Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 947230, 21 pages
doi:10.1155/2011/947230
Research Article
A Jacobi Dual-Petrov-Galerkin Method for Solving
Some Odd-Order Ordinary Differential Equations
E. H. Doha,1 A. H. Bhrawy,2, 3 and R. M. Hafez4
1
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
4
Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo 11931, Egypt
2
Correspondence should be addressed to A. H. Bhrawy, alibhrawy@yahoo.co.uk
Received 31 October 2010; Revised 13 February 2011; Accepted 16 February 2011
Academic Editor: Simeon Reich
Copyright q 2011 E. H. Doha et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A Jacobi dual-Petrov-Galerkin JDPG method is introduced and used for solving fully integrated
reformulations of third- and fifth-order ordinary differential equations ODEs with constant
coefficients. The reformulated equation for the Jth order ODE involves n-fold indefinite integrals
for n 1, . . . , J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the
Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the
reliability of the proposed method for some constant and polynomial coefficients ODEs.
1. Introduction
A well-known advantage of spectral methods is high accuracy with relatively fewer
unknowns when compared with low-order finite-difference methods 1, 2. On the other
hand, spectral methods typically give rise to full matrices, partially negating the gain in
efficiency due to the fewer degrees of freedom. In general, the use of the Jacobi polynomials
α,β
with α, β ∈ −1, ∞ and n is the polynomial degree has the advantage of obtaining
Pn
solutions of ordinary differential equations ODEs in terms of the Jacobi indices see for
instance, 3–5. Several such pairs α, β have been used for approximate solutions of ODEs
see 6–10. We avoid developing approximation results for each particular pair of indices
and instead carry out a study with general indices. With this motivation, we introduce in this
paper a family of the Jacobi polynomials with general indices.
Third-order differential equations have applications in many engineering models, see
for instance 11–14. Fifth-order differential equations generally arise in the mathematical
2
Abstract and Applied Analysis
modeling of viscoelastic flows and other branches of mathematical physics and engineering
sciences, see 15–17. Existence and uniqueness of solutions of such boundary value problems
are discussed, for instance, in 18.
In this paper, the proposed differential equations are integrated q times, where q is the
order of the equation, and we make use of the formulae relating the expansion coefficients
of integration appearing in these integrated forms of the proposed differential equations
to the Jacobi polynomials themselves see, Doha 19. An advantage of this approach is
that the general equation in the algebraic system then contains a finite number of terms.
We, therefore, motivated our interest in integrated forms of these differential equations. The
interested reader is referred to Doha and Bhrawy 7, 20.
The main aim of this paper is to propose a suitable way to approximate some
integrated forms of third- and fifth-order ODEs with constant coefficients using a spectral
method, based on the Jacobi polynomials such that it can be implemented efficiently and
at the same time has a good convergence property. It is worthy to note here that oddorder problems lack the symmetry of even-order ones, so we propose a Jacobi dual-PetrovGalerkin JDPG method. The method leads to systems with specially structured matrices
that can be efficiently inverted. We apply the method for solving the integrated forms
of third- and fifth-order ODEs by using compact combinations of the Jacobi polynomials,
which satisfy essentially all the underlying homogeneous boundary conditions. To be more
precise, for the JDPG we choose the trial functions to satisfy the underlying boundary
conditions of the differential equations, and we choose the test functions to satisfy the dual
boundary conditions. Extension of the JDPG for polynomial coefficient ODEs is obtained by
approximating the weighted inner products in the JDPG by using the Jacobi-Gauss-Lobatto
quadrature. Finally, examples are given to illustrate the efficiency and implementation of the
method. Comparisons are made to confirm the reliability of the method.
The remainder of this paper is organized as follows. In Section 2 we give an overview
of the Jacobi polynomials and their relevant properties needed hereafter. Section 3 is devoted
to the theoretical derivation of the JDPG method for third-order differential equations with
constant and variable polynomial coefficients. Section 4 gives the corresponding results
for those obtained in Section 3, but for the fifth-order differential equations with constant
coefficient and two choices of boundary conditions. In Section 5, we present some numerical
results exhibiting the accuracy and efficiency of our numerical algorithms. Some concluding
remarks are given in the final section.
2. Preliminaries
Let SN I be the space of polynomials of degree at most N on the interval I −1, 1. We set
WN u ∈ SN : u±1 u1 1 0 ,
∗
WN
α,β
u ∈ SN
: u±1 u −1 0 ,
1
2.1
and let Pn x n 0, 1, 2, . . . be the Jacobi polynomials orthogonal with the weight
functions wα,β x 1 − xα 1 xβ , where α, β > −1.
