Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient.
Trang 1ORIGINAL ARTICLE
New algorithms for solving third- and fifth-order
two point boundary value problems based
on nonsymmetric generalized Jacobi
Petrov–Galerkin method
E.H Doha a, W.M Abd-Elhameed a,b, Y.H Youssri a,*
a
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
b
Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia
A R T I C L E I N F O
Article history:
Received 19 September 2013
Received in revised form 9 March 2014
Accepted 10 March 2014
Available online 17 March 2014
Keywords:
Dual-Petrov–Galerkin method
Generalized Jacobi polynomials
Nonhomogeneous Dirichlet
conditions
Convergence analysis
A B S T R A C T
Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems gov-erned by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions The resulting linear systems from the application of our method are specially struc-tured and they can be efficiently inverted The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient.
ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University.
Introduction
Techniques for finding approximate solutions for differential
equations, based on classical orthogonal polynomials are
popularly known as spectral methods The term ‘‘spectral’’
was probably originated from the fact that the trigonometric
functions fei k xg are the eigenfunctions of the Laplace
operator with the periodic boundary conditions This fact and the availability of Fast Fourier Transform (FFT) are the main advantages of the Fourier spectral method Thus, using Fourier series to solve partial differential equations, with principal differential operator being the Laplace operator (or its power) with periodic boundary conditions, results in very alternative numerical algorithms[1–6]
The spectral methods aim to approximate functions (solutions of differential equations) by means of truncated series of orthogonal polynomials There are three well-known methods of spectral methods, namely, tau, collocation and Galerkin methods[2,4] The choice of the suitable spectral method suggested for solving the given equation depends certainly on the type of the differential equation and the type
of the boundary conditions governed by it The spectral
* Corresponding author Tel.: +20 1001875669; fax: +23 35676509.
E-mail address: youssri@sci.cu.edu.eg (Y.H Youssri).
Peer review under responsibility of Cairo University.
Production and hosting by Elsevier
Journal of Advanced Research (2015) 6, 673–686
Cairo University Journal of Advanced Research
2090-1232 ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University.
http://dx.doi.org/10.1016/j.jare.2014.03.003
Trang 2methods take various orthogonal polynomials as trial
func-tions The use of the Chebyshev, Legendre, ultraspherical
and classical Jacobi polynomials is suitable for non-periodic
problems, while the use of Laguerre and Hermite polynomials
is suitable for handling the problems defined respectively on
the half line, and on the whole line[4,6,7]
Standard spectral methods are capable of providing very
accurate approximations to well-behaved smooth functions
with significantly less degrees of freedom when compared with
finite difference or finite element methods[1,2,4,8]
Classical orthogonal polynomials are used successfully and
extensively for the numerical solution of differential equations
in spectral and pseudospectral methods[2,8–13]
The classical Jacobi polynomials Pða;bÞn ðxÞ have great
impor-tance in analysis from both theoretical and practical points of
view[14] The six special polynomials of the classical Jacobi
polynomials namely, ultraspherical, Chebyshev polynomials
