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Abstract

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 $\nu \to 0$, in the absence of any adaptive strategies typically required for adaptive refinements.
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. Elgindy1,2Sayed 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 ν0, 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. [39].
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. [10,2035]. 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. [3648]. 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 +uu
∂x =ν2u
∂x2,(x, t)∈ D2
a,b,T ,(2.1)
subject to the initial condition
u(x, 0) = f(x), a xb,
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),0tT,
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 nZ+, 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(α1/21/2)
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)
n1(x), n = 1,2,3, ...,
or in terms of the hypergeometric functions [49]:
G(α)
n(x) = 2F1n, 2α+n;α+1
2;1x
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= 2n1Γ(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)α1/2. 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)= 212απΓ(n+ 2α)
n!(n+α2(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)
n1(x), n 1,(3.1)
ZG(α)
n(x)dx =(2α1)
(n+ 1)(n+ 2α1)G(α1)
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)
i1(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+ 2α1i.
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
xx(α)
n,j ,n
X
i=0
ξ(α)
n,i
xx(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 cos1x(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 ,0i, 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
xx(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
xx(α)
n,i
xx(α)
n,k
=ξ(α)
n,j
xx(α)
n,i
xx(α)
n,j
.
It follows with S(x) =
n
P
k=0
ξ(α)
n,k xx(α)
n,i .xx(α)
n,kthat
S(x)DxL(α)
B,n,j (x) + L(α)
B,n,j (x)S0(x) = ξ(α)
n,j xx(α)
n,i
xx(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 m1; 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 fCn+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;n1.
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 =ν2φ
∂x2,(x, t)∈ D2
a,b,T ,
with the initial condition
φ(x, 0) = exp 1
2νZx
a
f(ξ), a xb,
and the mixed Neumann and Robin boundary conditions
φx(a, t) = h(t)
2ν, φ(b, t)k(t)+2νφx(b, t)=0,0tT.
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= (2xab)/(ba), τ =2
Tt1,
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((ba)ξ+a+b),1y1,(4.2b)
and the boundary conditions
∂φS(1, τ )
∂y =λνh(T(τ+ 1) /2) ,1τ1,(4.2c)
φS(1, τ )k(T(τ+ 1) /2) 1
λν
∂φS(1, τ )
∂y = 0,1τ1,(4.2d)
where
µν=2νT
(ba)2, λν=ab
4ν,
and
φS(y, τ ) = φba
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)
∂y2= exp λνZy
1
f1
2((ba)ξ+a+b).(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 )+ϑ1(τ),(4.5)
φS(y, τ ) = Zy
1Zσ
1
ψ(σ1, τ )1+ (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 )1
1
σ
1
ψ(σ1, τ )1+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 )11
1
σ
1
ψ(σ1, τ )1µν
τ
1
ψ(y, σ)+ϑ3(τ)1
1
ψ(σ, τ )= Ω(y, τ),(4.10)
where
Ω(y, τ ) = exp λν
y
1
f1
2((ba)ξ+a+b)+ϑ1(τ) (1 yϑ3(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)
B=p(q)
B,r,sand
P(q),y
OB =p(q),y
OB,r,k , r, s = 0, . . . , Nθ;k= 0, . . . , MyZ+;qZ+;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)
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,ss,˜
ψ(α)
Ny,Nτ,r,s ψy(α)
Ny,r, τ (α)
Nτ,sr, s, and
(α)
r,s = exp
λν
My
X
k=0
p(1),y
OB,r,k f1
2((ba)z(α
r)
Ny,r,k +a+b)
+ϑ1,s 1y(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 =,(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 (α)
r,s , r = 0, . . . , Ny;s= 0, . . . , Nτin the form of
vector arrays, Ψ=˜
ψnL
n=0 and ={n}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)
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)
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 )=ϑ1(τ).(4.17)
Through Eqs. (4.3), (4.4), and (4.6) we have
y
1
σ
1
ψ(σ1, τ )1+ϑ2(τ)µν
τ
1
ψ(y, σ)=η(y , τ),(4.18)
where
η(y, τ ) = exp λν
y
1
f1
2((ba)ξ+a+b)(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)
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((ba)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)
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)
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,sˆ
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(ba)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
ba
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 2xab
ba, i = 0,...Ny;x[a, b],
L(α),t
S,Nτ,j (t) = L(α)
Nτ,j 2
Tt1, 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 NtZ+. 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
Ttk1i, 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 fCn+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 fCn+1 [1,1], and
dn+1
dxn+1 x(α)
n,j xf
L[1,1]
AR+
0=R∪ {0},
for some number nZ+
0, where the constant Ais independent of n. Moreover, let Rx(α)
n,j
1Rσ2
1f(σ1)12be
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
A2nx(α)
n,j + 1Γ (n+ 2α+ 1) Γ (α+ 1)
(n+ 1)! Γ (n+α+ 1) Γ (2α+ 1) , n 0α0,
A2n1x(α)
n,j + 1Γn+1
2+α
Γn+3
2Γ (n+α+ 1) ,n+ 1
2Z+∧ −1
2α < 0,
E(α)
2,n x(α)
n,j , ξ(α)
n,j <
A2nx(α)
n,j + 1Γn
2+α+ 1
n
2!p(n+ 1) (n+ 2α+ 1) Γ (n+α+ 1) ,n
2Z+
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 + 1nαn3
2, α 0,
B(α)
2e
2nx(α)
n,j + 1nn3
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((ba)ξ+a+b)r,
using the first-order optimal Gegenbauer matrix, as the number of its columns grows large.