Abstract and Applied Analysis
3
α,β
α,β
α,β
Let xN,j , 0 ≤ j ≤ N, be the zeros of 1 − x2 ∂x PN . Denote by N,j , 0 ≤ j ≤ N,
the weights of the corresponding Gauss-Lobatto quadrature formula, which are arranged in
decreasing order. We define the discrete inner product and norm of weighted space L2wα,β I
as follows:
u, vwα,β ,N N α,β α,β
α,β
u xN,k v xN,k N,k ,
uwα,β ,N u, uwα,β ,N .
2.2
k0
Obviously see, e.g., formula 2.25 of 21,
u, vwα,β ,N u, vwα,β
∀u, v ∈ S2N−1 .
2.3
Thus, for any u ∈ SN , the norms uwα,β ,N and uwα,β coincide.
α,β
P
For any u ∈ CI, the Jacobi-Gauss-Lobatto interpolation operator IN
satisfying
α,β
P
IN
α,β α,β u xN,k u xN,k ,
−1/2,−1/2
ux ∈ SN ,
0 ≤ k ≤ N.
2.4
0,0
c
l
P
P
IN
and IN
IN
the Chebyshev-Gauss-Lobatto and LegendreWe also denote by IN
Gauss-Lobatto interpolation operators, respectively.
For any real numbers α, β > −1, the set of the Jacobi polynomials forms a complete
L2wα,β I-orthogonal system, and
α,β
Pk
α,β
, Pj
wα,β
hk δk,j ,
2.5
where δk,j is the Kronecker function and
2αβ1 Γk α 1Γ k β 1
hk .
2k α β 1 Γk 1Γ k α β 1
2.6
The following special values will be of fundamental importance in what follows see,
22–24
α,β
Pn 1
α 1n
,
n!
α,β
Pn −1
q−1
Γn α 1n λ i
α,β
Dq Pn 1 ,
q
i0 2 n − q !Γ q α 1
where ak Γa k/Γa and λ 1 β α.
−1n β 1 n
,
n!
2.7
α,β
D q Pn
β,α
−1 −1nq Dq Pn
1,
4
Abstract and Applied Analysis
α,β
If we define the q times repeated differentiation and integration of Pn
D
q
α,β
Pn x
and
q,α,β
In
x,
x by
respectively, then cf. Doha 19, 22
α,β
D q Pn
x 2−q n λq
n−q
α,β
Cn−q,i q, α, β Pi x,
2.8
i0
q,α,β
In
q ≥ 0,
nq
α,β
2q
Cnq,i −q, α, β Pi x πq−1 x,
x
n − q λ q iq
1
n ≥ q 1 for α β − ,
2
q ≥ 0,
n≥q
for α /
−
2.9
1
1
or β /
− ,
2
2
where
2q λ i i q α 1 −i Γi λ
C,i q, α, β − i!Γ2i λ
× 3F2
− i
2q i λ
iqα1
iα1
2i λ 1
1
2.10
,
q,α,β
with πq−1 x being a polynomial of degree at most q − 1. It is to be noted that In
q
x may
α,β
Pn x
be obtained from D
by replacing q with negative q. In general, the hypergeometric
series 3 F 2 cannot be summed in explicit form, but it can be summed by Watsons identity 25,
if α β. The following two lemmas will be of fundamental importance in what follows.
Lemma 2.1 see, 19, 26. One has
x γ,δ
γ,δ
Dj Pj
x,
2.11
j0
where
γ,δ
Dj
−j
−2i 1 γ j i
.
2 ! 1 γ δ 2j
i0 i! − i − j ! 1 γ δ j i1 2 γ δ i j j
j
2.12
Lemma 2.2. If one writes
q,α,β
Ik
x
kq
α,β
Sq k, i, α, β Pi x πq−1 x,
iq
2.13
Abstract and Applied Analysis
5
then
2q
Sq k, i, α, β Ckq, i −q, α, β ,
k−qλ q
q 1, 2, . . . .
2.14
Proof. It is immediately obtained from relation 2.9.
3. Third-Order Differential Equation
We are interested in using the JDPG method to solve the third-order differential equation
u3 x γ1 u2 x γ2 u1 x γ3 ux gx,
3.1
in I,
subject to
u±1 u1 1 0,
3.2
where γ1 , γ2 , and γ3 are constants and g is a given source function. In this paper, we consider
the fully integrated form of the ODE, given by
ux γ1
1
uxdx1 γ2
fx 2
2
uxdx2 γ3
3
uxdx3
3.3
α,β
di Pi x,
u±1 u 1 0,
i0
where
q
q times
uxdxq
q times
· · · ux dx dx · · · dx,
fx 3
gxdx3 .