of the four kinds, and Legendre polynomials have been
exten-sively employed in numerical analysis and in particular in
spec-tral methods[15] It is known that the Jacobi polynomials are
precisely the only polynomials arising as eigenfunctions of a
singular Sturm–Liouville problem ([2, Section 9.2])
The construction of the generalized Jacobi polynomials was
first introduced by Guo et al.[16] They extended the definition
of the classical Jacobi polynomials Pða;bÞ
n ðxÞ to allow their parameters a; b to take negative integers In Guo et al.[16],
it has been shown that the generalized Jacobi polynomials,
with parameters corresponding to the number of boundary
conditions in a given differential equation, are considered as
the natural basis functions for the spectral approximation of
this problem Moreover, it has been shown that the use of
gen-eralized Jacobi polynomials simplifies the numerical analysis
for the spectral approximations of boundary value problems
(BVPs) and also leads to very efficient numerical algorithms
Recently, Abd-Elhameed et al.[17] and Doha et al [18]
have analyzed in detail some numerical algorithms for solving
the differentiated and integrated forms of third and fifth-order
boundary value problems based on the application of the
spec-tral method namely Petrov–Galerkin method In these two
articles, the authors have employed two new families of general
parameters generalized Jacobi polynomials
A large number of books and research articles dealing with
the theory of ordinary differential equations, or their practical
applications in various fields, contain mainly results from the
theory of second-order linear differential equations, and some
results from the theory of some special linear differential
equa-tions of higher even order However, there are few studies for
handling third- and fifth- order BVPs This is due to that the
application of the collocation method on suck kinds of BVPs
leads to high condition numbers More precisely, it leads to
condition numbers of order N6 for third-order BVPs and N10
for fifth-order BVPs, respectively, where N is the number of
re-tained modes These high condition numbers will lead to
insta-bilities caused by rounding errors[9,19,20] In this paper, we
introduce some efficient spectral algorithms for reducing these
condition numbers to be of OðN2Þ and OðN4Þ for third- and
fifth-order BVPs, respectively, based on certain nonsymmetric
generalized Jacobi Petrov–Galerkin method
The study of odd-order equations is of mathematical and
physical interest As an example, third-order equation contains
a type of operator which appears in many physical
applica-tions such as the Kortweg–de Vries equation The oscillation
properties of third-order differential equations can be found
in the monographs of Mckelvey [21] For more applications
of odd-order differential equations, see the monograph by Gregus[22], in which many physical and engineering applica-tions of third-order differential equaapplica-tions are discussed[22]
In the sequence of papers of Abd-Elhameed[23], Doha and Abd-Elhameed[24,25], Doha and Bhrawy[26]and Doha et al
[27], the authors handled second-, fourth-, 2nth- and ð2n þ 1Þth-order two point boundary value problems In these articles, they suggested some numerical algorithms based on constructing combinations of various orthogonal polynomials together with the application of the Galerkin method Re-cently, Doha and Abd-Elhameed [28] have introduced and used a family of orthogonal polynomials called ‘‘symmetric generalized Jacobi polynomials’’ for handling multidimen-sional sixth-order two point boundary value problems by the Galerkin method For other studies on third- and fifth-order BVPs, one can be referred for example to[29,30]
