny→ ∞.

Proof

By Theorem 5.3,FNy(y)→F(y), and its derivative, F0

Ny(y)→F0(y)uniformly on [−1,1], as Ny, Nτ→ ∞.

Therefore, by deﬁnition, for any given two positive real numbers ε1, ε2there exist two positive integers N1=N1(ε1)

and N2=N2(ε2)such that

FNy(y)−F(y)< ε1if Ny> N1∀y∈[−1,1],

and

F0

Ny(y)−F0(y)< ε2if Ny> N2∀y∈[−1,1].

Since the sequences FNy(y)and F0

Ny(y)are uniformly convergent, they are uniformly bounded, and there

exist positive real numbers k1and k2such that F0

Ny(y)≤k1and FNy(y)≤k2∀Ny. We could choose k1and k2

such that

F0

Ny(y)

FNy(y)

<k1

k2

∀Ny.

Since Fis bounded and F(y)6= 0 ∀y∈[−1,1] by assumption, there exists a positive real number msuch that

kFkL∞([−1,1]) ≤m. Let

ε1=mk2

2k1

εand ε2=m

2ε,

for a relatively small positive number ε. We need to show that there exists a positive integer N=N(ε)such that

F0

Ny(y)

FNy(y)−F0(y)

F(y)

< ε if Ny> N.

Math Methods Appl Sci

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COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 17

Indeed, since

F0

Ny(y)

FNy(y)−F0(y)

F(y)

=

F0

Ny(y)F(y)−FNy(y)F0(y)

FNy(y)F(y)

=

F0

Ny(y)F(y)−F0

Ny(y)FNy(y) + F0

Ny(y)FNy(y)−FNy(y)F0(y)

FNy(y)F(y)

=

−F0

Ny(y)FNy(y)−F(y)+FNy(y)F0

Ny(y)−F0(y)

FNy(y)F(y)

≤F0

Ny(y)FNy(y)−F(y)

FNy(y)|F(y)|+F0

Ny(y)−F0(y)

|F(y)|

<k1

mk2mk2

2k1

ε+1

mm

2ε=εif Ny> N = max{N1, N2} ∀y∈[−1,1],

then the quotient F0

Ny(y).FNy(y)→F0(y)/F(y)uniformly on [−1,1].

Since the chosen number τwas arbitrary, it follows by Theorem 5.4 that P(α)

Ny,Nτu→uuniformly on D2

a,b,T with the

stated exponential order of convergence. We shall verify our claim numerically later in Section 6through numerical

simulations.

5.3. Convergence and error analysis of the CHBGPM under Assumption 2

The following two theorems give in respective order some useful upper bounds for the truncation error and the

asymptotic truncation error associated with the orthogonal collocation discretization of the augmented system of

equations formed by the integral formulation of Burger’s equation deﬁned by Eq. (4.18) and the integral boundary

condition (4.17) at the interior GG nodes using the CHBGPM and under Assumption 2.

Theorem 5.5 (Truncation error upper bounds under Assumption 2)

Consider the integral formulation of Burger’s equation (4.18) provided with the integral boundary condition (4.17),

and their orthogonal collocation discretization at the interior GG nodes y(α)

Ny,r, τ (α)

Nτ,s,r= 0, . . . , Ny, and s=

0, . . . , Nτ, Eqs. (4.19a) and (4.19b), using the CHBGPM. Moreover, suppose that Assumptions (5.2a), (5.2b), and

(5.2c) hold. If we denote the corresponding truncation errors associated with Eqs. (4.17) and (4.18) at each GG node

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

18 KAREEM T. ELGINDY AND SAYED A. DAHY

by Eψ

1,s and Eψ

2,r,s, respectively, then Eψ

1,s is bounded by Inequalities (5.6a) and (5.6b), and

Eψ

2,r,s≤

2−NyA4Γ(α+ 1)Γ(Ny+ 2α+ 1) y(α)

Ny,r + 1

Γ(2α+ 1)(Ny+ 1)!Γ(Ny+α+ 1) , Ny≥0∧α≥0

2−Ny−1A4ΓNy+1

2+αy(α)

Ny,r + 1

ΓNy+3

2Γ(Ny+α+ 1)

,Ny+ 1

2∈Z+∧ −1

2< α < 0

2−NyA4ΓNy

2+α+ 1y(α)