3.4
In this work we assume that f, the three-fold indefinite integral form of g, can be evaluated
analytically. We set
α,β
α,β
α,β
SN span P0 x, P1 x, . . . , PN x ,
3.5
then the dual-Petrov-Galerkin approximation to 3.3 is to find uN ∈ WN such that
uN , vwα,β γ1
1
1
γ2
uN dx , v
2
2
uN dx , v
wα,β
γ3
3
uN dx3 , v
wα,β
f
wα,β
2
i0
α,β
di P i
∗
∀v ∈ WN
,
,v
wα,β
3.6
6
Abstract and Applied Analysis
where wα,β x 1 − xα 1 xβ and u, vw I uvw dx is the inner product in the weighted
space L2wα,β I. The norm in L2wα,β I will be denoted by · wα,β .
3.1. The Jacobi Dual-Petrov-Galerkin Method
We choose compact combinations of the Jacobi polynomials as basis functions to minimize the
bandwidth hoping to improve the condition number of the coefficient matrix corresponding
to 3.6. We choose the test basis and trial functions of expansions φk x and ψk x to be of
the form
α,β
φk x Pk
α,β
ψk x Pk
α,β
α,β
α,β
3.7
α,β
3.8
x k Pk1 x εk Pk2 x ζk Pk3 x,
α,β
α,β
x ρk Pk1 x k Pk2 x σk Pk3 x,
where k , εk , ζk , ρk , k , and σk are the unique constants such that φk x ∈ WN and ψk x ∈
1
∗
WN
. From the boundary conditions, φk ±1 φk 1 0 and 2.7, hence k , εk and ζk can be
uniquely determined by using mathematica to give see, 27
−k 12k λ 2 k − α 2β 1
k ,
k α 1 k β 1 2k λ 4
εk −k 12 2k λ 1 k − β 2α 3
,
k α 12 k β 1 2k λ 5
ζk k 13 2k λ 12
.
k α 12 k β 1 2k λ 42
3.9
1
Using 2.7, and ψk ±1 ψk −1 0, one verifies readily that
k 12k λ 2 k − β 2α 1
ρk ,
k α 1 k β 1 2k λ 4
−k 12 2k λ 1 k − α 2β 3
k ,
k β 1 2 k α 12k λ 5
σk −k 13 2k λ 12
.
k β 1 2 k α 12k λ 42
3.10
Abstract and Applied Analysis
7
Now it is clear that 3.6 is equivalent to
uN , ψk x
wα,β
γ1
1
1
γ2
uN dx , ψk x
2
2
uN dx , ψk x
wα,β
γ3
3
3
uN dx , ψk x
fx wα,β
2
α,β
di Pi x, ψk x
i0
wα,β
k 0, 1, . . . , N,
,
wα,β ,N
3.11
where ·, ·wα,β ,N is the discrete inner product associated with the Jacobi-Gauss-Lobatto
quadrature. The constants d0 , d1 , and d2 would not appear if we take k ≥ 3 in 3.11, therefore
we get
uN , ψk x
wα,β
γ1
1
1
γ2
uN dx , ψk x
2
2
uN dx , ψk x
wα,β
γ3
3
wα,β
3.12
fx, ψk x wα,β ,N ,
uN dx3 , ψk x
k 3, 4, . . . , N.