Since, the main differential operator in odd-order differen-tial equations is not symmetric, it is convenient to use a Pet-rov–Galerkin method The main difference between the two spectral mthods namely, Galerkin and Petrov–Galerkin meth-ods, is that in case of Galerkin method, the test functions coin-cide with the trial functions, while in Petrov–Galerkin method, the trial and test functions are chosen in a way such that they satisfy respectively, the boundary conditions and their dual conditions of the differential equation
The main objective in this article is to introduce new algo-rithms for handling third- and fifth-order BVPs, based on applying the nonsymmetric generalized Jacobi Petrov–Galerkin method (GJPGM) The linear systems resulted from the appli-cation of GJPGM are band and hence they can be efficiently inverted
The structure of the paper is as follows In ‘‘Some peoper-ties of classical and generalized Jacobi polynomials’’ Section, some properties of classical and generalized Jacobi polynomi-als are given In ‘‘Dual Petrov-Galerkin algorithms for third-order elliptic linear differential equations’’ and ‘‘Dual Petrov-Galerkin algorithms for fifth-order elliptic linear differ-ential equations’’ Sections, GJPGM is applied for the sake of solving third- and fifth-order linear BVPs with constant coeffi-cients governed by homogenous boundary conditions In
‘‘Structure of the coefficients matrices in the linear systems
(23) and (32)’’ Section, the linear systems resulting from the application of GJPGM are investigated In ‘‘Condition num-ber of the resulting matrices’’ Section, we discuss the condition numbers of the obtained systems In ‘‘Convergence analysis’’ Section, we state and prove two theorems for the convergence
of the proposed algorithms In ‘‘Numerical results’’ Section, some numerical results accompanied by some comparisons with the other available algorithms appeared in literature are given Conclusions are given in ‘‘Concluding remarks’’ Section
Some properties of classical and generalized Jacobi polynomials Classical Jacobi polynomials
The classical Jacobi polynomials associated with the real parameters ða > 1; b > 1Þ [14,31,32], are a sequence of
Trang 3polynomials Pða;bÞn ðxÞ; x 2 ð1; 1Þ; ðn ¼ 0; 1; 2; Þ, each
respectively of degree n Now, and for the sake of simplicity
in the upcoming computations, it is useful to define the
follow-ing normalized classical Jacobi polynomials by Rða;bÞn ðxÞ ¼
Pða;bÞn ðxÞ
Pða;bÞn ð1Þ This means that, Rða;bÞ
n ðxÞ ¼Cðnþaþ1Þn!Cðaþ1ÞPða;bÞn ðxÞ In such case Ra
1 ;a 1
n ðxÞ ¼ CðaÞ
n ðxÞ, where CðaÞ
n ðxÞ is the ultraspherical polynomial Moreover, Rða;bÞn ðxÞ may be generated with the aid
of the following three term recurrence relation:
2ðn þ kÞðn þ a þ 1Þð2n þ k 1ÞRða;bÞnþ1ðxÞ
¼ ð2n þ k 1Þ3xRða;bÞ
n ðxÞ þ ða2 b2Þð2n þ kÞRða;bÞ
2nðn þ bÞð2n þ k þ 1ÞRða;bÞn1ðxÞ; n¼ 1; 2; ;
starting from Rða;bÞ0 ðxÞ ¼ 1 and Rða;bÞ1 ðxÞ ¼ 1
2ðaþ1Þ
½a b þ ðk þ 1Þx, or obtained from Rodrigue’s formula
Rða;bÞn ðxÞ ¼ 1
2
n
Cða þ 1Þ Cðn þ a þ 1Þð1 xÞ
að1 þ xÞbDn
½ð1 xÞaþnð1 þ xÞbþn;
where
k¼ a þ b þ 1; ðzÞk¼Cðz þ kÞ
CðzÞ ; D
d
dx; The orthogonality relation of Rða;bÞ
n ðxÞ is
Z 1
1
ð1 þ xÞað1 þ xÞbRða;bÞ
m ðxÞRða;bÞ
n ðxÞdx ¼ 0; m–n;
ha;bn ; m¼ n;
ð1Þ where
ha;b
kn!Cðn þ b þ 1ÞðCða þ 1ÞÞ2
ð2n þ kÞCðn þ kÞCðn þ a þ 1Þ:
The polynomials Rða;bÞ
n ðxÞ are eigenfunctions of the singular Sturm–Liouville equation:
ð1 x2Þ/00ðxÞ þ ½b a ðk þ 1Þx/0ðxÞ þ nðn þ kÞ/ðxÞ ¼ 0:
The following relations are useful in the sequel
Rða;bÞk ðxÞ ¼ 1
kþ 1 ðk þ a þ 1ÞR
ða;b1Þ kþ1 ðxÞ aRða1;bÞkþ1 ðxÞ
Rða;bÞk ðxÞ ¼ 1
kþ a þ b ðk þ bÞR
ða;b1Þ
k ðxÞ þ aRða1;bÞk ðxÞ