Ny,r + 1

Ny

2!p(Ny+ 1)(Ny+ 2α+ 1)Γ(Ny+α+ 1) ,Ny

2∈Z+

0∧ −1

2< α < 0

+µν

2−NτA3Γ(α+ 1)Γ(Nτ+ 2α+ 1) τ(α)

Nτ,s + 1

(Nτ+ 1)!Γ(Nτ+α+ 1)Γ(2α+ 1) , Nτ≥0∧α≥0

2−Nτ−1A3ΓNτ+1

2+ατ(α)

Nτ,s + 1

ΓNτ+3

2Γ(Nτ+α+ 1) ,Nτ+ 1

2∈Z+∧ −1

2< α < 0

2−NτA3ΓNτ

2+α+ 1τ(α)

Nτ,s + 1

Nτ

2!p(Nτ+ 1)(Nτ+ 2α+ 1)Γ(Nτ+α+ 1) ,Nτ

2∈Z+

0∧ −1

2< α < 0

+c(α∗

r),ν

4M−My+α∗

r−3

2

ye

2My

1, α∗

r≥0

c(α∗

r)

2M−α∗

r

y,−1

2< α∗

r<0

.

(5.10)

Proof

Following the mathematical convention made in the proof of Theorem 5.1, we directly ﬁnd that |E1ψs|=E3,s ∀s,

and the rest of the proof is established by combining the following useful inequality

|E2ψr,s| ≤ E1,r,s +µνE2,r,s +ERHS,r ∀r, s,

with Inequalities (5.4a), (5.4b), (5.5a), (5.5b), and (5.7).

Theorem 5.6 (Asymptotic truncation error upper bounds under Assumption 2)

Consider the integral formulation of Burger’s equation (4.18) provided with the integral boundary condition (4.17),

and their orthogonal collocation discretization at the interior GG nodes y(α)

Ny,r, τ (α)

Nτ,s,r= 0, . . . , Ny, and s=

0, . . . , Nτ, Eqs. (4.19a), (4.19b), and (4.20), using the CHBGPM. If we denote the corresponding truncation errors

associated with Eqs. (4.17) and (4.18) at each GG node by Eψ

1,s and Eψ

2,r,s, respectively, then there exist some non-

negative constants B(α)

1and B(α)

2such that

Eψ

1,s≤(B(α)

1e

2NyNα−Ny−3

2

y, α ≥0,

B(α)

2e

2NyN−Ny−3

2

y,−1

2< α < 0,

and Eψ

2,r,s is bounded by inequalities of the form (5.8), as Ny, Nτ, and My→ ∞.

Proof

The proof is straightforward using the asymptotic results of Theorem 5.2 and [51, Theorem 3.2].

Theorem 5.6 shows that the truncation errors in the discretization of Eqs. (4.17) and (4.18) converge uniformly to

zero on D(α)

Ny,Nτwith exponential rates of convergence.

Math Methods Appl Sci

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COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 19

5.4. Uniform convergence of the CHBGPM under Assumption 2

Similar to the argument presented in Section 5.2, we introduce the following rule of thumb.

Rule of Thumb 2. “For smooth solutions, and for collocations performed at D(α),G

Ny,Nτ, as prescribed by Elgindy and

Refat [55], the CHBGPM generates sequences of approximate smooth functions, ˜

ψ(α)

Ny,Nτand ˜

ϑ(α)

2,Ny,Nτ, that

converge uniformly to ψand ϑ2, respectively, on D(α),G

Ny,Nτwith the same orders of exponential rates of convergence

as that stated in Theorem 5.6.”

Based on Rule of Thumb 2 and an argument similar to that presented in Section 5.2, we can also show that

P(α)

Ny,Nτu→uuniformly on D2

a,b,T with an exponential order of convergence as given by Inequalities (5.8).

6. NUMERICAL EXPERIMENTS

In this section we apply the CHBGPM on three well-studied test examples in the literature with known exact solutions.