wα,β
If we take φk x and ψk x as defined in 3.7 and 3.8, respectively, and if we denote
fk f, ψk x wα,β ,N ,
uN x N−3
T
f f3 , f4 , . . . , fN ,
v v0 , v1 , . . . , vN−3 T ,
vn φn x,
n0
akj
φj−3 x, ψk x wα,β ,
bkj 1
3.13
1
φj−3 xdx , ψk x
,
wα,β
ckj 2
2
φj−3 xdx , ψk x
,
wα,β
dkj 3
3
φj−3 xdx , ψk x
,
wα,β
then
WN span φ0 x, φ1 x, . . . , φN−3 x ,
∗
span ψ3 x, ψ4 x, . . . , ψN x ,
WN
3.14
8
Abstract and Applied Analysis
and the nonzero elements akj , bkj , ckj , and dkj for 3 ≤ k, j ≤ N are given as follows:
akk ζk−3 hk ,
ak,k1 εk−2 hk ζk−2 ρk hk1 ,
ak,k2 k−1 hk εk−1 ρk hk1 ζk−1 k hk2 ,
ak,k3 hk k ρk hk1 εk k hk2 ζk σk hk3 ,
ak,k4 ρk hk1 k1 k hk2 εk1 σk hk3 ,
ak,k5 k hk2 k2 σk hk3 ,
ak,k6 σk hk3 ,
bk,j R1 j, k, α, β hk R1 j, k 1, α, β ρk hk1 R1 j, k 2, α, β k hk2
R1 j, k 3, α, β σk hk3 , j k − 1, 0, 1, . . . , 8,
ck,j R2 j, k, α, β hk R2 j, k 1, α, β ρk hk1 R2 j, k 2, α, β k hk2
R2 j, k 3, α, β σk hk3 , j k − 2, 0, 1, . . . , 10,
dk,j R3 j, k, α, β hk R3 j, k 1, α, β ρk hk1 R3 j, k 2, α, β k hk2
R3 j, k 3, α, β σk hk3 , j k − 3, 0, 1, . . . , 12,
3.15
where
Ri j, k, α, β Si j − 3, k, α, β j−3 Si j − 2, k, α, β
εj−3 Si j − 1, k, α, β ζj−3 Si j, k, α, β .
3.16
Hence by setting
A akj ,
B bkj ,
C ckj ,
D dkj ,
3 ≤ k, j ≤ N,
3.17
then 3.12 is equivalent to the following matrix equation:
A γ1 B γ2 C γ3 D v f.
3.18
All the analytical formulae of the nonzero elements of matrices A, B, C and D can be obtained
by direct computations using the properties of the Jacobi polynomials for details see,
27, 28.
In the case of α, β / 0, α, β ∈ −1, ∞, we can form explicitly the LU factorization, that
is, A γ1 B γ2 C γ3 D LU. In general, the expense of calculating LU factorization of an
N × N dense matrix M is ON 3 operations, and the expense of solving Ax b, provided
Abstract and Applied Analysis
9
that the factorization is known, is ON 2 see, 27. However, in the case of banded matrix
A of bandwidth r, we need just Or 2 N operations to factorize and OrN operations to
solve a linear system. In the case of γi / 0, i 0, 1, 2, the square matrix A γ1 B γ2 C γ3 D
has bandwidth of 13. So we need just O13N operations to solve the linear system 3.18. If
r N, this represents a very substantial saving.
It is worthy to note that, for α, β /
0, the algebraic system 3.18 resulting from fully
integrated reformulation of 3.1 is sparse and is therefore cheaper to solve than those
obtained from the differentiated form see 27, Theorem 3.1. Moreover, the savings in
computational effort increase as the size of the systems grow. Thus, we have demonstrated the
advantage of using the integrated forms over the differentiated ones for constant coefficients
ODEs.
3.2. A Quadrature JDPG Method
The JDPG can be extended for ODEs with polynomial coefficients because of analytical form
of a product of an algebraic polynomial, and the Jacobi polynomials are known.
Now the formula of the Jacobi coefficients of the moments of one single Jacobi
polynomial of any degree see, Doha 22 is
α,β
x m Pj
x 2m
α,β
Θm,n j Pjm−n x
∀m, j ≥ 0,
3.19
n0
α,β
with P−r x 0, r ≥ 1, where
−1n 2jm−n m! 2j 2m − 2n λ Γ j m − n λ Γ j α 1 Γ j β 1
Θm,n j Γ j m−nα1 Γ j m−nβ1 Γ j λ
×
minjm−n,j
j m−n
kmax0,j−n
k
Γ j kλ
2k n k − j !Γ 3j 2m − 2n − k λ 1
3.20
j−k
−1 Γ 2j m − n − k − α 1 Γ j m − n β 1
×
! j − k − !Γ j − α 1 Γ k β 1
0
× 2 F 1 j − k − n, j m − n β 1; 3j 2m − 2n − k λ 1; 2 .
This formula can be used to facilitate greatly the setting up of the algebraic systems to be
obtained by applying the spectral methods for solving differential equations with polynomial
coefficients of any order.
10
Abstract and Applied Analysis
Let us consider the following integrated form of the third-order differential equation:
1
ux 2
γ1 xuxdx1 fx 2
γ2 xuxdx2 3
γ3 xuxdx3
3.21
α,β
di P i
in I −1, 1, u±1 u1 1 0,
x,
i0
where γ1 x, γ2 x, and γ3 x are the variable polynomial coefficients of the differential
equation. The quadrature dual-Petrov-Galerkin method for 3.21 is to find uN ∈ WN such
that
uN , vN wα,β ,N 1
1
2
γ1 xuN dx , vN
2
γ2 xuN dx , vN
wα,β ,N
3
3
γ3 xuN dx , vN
f
wα,β ,N
2
α,β
di Pi , vN
i0
wα,β ,N
∗
∀vN ∈ WN
,
wα,β ,N
3.22
where u, vwα,β ,N is the discrete inner product of u and v associated with the Jacobi-GaussLobatto quadrature.