Lemma 5.3
Assume that
f(k)
LAk/(My+1) R+
0, k = 0, My+ 1. Then exp My
P
k=0
p(1),y
OB,r,k f1
2(ba)z(α
r)
Ny,r,k +a+b!
is an approximation to exp Ry(α)
Ny,r
1f1
2((ba)ξ+a+b)rwith an approximate truncation error upper
bound given by
c(α
r)
12MyMyα
rMy3
2eMy
1, α
r0,
c(α
r)
2Myα
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((ba)ξ+a+b)and ˜gy(α)
Ny,r=
My
P
k=0
p(1),y
OB,r,k f1
2((ba)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(ba)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
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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,rk
< 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)
32My1 + y(α)
Ny,rMyα
rMy3
2eA01+y(α)
Ny,r+My (1, α
r0,
c(α
r)
2Myα
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 fλνf, 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
A2R+
0,
Nτ+1ψ
∂τ Nτ+1
L
A3R+
0,(5.2b)
Ny+1
∂yNy+1 y(α)
Ny,r yψ
L
A4R+
0r, (5.2c)
Math Methods Appl Sci
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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
2NyΓ(α+ 1)Γ(Ny+ 2α+ 1) A4y(α)
Ny,r + 3+ 2A2|ϑ3,s|
Γ(2α+ 1)(Ny+ 1)!Γ(Ny+α+ 1) , Ny0α0
2Ny1ΓNy+1
2+αA4y(α)
Ny,r + 3+ 2A2|ϑ3,s|
ΓNy+3
2Γ(Ny+α+ 1)
,Ny+ 1
2Z+∧ −1
2< α < 0
2NyΓNy
2+α+ 1A4y(α)
Ny,r + 3+ 2A2|ϑ3,s|
Ny
2!p(Ny+ 1)(Ny+ 2α+ 1) Γ(Ny+α+ 1) ,Ny
2Z+
0∧ −1
2< α < 0
+µν
2NτA3τ(α)
Nτ,s + 1Γ(α+ 1)Γ(Nτ+ 2α+ 1)
(Nτ+ 1)!Γ(Nτ+α+ 1)Γ(2α+ 1) , Nτ0α0
2Nτ1A3τ(α)
Nτ,s + 1ΓNτ+1
2+α
ΓNτ+3
2Γ(Nτ+α+ 1) ,Nτ+ 1
2Z+∧ −1
2< α < 0
2NτA3τ(α)
Nτ,s + 1ΓNτ
2+α+ 1
Nτ
2!p(Nτ+ 1)(Nτ+ 2α+ 1) Γ(Nτ+α+ 1) ,Nτ
2Z+
0∧ −1
2< α < 0
+c(α
r)
4MMy+α
r3
2
ye
2My
1, α
r0
c(α
r)
2Mα
r
y,1
2< α
r<0
,
(5.3)
where c(α
r)
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 )1dσ, Zτ(α)
Nτ,s
1
ψ(y, σ)dσ , Z1
1
ψ(σ, τ )dσ, and exp λνZy(α)
Ny,r
1
f1
2((ba)ξ+a+b)!
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
A42Nyy(α)
Ny,r + 1Γ (Ny+ 2α+ 1) Γ (α+ 1)
(Ny+ 1)! Γ (Ny+α+ 1) Γ (2α+ 1) , Ny0α0,
A42Ny1y(α)
Ny,r + 1ΓNy+1
2+α
ΓNy+3
2Γ (Ny+α+ 1)
,Ny+ 1
2Z+∧ −1
2< α < 0,
(5.4a)
E1,r,s <
A42Nyy(α)
Ny,r + 1ΓNy
2+α+ 1
Ny
2!p(Ny+ 1) (Ny+ 2α+ 1) Γ (Ny+α+ 1),Ny
2Z+
0∧ −1
2< α < 0,(5.4b)
E2,r,s
A32Nττ(α)
Nτ,s + 1Γ (Nτ+ 2α+ 1) Γ (α+ 1)
(Nτ+ 1)! Γ (Nτ+α+ 1) Γ (2α+ 1) , Nτ0α0,
A32Nτ1τ(α)
Nτ,s + 1ΓNτ+1
2+α
ΓNτ+3
2Γ (Nτ+α+ 1) ,Nτ+ 1
2Z+∧ −1
2< α < 0,
(5.5a)
Math Methods Appl Sci
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14 KAREEM T. ELGINDY AND SAYED A. DAHY
E2,r,s <
A32Nττ(α)
Nτ,s + 1ΓNτ
2+α+ 1
Nτ
2!p(Nτ+ 1) (Nτ+ 2α+ 1) Γ (Nτ+α+ 1),Nτ
2Z+
0∧ −1
2< α < 0,(5.5b)
E3,s
A221NyΓ (Ny+ 2α+ 1) Γ (α+ 1)
(Ny+ 1)! Γ (Ny+α+ 1) Γ (2α+ 1) , Ny0α0,
A22NyΓNy+1
2+α
ΓNy+3
2Γ (Ny+α+ 1)
,Ny+ 1
2Z+∧ −1
2< α < 0,
(5.6a)
E3,s <
A221NyΓNy
2+α+ 1
Ny
2!p(Ny+ 1) (Ny+ 2α+ 1) Γ (Ny+α+ 1),Ny
2Z+
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)
42MyMyα
rMy3
2eMy
1, α
r0,
c(α
r)
2Myα
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)
1NαNy3/2
ye
2Ny+B(α)
2NαNτ3/2
τe
2Nτ, α 0
B(α)
3NNy3/2
ye
2Ny+B(α)
4NNτ3/2
τe
2Nτ,1
2< α < 0
+c(α
r)
4e
2MyMα
rMy3/2
y
1, α
r0
c(α
r)
2Mα
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
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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 [4851,5456] 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 + εαr, 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 nso 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:αIG
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:αIG