The numerical experiments were conducted on a personal laptop equipped with an Intel(R) Core(TM) i5-4210U CPU

@1.70GHz (4CPUs), ∼2.4GHZ speed running on a Windows 10 64-bit operating system. The resulting algebraic

linear system of equations were solved using MATLAB mldivide algorithm provided with MATLAB V. R2015a

(8.5.0.197613). Collocations were carried out for values of αchosen optimally based on experimental testing among

the range of candidate values −0.49(0.01)2. We report the exact and approximate solutions obtained by the CHBGPM

using several values of Ny, Nτ, α, and ν. We use the symbol “N” when Ny=Nτ=N∈Z+. To further verify

the accuracy of our plots, we report also the cross sections of the exact and approximate solution surfaces at the

average value of x∀t. Moreover, we support our numerical results by reporting the absolute error matrix Eu,(α)

N,N

whose elements are deﬁned by

Eu,(α)

N,N r,s =ux(α)

N,r , t(α)

N,s−P(α)

N,N ux(α)

N,r , t(α)

N,s,x(α)

N,r , t(α)

N,s∈ˆ

D(α),G

N,N ∀r, s,

the maximum absolute errors at different times; in addition to the 1-, 2-, and inﬁnity-norms of Eu,(α)

N,N . To demonstrate

the rapid exponential convergence achieved by the CHBGPM in all test examples, we report in Figure 6the relation

between the number Nand the natural logarithm of the maximum absolute error,

vec Eu,(α)

N,N

∞

, for various

values of αin each case, where “vec” denotes the vectorization of a matrix. Comparisons with other competitive

numerical schemes are also presented to assess further the accuracy of the CHBGPM.

Example 6.1

Consider Burger’s problem P1with ν= 0.5and 1, subject to the initial condition

u(x, 0) = x

σ,0≤x≤1,

and the boundary conditions

u(0, t)=0, u (1, t) = 1

σ+t,0≤t≤1,

where σ > 1is a parameter. The exact solution of this problem is

u(x, t) = x

σ+t,0≤x≤1,0≤t≤1.

The numerical results are reported in Figure 1and Table Ishowing excellent approximations using relatively small

values of N. It is interesting to observe in Table Imore accuracy in the approximations for larger values of σ; but

Math Methods Appl Sci

Prepared using speauth.cls DOI: 10.1002/mma.5135

20 KAREEM T. ELGINDY AND SAYED A. DAHY

why? The answer to this question can be drawn from Theorem 5.1. In fact, we can show in this example that

ψ(y, τ ) = 1

64ν2σ2e−(1+y)2

16νσ (1 + y)2−8νσ∀y, τ ∈,D2,

lim

σ→∞

∂13ψ

∂y13 = lim

σ→∞

∂13

∂y13 y(α)

Ny,r −yψ= 0 ∀r,

∂Nτ+1ψ

∂τ Nτ+1 = 0 ∀Nτ.

Hence, A3= 0, and the upper bounds A2and A4monotonically decrease for increasing values of σso that the

truncation error upper bound, Eψ

r,s, vanishes quickly at any collocation point as shown by Inequality (5.3), assuming

that the CHBGPM was implemented using a certain value of α∈IG

ε,r. The reader should notice though that Table I

was constructed using two different values of αwith a slight difference– too small to cause any change in the above

argument.

0

11

0.005

(a)

0.5

0.01

0.5

00

0

11

0.005

(b)

0.5

0.01

0.5

00

0 0.2 0.4 0.6 0.8 1

4.94

4.96

4.98

510-3 (d)

u(0.5,t)

P4,4

(2) u(0.5,t)

0

11

0.5

10-9 (c)

0.5

1

0.5

00

Figure 1. The numerical simulation of Example 6.1 using the CHBGPM. Figure (a) shows the exact solution on D2

0,1,1. Figure

(b) shows the approximate solution on the same region obtained using N= 4, for the parameters ν= 1,σ= 100, and α= 2.

The ﬁgure was generated using 101 linearly spaced nodes in the x- and t-directions from 0 to 1. Figure (c) shows the absolute

errors at the SGG nodes. Figure (d) shows the cross sections of the exact and approximate solution surfaces at x= 0.5∀t.

The reported 1-, 2-, and inﬁnity-norms of the absolute error matrix constructed at the SGG mesh grid were in respective order

1.553e−09,2.259e−09, and 4.811e−09.

Table I. The maximum absolute errors of Example 6.1 for N= 12, ν = 0.1,0.5,1,2, and σ= 2,100 at different times

t= 0.1,0.5,1. The corresponding values of αfor σ= 2 and 100 were −0.49 and −0.2, respectively.