Let us consider
uN N−3
ak φk ,
fk f, ψk wα,β ,N ,
a a0 , a1 , . . . , aN−3 T ,
T
f f3 , f1 , . . . , fN , 3.23
k0
and using Lemma 2.2 and formula 3.19, we can obtain
aij φj−3 , ψi
wα,β ,N
bij ,
1
1
γ1 xφj−3 xdx , ψi
,
wα,β ,N
cij 2
2
γ2 xφj−3 xdx , ψi
,
dij 3
3
γ3 xφj−3 xdx , ψi
wα,β ,N
.
wα,β ,N
3.24
And by setting
akj ,
A
B bkj ,
ckj ,
C
dkj ,
D
3 ≤ k, j ≤ N,
3.25
then the linear system of 3.22 becomes
B C
D
a f.
A
3.26
Abstract and Applied Analysis
11
4. Fifth-Order Differential Equation
In this section, we consider the fifth-order differential equation of the form
u5 γ1 u3 γ2 u1 γ3 u gx,
4.1
x ∈ I,
but by considering its integrated form, namely,
ux γ1
2
uxdx
2
γ2
4
uxdx
4
γ3
5
uxdx5 fx 4
α,β
di P i
i0
x.
4.2
4.1. First Choice of Boundary Conditions
Here, we apply the dual-Petrov-Galerkin approximation to 4.2 subject to the boundary
conditions
u±1 u1 ±1 u2 1 0.
4.3
We set
VN v ∈ SN : v±1 v1 ±1 v2 1 0 ,
VN∗
v ∈ SN
4.4
: v±1 v ±1 v −1 0 ;
1
2
then the Jacobi dual-Petrov-Galerkin approximation to 4.2 is to find uN ∈ VN such that
uN , vwα,β γ1
2
2
γ2
uN dx , v
4
4
uN dx , v
wα,β
γ3
5
wα,β
uN dx5 , v
f
4
α,β
di P i
∀v ∈ VN∗ .
,v
i0
wα,β
4.5
wα,β ,N
We consider the following the Jacobi dual-Petrov-Galerkin procedure for 4.1: find uN ∈ VN
such that
uN , vwα,β γ1
2
2
γ2
uN dx , v
4
4
uN dx , v
wα,β
γ3
5
uN dx5 , v
wα,β
f
wα,β
4
i0
4.6
α,β
di P i
∀v ∈ VN∗ .
,v
wα,β
12
Abstract and Applied Analysis
Now, we choose the basis and the dual basis functions Φk x and Ψk x to be of the form
α,β
Φk x Pk
α,β
α,β
α,β
x k Pk1 x εk Pk2 x ζk Pk3 x
α,β
α,β
μ
k Pk4 x υk Pk5 x,
k 0, 1, . . . , N − 5,
4.7
Ψk x α,β
Pk x
α,β
α,β
ρk Pk1 x
α,β
α,β
k Pk2 x
ςk Pk4 x τk Pk5 x,
α,β
σk Pk3 x
k 0, 1, . . . , N − 5.
It is not difficult to show that the basis functions Φk x ∈ Vk5 and the dual basis functions
∗
Ψk x ∈ Vk5
.
We choose the coefficients k , εk , ζk , μk , and υk such that Φk x verifies the boundary
conditions 4.3. Making use of 2.7, then the boundary conditions 4.3 lead to linear system
for these coefficients. The computation of the exact solution of such linear system for the
unknown coefficients is extremely tedious by hand, and we have resorted to the symbolic
computation software mathematica version 6, hence k , εk , ζk , μk , and υk can be uniquely
determined to give
k −
k 12k λ 2 k − 2α 3β 1
,
k α 1 k β 1 2k λ 6
εk −
k 12 2k λ 12k λ 4
k α 12 k β 1 2 2k λ 62
× 2k2 4α 3k −α2 9α 3β2 32α − 1β 16 ,
ζk k 13 2k λ 12
k α 13 k β 1 2 2k λ 72
× 2k2 4 β 3 k −3α2 − 3α − β2 32α 5β 16 ,
μ
k k 14 k 3α − 2β 5 2k λ 13
,
k α 13 k β 1 2 2k λ 62 2k λ 9
υk −
k 15 2k λ 14
.