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,NτφSand
∂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 )1+ϑ3(τ)Z1
1
ψ(σ, τ )+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 )1+ϑ3(τ)Z1
1
˜
ψ(α)
Ny,Nτ(σ, τ )+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,sr, s. Since integration preserves uniform convergence, and
by assumption, ˜
ψ(α)
Ny,Nτψuniformly on D(α),G
Ny,Nτ, and the approximate function ˜
ψ(α)
Ny,Nτis continuous on D2
a,b,T ,
then ¯
φSφSuniformly on D(α),G
Ny,Nτwith the referred exponential order of convergence. Now, to prove that
˜
φ(α)
S,Ny,NτφSuniformly 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
2NyNαNy3
2
y, α 0,
B(α)
2e
2NyNNy3
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,NτφSuniformly 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> N1y[1,1],
and
F0
Ny(y)F0(y)< ε2if Ny> N2y[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)k2Ny. 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
Prepared using speauth.cls DOI: 10.1002/mma.5135
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,Nτuuuniformly 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
2NyA4Γ(α+ 1)Γ(Ny+ 2α+ 1) y(α)
Ny,r + 1
Γ(2α+ 1)(Ny+ 1)!Γ(Ny+α+ 1) , Ny0α0
2Ny1A4ΓNy+1
2+αy(α)
Ny,r + 1
ΓNy+3
2Γ(Ny+α+ 1)
,Ny+ 1
2Z+∧ −1
2< α < 0
2NyA4ΓNy
2+α+ 1y(α)
Ny,r + 1
Ny
2!p(Ny+ 1)(Ny+ 2α+ 1)Γ(Ny+α+ 1) ,Ny
2Z+
0∧ −1
2< α < 0
+µν
2NτA3Γ(α+ 1)Γ(Nτ+ 2α+ 1) τ(α)
Nτ,s + 1
(Nτ+ 1)!Γ(Nτ+α+ 1)Γ(2α+ 1) , Nτ0α0
2Nτ1A3ΓNτ+1
2+ατ(α)
Nτ,s + 1
ΓNτ+3
2Γ(Nτ+α+ 1) ,Nτ+ 1
2Z+∧ −1
2< α < 0
2NτA3ΓNτ
2+α+ 1τ(α)
Nτ,s + 1
Nτ
2!p(Nτ+ 1)(Nτ+ 2α+ 1)Γ(Nτ+α+ 1) ,Nτ
2Z+
0∧ −1
2< α < 0
+c(α
r)
4MMy+α
r3
2
ye
2My
1, α
r0
c(α
r)
2Mα
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
2NyNαNy3
2
y, α 0,
B(α)
2e
2NyNNy3
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
Prepared using speauth.cls DOI: 10.1002/mma.5135
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,Nτuuuniformly 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=NZ+. 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 xt. 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,sP(α)
N,N ux(α)
N,r , t(α)
N,s,x(α)
N,r , t(α)
N,sˆ
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,0x1,
and the boundary conditions
u(0, t)=0, u (1, t) = 1
σ+t,0t1,
where σ > 1is a parameter. The exact solution of this problem is
u(x, t) = x
σ+t,0x1,0t1.
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)28νσy, 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 αIG
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= 0.5t.
The reported 1-, 2-, and infinity-norms of the absolute error matrix constructed at the SGG mesh grid were in respective order
1.553e09,2.259e09, and 4.811e09.
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),0x1,
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) = 2νπeπ2νt sin (πx)
σ+eπ2νt cos (πx),0x1, t > 0.
The numerical results are reported in Figures 24, and in Tables IIVI. Table II manifests again the superior
accuracy of the CHBGPM even when the viscosity paramter ν0, 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 IIIVI 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= 0.5t.
The reported 1-, 2-, and infinity-norms of the absolute error matrix constructed at the SGG mesh grid were in respective order
3.084e07,3.274e07, and 6.461e07.
Math Methods Appl Sci
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