ν t = 0.1t= 0.5t= 1

σ= 2 σ= 100 σ= 2 σ= 100 σ= 2 σ= 100

0.1 1.70e-09 1.11e-15 8.19e-10 1.26e-15 1.65e-10 3.83e-15

0.5 1.29e-13 5.90e-15 6.56e-14 4.38e-15 2.53e-14 3.03e-14

1.0 9.09e-14 4.97e-14 6.51e-14 2.33e-14 2.50e-14 2.36e-14

2.0 7.76e-14 7.34e-14 7.35e-14 2.28e-14 8.03e-14 7.62e-14

Example 6.2

Consider Burger’s problem P1with ν= 1,0.5, and 0.1subject to the initial condition

u(x, 0) = 2νπ sin (πx)

σ+ cos (πx),0≤x≤1,

Math Methods Appl Sci

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COLE-HOPF BARYCENTRIC GEGENBAUER INTEGRAL PSEUDOSPECTRAL METHOD 21

together with the boundary conditions

u(0, t) = u(1, t) = 0, t > 0,

where σ > 1is a parameter. The exact solution of this problem is

u(x, t) = 2νπe−π2νt sin (πx)

σ+e−π2νt cos (πx),0≤x≤1, t > 0.

The numerical results are reported in Figures 2–4, and in Tables II–VI. Table II manifests again the superior

accuracy of the CHBGPM even when the viscosity paramter ν→0, in the absence of any adaptive strategies required

for adaptive reﬁnements, whereas most of the numerical methods available in the literature fail to capture the physical

behavior of the solutions at this limit. Table II shows another quite interesting behavior. In particular, we observe

that the error produced by the CHBGPM monotonically decay for decreasing values of the viscosity parameter, ν– a

behavior that was not captured in Table I. This can be directly linked to Theorem 5.5, which shows that the truncation

error upper bound, Eψ

2,r,s, is dominated by the second term including the parameter µνas Ngrows larger, as clearly

shown by Inequality (5.10). The rest of the story stems from the linear dependence of µνwith ν. This result was not

salient in Table I, because the second term vanishes, and the values of νhave no direct effect on the truncation error

upper bound.

Example 6.2 was previously solved in a series of papers. For instance, Mittal and Jain [57] applied a collocation

scheme using modiﬁed cubic B-splines over ﬁnite elements for the spatial discretization followed by a Runge-Kutta

scheme in time to solve the reduced ﬁrst-order ordinary differential equations system. Jiwari [58] treated the problem

using uniform Haar wavelets combined with a quasilinearization approach; however, the method failed for ν < 0.003.

In attempt to overcome this drawback, Jiwari [59] considered later a hybrid numerical scheme based on Euler implicit

method, quasilinearization, and uniform Haar wavelets to solve the problem. Ganaie and Kukreja [60] further solved

the problem using a cubic Hermite collocation method to produce continuous approximations to the solution and its

derivative throughout the solution range. The method was shown through linear stability analysis to be unconditionally

stable when combined with Crank-Nicolson approximation in time. Tables III–VI show the superior accuracy and the

exponential convergence achieved by the present method using relatively small values of Nyand Nτ.

0

11

10-3 (b)

5

0.5 0.5

00

0 0.2 0.4 0.6 0.8 1

2

4

6

810-3 (d)

u(0.5,t)

P8,8

(2) u(0.5,t)

0

1

1

1

10-7 (c)

0.5

2

0.5

00

0

11

10-3 (a)

5

0.5 0.5

00

Figure 2. The numerical simulation of Example 6.2 using the CHBGPM. Figure (a) shows the exact solution on D2

0,1,1. Figure

(b) shows the approximate solution on the same region obtained using N= 8, for the parameters ν= 0.1,σ= 100, and α= 2.

The ﬁgure was generated using 101 linearly spaced nodes in the x- and t-directions from 0 to 1. Figure (c) shows the absolute

errors at the SGG nodes. Figure (d) shows the cross sections of the exact and approximate solution surfaces at x= 0.5∀t.

The reported 1-, 2-, and inﬁnity-norms of the absolute error matrix constructed at the SGG mesh grid were in respective order

3.084e−07,3.274e−07, and 6.461e−07.

Math Methods Appl Sci

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