k α 13 k β 1 2 2k λ 64
4.8
Abstract and Applied Analysis
13
Since the dual basis functions Ψk x satisfy the dual boundary conditions, and making
use of 2.7 then the unknown coefficients ρk , k , σk , ςk , and τk are determined by using
Mathematica to give
k 12k λ 2 k 3α − 2β 1
ρk ,
k α 1 k β 1 2k λ 6
k −
k 12 2k λ 12k λ 4
k α 12 k β 1 2 2k λ 62
× 2k2 4 β 3 k −3α2 3α − β2 32α 3β 16 ,
σk −
k 13 2k λ 12
k α 12 k β 1 3 2k λ 72
4.9
× 2k2 4α 3k −α2 15α − 3β2 32α − 1β 16 ,
ςk k 14 k 3β − 2α 5 2k λ 13
,
k α 12 k β 1 3 2k λ 62 2k λ 9
τk k 15 2k λ 14
.
k α 12 k β 1 3 2k λ 64
It is clear that 4.6 is equivalent to
uN , Ψk xwα,β γ1
2
2
uN dx , Ψk x
γ2
4
4
uN dx , Ψk x
wα,β
γ3
5
wα,β
5
uN dx , Ψk x
fx 4
α,β
di Pi x, Ψk x
i0
wα,β
,
k 0, 1, . . . , N.
wα,β
4.10
The constants d0 , d1 , d2 , d3 , and d4 would not appear if we take k ≥ 5 in 4.10, therefore we
get
uN , Ψk xwα,β γ1
2
2
uN dx , Ψk x
γ2
4
4
uN dx , Ψk x
wα,β
γ3
5
uN dx5 , Ψk x
wα,β
wα,β
4.11
fx, Ψk x wα,β ,
k 5, 6, . . . , N.
14
Abstract and Applied Analysis
If we take Φk x and Ψk x as defined in 4.7 and if we denote
fk f, Ψk x wα,β ,
uN x T
f f5 , f6 , . . . , fN ,
N−5
v v0 , v1 , . . . , vN−5 T ,
vn Φn x,
n0
pkj Φj−5 x, Ψk x
qkj ,
wα,β
2
2
Φj−5 xdx , Ψk x
4.12
,
wα,β
skj 4
4
Φj−5 xdx , Ψk x
,
tkj 5
5
Φj−5 xdx , Ψk x
wα,β
,
wα,β
then
VN span{Φ0 x, Φ1 x, . . . , ΦN−5 x},
VN∗ span{Ψ5 x, Ψ6 x, . . . , ΨN x},
4.13
and the nonzero elements pkj , qkj , skj , and tkj for 5 ≤ k, j ≤ N are given as follows:
pkk υk−5 hk ,
k−4 hk υk−4 ρk hk1 ,
pk,k1 μ
pk,k2 ζk−3 hk μ
k−3 ρk hk1 υk−3 k hk2 ,
pk,k3 εk−2 hk ζk−2 ρk hk1 μk−2 k hk2 υk−2 σk hk3 ,
pk,k4 k−1 hk εk−1 ρk hk1 ζk−1 k hk2 μk−1 σk hk3 υk−1 ςk hk4 ,
pk,k5 hk k ρk hk1 εk k hk2 ζk σk hk3 μ
k ςk hk4 υk τk hk5 ,
pk,k6 ρk hk1 k1 k hk2 εk1 σk hk3 ζk1 ςk hk4 μ
k1 τk hk5 ,
pk,k7 k hk2 k2 σk hk3 εk2 ςk hk4 ζk2 τk hk5 ,
pk,k8 σk hk3 k3 ςk hk4 εk3 τk hk5 ,
pk,k9 ςk hk4 k4 τk hk5 ,
pk,k10 τk hk5 ,
qk,j R2 j, k, α, β hk R2 j, k 1, α, β ρk hk1 R2 j, k 2, α, β k hk2
R2 j, k 3, α, β σk hk3 R2 j, k 4, α, β ςk hk4 R2 j, k 5, α, β τk hk5 ,
j k − 2,
0, 1, . . . , 14,
4.14
Abstract and Applied Analysis
15
sk,j R4 j, k, α, β hk R4 j, k 1, α, β ρk hk1 R4 j, k 2, α, β k hk2
R4 j, k 3, α, β σk hk3 R4 j, k 4, α, β ςk hk4 R4 j, k 5, α, β τk hk5 ,
tk,j
j k − 4, 0, 1, . . . , 18,
R5 j, k, α, β hk R5 j, k 1, α, β ρk hk1 R5 j, k 2, α, β k hk2
R5 j, k 3, α, β σk hk3 R5 j, k 4, α, β ςk hk4 R5 j, k 5, α, β τk hk5 ,
j k − 5,
0, 1, . . . , 20,
4.15
where
Ri j, k, α, β Si j − 5, k, α, β j−5 Si j − 4, k, α, β εj−5 Si j − 3, k, α, β
ζj−5 Si j − 2, k, α, β μj−5 Si j − 1, k, α, β υj−5 Si j, k, α, β .
4.16
Then 4.11 is equivalent to the following matrix equation:
P γ1 Q γ2 S γ3 T v f.
4.17
4.2. Second Choice of Boundary Conditions
In this subsection, we consider the fifth-order differential equation 4.1 with the following
boundary conditions:
u±1 u1 ±1 u3 −1 0.
4.18
Equation 4.1 subject to the boundary conditions 4.18 has been considered in 29, 30. Let
us denote
ZN v ∈ SN : v±1 v1 ±1 v3 −1 0 ,
4.19
N v ∈ SN : v±1 v1 ±1 v3 1 0 ;
Z
then the Jacobi dual-Petrov-Galerkin approximation of 4.1 subject to 4.18 consists of
finding uN ∈ ZN such that
uN , vwα,β γ1
2
2
γ2
uN dx , v
4
4
uN dx , v
wα,β
γ3
5
uN dx5 , v
wα,β
f
wα,β
4
i0
α,β
di P i , v
N .
∀v ∈ Z
wα,β
4.20
16
Abstract and Applied Analysis
We consider the following choice of basis functions:
α,β
ϕk x Pk
α,β
α,β
α,β
α,β
α,β
α,β
α,β
α,β
α,β
x ξ1,k Pk1 x ξ2,k Pk2 x ξ3,k Pk3 x ξ4,k Pk4 x ξ5,k Pk5 x,
4.21
and dual basis functions:
α,β
ϕk x Pk
α,β
x δ1,k Pk1 x δ2,k Pk2 x δ3,k Pk3 x δ4,k Pk4 x δ5,k Pk5 x,
4.22
where ξ1,k , ξ2,k , ξ3,k , ξ4,k , ξ5,k , δ1,k , δ2,k , δ3,k , δ4,k , and δ5,k are chosen to be the unique constants
N , for all k 0, 1, . . . , N − 5.
such that ϕk x ∈ ZN and ϕk x ∈ Z
Equation 4.20 is equivalent to the following matrix equation:
γ2 S γ3 T v f,
P γ1 Q
4.23
S,
and T can be obtained similarly as in the previous
where the elements of the matrices P , Q,
sections, but details are not given here.
5. Numerical Results
In this section some examples are considered aiming to illustrate how one can apply the
proposed algorithms presented in the previous sections. Comparisons between JDPG method
and other methods proposed in 29–32 are made.
Example 5.1. Consider the one-dimensional third-order equation
u3 x πu2 x 2πu1 x 3πux fx,
1
u±1 ± √ ,
2
x ∈ I,
π
u1 1 √ ,
4 2
5.1
with an exact smooth solution
ux sin
πx 4
5.2
.
Table 1 lists the maximum pointwise error of u − uN , using the JDPG with various
choices of α, β, and N.
Example 5.2. Consider the one-dimensional fifth-order differential problem
u5 − γ1 u3 − γ2 u1 − γ3 u fx,
u±1 0,
π
u1 ±1 ∓ ,
2
with the exact solution ux x2 cosπ/2x.
x ∈ I,
u2 1 −2,
5.3
Abstract and Applied Analysis
17
Table 1: Maximum pointwise error using JDPG method for N 8, 16, 24, for Example 5.1.
N
α
β
JDPG
16
1
2
1
2
1
2
1
−
2
1.404 · 10−11
−
1.249 · 10−16
24
1.165 · 10−1
8
16
α
8.278 · 10−2
8
2.220 · 10−16
JDPG
3.645 · 10−2
1
2
1
2
3.368 · 10−12
2.220 · 10−16
5.644 · 10−2
1
−
2
1
−
2
2.439 · 10−11
24
β
6.311 · 10−12
2.220 · 10−16
Table 2: Maximum pointwise error using JDPG method for N 12, 24, 36, for Example 5.2.
N
α
β
γ1
γ2
γ3
JDPG
24
1
2
1
2
γ1
1
1
2.650 · 10−13
1
36
2.116 · 10−16
12
3.024 · 10−1
24
−13
0
0
γ2
γ3
3.294 · 10−1
12
1
1
1.982 · 10
1
JDPG
9.117 · 10−1
N
N2
N3
1.052 · 10−10
4.770 · 10−16
8.366 · 10−1
N
N
2
N
3
7.947 · 10−11
36
3.053 · 10−16
3.053 · 10−16
12
−1
7.598 · 10−1
24
1
−
2
1
−
2
2.748 · 10
1
1
1.437 · 10−13
1
N
N2
N3
2.116 · 10−16
36
5.828 · 10−11
1.387 · 10−16
Table 2 lists the maximum pointwise error, using the JDPG method with various
choices of α, β, γ1 , γ2 , γ3 , and N. Numerical results of this example show that the JDPG method
converges exponentially.
Example 5.3. Consider the following fifth-order boundary value problem see 29–32:
u5 x − ux −15 10xex ,
u0 0,
u1 0,
1
u 0 1,
1
x ∈ 0, 1,
u 1 −e,
3
5.4
u 0 −3.
The analytic solution of this problem is ux x1 − xex . Approximate solutions uN x
N 10, 13, 20, 26, 40 are obtained by using our proposed method. Table 3 exhibits a
comparison between the error obtained by using JDPG method and the sixth-degree B-spline
31, sextic spline 32, nonpolynomial sextic spline 29, and the computational method
in 30. The numerical results show that JDPG method is more accurate than the existing
methods.
18
Abstract and Applied Analysis
Table 3: Maximum pointwise error of u − uN for N 15, 18, 25, 31, for Example 5.3.
N−5
10
13
20
26
α
β
Our algorithm
1
2
0
1
−
2
1
2
0
1
−
2
2.50 · 10−6
1
2
0
1
−
2
1
2
0
1
−
2
1.35 · 10−10
1
2
0
1
−
2
1
2
0
1
−
2
1
2
0
1
−
2
1
2
0
1
−
2
2.28 · 10−6
In 31
In 32
0.1570
2.2593 · 10−4
In 29
In 30
6.29887 · 10−5
2.05 · 10−6
1.17 · 10−10
1.3767 · 10−4
5.91739 · 10−5
1.00 · 10−10
2.49 · 10−16
2.22 · 10−16
0.0747
1.3300 · 10−5
2.14116 · 10−6
−16
1.87 · 10
2.22 · 10−16
1.18 · 10−16
7.1273 · 10−6
3.40705 · 10−7
1.94 · 10−16
Example 5.4. Consider the following third-order ODE with polynomial coefficients:
u3 x x2 u2 x x3 4x u1 x 3x2 x 2 ux gx,
x ∈ −1, 1,
5.5
subject to
u±1 u1 1 0,
5.6
where g is selected such that exact solution is ux 1 − x2 1 − xe2x .
Equation 5.5 can be rearranged to take the form
2 1
x3 ux
xux gx,
u3 x x2 ux
5.7
and accordingly its fully integrated form is
ux 1
x2 uxdx1 fx 2
i0
2
x3 uxdx2 3
xuxdx3
5.8
α,β
di P i
x.
Abstract and Applied Analysis
19
Table 4: Maximum pointwise error using quadrature JDPG method for N 12, 16, 20, 24, for Example 5.4.
N
α
β
16
α
β
−2
0
0
20
−5
1.496 · 10
2.513 · 10−9
−14
3.613 · 10
Quadrature JDPG
3.280 · 10−2
2.643 · 10
8
12
Quadrature JDPG
1
2
1
2
2.060 · 10−5
2.920 · 10−9
5.847 · 10−14
Using the quadrature dual-Petrov-Galerkin method described in Section 3.2, we
evaluate the maximum pointwise error of u − uN with various choices of α, β, and N in
Table 4. Numerical results show that there is a very good agreement between the approximate
solution obtained by the quadrature JDPG method and the exact solution and at the same
time ascertain that the JDPG method converges exponentially.
6. Concluding Remarks
In this paper, we described a JDPG method for fully integrated forms of third- and fifth-order
ODEs with constant coefficients. Because of the constant coefficients, the matrix elements of
the discrete operators are provided explicitly, and this in turn greatly simplifies the steps and
the computational effort for obtaining solutions. However, the integrated form of the source
function involving severalfold indefinite integrals should be known analytically, and the
right hand side vector require quadrature approximations. This approach is also considered
for ODEs with polynomial coefficients. Numerical results exhibit the high accuracy of the
proposed numerical methods of solutions.
Acknowledgment
The authors are very grateful to the referees for carefully reading the paper and for their
comments and suggestions which have improved the paper.
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