ð1 xÞRðaþ1;bÞk ðxÞ ¼ 2ða þ 1Þ
2kþ a þ b þ 2 R
ða;bÞ
k ðxÞ Rða;bÞkþ1ðxÞ
; ð4Þ
ð1 x2Þ Rðaþ1;bþ1Þk1 ðxÞ ¼ 4ða þ 1Þ
ð2k þ k 1Þ3
ðk þ bÞð2k þ k þ 1ÞRh ða;bÞk1ðxÞ
k þ a þ 1Þð2k þ k 1ÞR ða;bÞkþ1ðxÞ
þ a bÞð2k þ kÞR ða;bÞk ðxÞi
DqRða;bÞk ðxÞ ¼ðk q þ 1Þq ðk þ kÞq
2qða þ 1Þq Rðaþq;bþqÞkq ðxÞ: ð6Þ
Note 1 It is worth noting that Ra
1 ;a 1
n ðxÞ is identical to the ultraspherical polynomials, CðaÞn ðxÞ, which is explicitly defined
by
CðaÞn ðxÞ ¼ 1
2
n C aþð 1Þ
C nþaþð 1Þð1 x
2Þ1aDnhð1 x2Þaþnþ1i
;
CðaÞn ð1Þ ¼ 1; n¼ 0; 1; 2; : This definition has the desirable properties that Cð0Þn ðxÞ is iden-tical with the Chebyshev polynomials of the first kind TnðxÞ,
Cðn1ÞðxÞ is the Legendre polynomials PnðxÞ, and Cð1Þ
n ðxÞ is equal
to 1 þ1UnðxÞ is the Chebyshev polynomials of the second kind (see,[33])
Now, the following theorem is useful in what follows Theorem 1 The qth derivative of the normalized Jacobi poly-nomial Rða;bÞn ðxÞ is given explicitly by
DqRða;bÞn ðxÞ ¼ ðn þ kÞq2qn!Xnq
i¼0
Cnq;iða þ q; b þ q; a; bÞRða;bÞi ðxÞ; where
Cnq;iða þ q; b þ q; a; bÞ
¼ðn þ q þ kÞi ði þ q þ a þ 1ÞniqCði þ kÞ
ðn i qÞ! Cð2i þ kÞi!ði þ a þ 1Þni
3F2
n þ q þ i; nþ i þ q þ k; i þ a þ 1
iþ q þ a þ 1; 2iþ k þ 1 ;1
: (For the proof of Theorem1, Doha[34])
Nonsymmetric generalized Jacobi polynomials Following [16], a family of generalized Jacobi polynomials/ functions with indexes a, b2 R can be defined
Let wa;bðxÞ ¼ ð1 xÞað1 þ xÞb
We denote by L2
w a;bð1; 1Þ the weighted L2 space with inner product:
ðu; vÞwa;bðxÞ :¼
Z
I
uðxÞvðxÞwa;bðxÞdx;
and the associated normjjujjwa;b¼ ðu; uÞ1wa;b Now, we aim to define Jacobi polynomials with parameters a and/or b 61, which will be called ‘‘nonsymmetric generalized Jacobi polyno-mials (GJPs)’’ These polynopolyno-mials will satisfy some selected properties that are essentially relevant to spectral approxima-tions In this work, the values of the two parameters a and b are restricted to take negative integers
Now, and if we assume that ‘; m are two integers, then we can define GJPS by
Jð‘;mÞk ðxÞ ¼
ð1 xÞ‘ð1 þ xÞmRð‘;mÞkk
0 ðxÞ; k 0 ¼ ð‘ þ mÞ; ‘;m 6 1; ð1 xÞ‘Rð‘;mÞkk
0 ðxÞ; k 0 ¼ ‘;‘ 6 1;m > 1; ð1 þ xÞmRð‘;mÞkk
0 ðxÞ; k 0 ¼ m;‘ > 1;m 6 1;
Rð‘;mÞkk0ðxÞ; k 0 ¼ 0; ‘;m > 1:
8
>
>
>
>
It should be noted here that the GJPs have the characteriza-tion that for ‘; m2 Z and ‘; m P 1,
DiJð‘;mÞk ð1Þ ¼ 0; i¼ 0; 1; ; ‘ 1;
DjJð‘;mÞk ð1Þ ¼ 0; j¼ 0; 1; ; m 1:
It is not difficult to verify that
ðk 1Þð2k 3Þ Lk3ðxÞ
2k 3 2k 1Lk2ðxÞ
Lk1ðxÞ þ2k 3
2k 1LkðxÞ
; kP 3;
Trang 4Jð1;2Þk ðxÞ ¼ 2
2k 3 Lk3ðxÞ þ
2k 3 2k 1Lk2ðxÞ
Lk1ðxÞ 2k 3
2k 1LkðxÞ
; kP 3;
ð2k 5Þð2k 7Þðk 2Þ Lk5ðxÞ
ð2k 7Þ 2k 3 Lk4ðxÞ
2ð2k 5Þ
2k 3 Lk3ðxÞ þ
2ð2k 7Þ 2k 1 Lk2ðxÞ
þ2k 7
2k 3Lk1ðxÞ
ð2k 5Þð2k 7Þ ð2k 1Þð2k 3ÞLkðxÞ
; kP 5;
ð2k 5Þð2k 7Þ Lk5ðxÞ þ
2k 7 2k 3Lk4ðxÞ
2ð2k 5Þ
2k 3 Lk3ðxÞ
2ð2k 7Þ 2k 1 Lk2ðxÞ
þ2k 7
2k 3Lk1ðxÞ þ
ð2k 5Þð2k 7Þ ð2k 1Þð2k 3ÞLkðxÞ
; kP 5;
where LkðxÞ is the Legendre polynomial of the kth degree
fJð‘;mÞk ðxÞg are natural candidates as basis functions for
PDFs with the following boundary conditions:
Diuð1Þ ¼ ai; i¼ 0; 1; ; ‘ 1;
Djuð1Þ ¼ bj; j¼ 0; 1; ; m 1:
Dual Petrov–Galerkin algorithm for third-order elliptic linear
differential equations
This section is concerned with using GJPGM for solving the
following third-order elliptic linear differential equation
uð3ÞðxÞ a1uð2ÞðxÞ b1uð1ÞðxÞ þ c1uðxÞ ¼ fðxÞ;
governed by the homogeneous boundary conditions
We define the space
V¼ fu 2 Hð2ÞðIÞ : uð1Þ ¼ uð1Þð1Þ ¼ 0g;
and its dual space
V¼ fu 2 Hð2ÞðIÞ : uð1Þ ¼ uð1Þð1Þ ¼ 0g;
where
Hð2ÞðIÞ ¼ fu : kuk2;wa;b<1g;
kuk2;wa;b¼ X2
k¼0
kdkxuk2waþk;bþk
!1
:
Let PNbe the space of all polynomials of degree less than or
equal to N Setting VN¼ V \ PNand VN¼ V\ PN We
ob-serve that:
VN¼ span Jn ð2;1Þ3 ðxÞ; Jð2;1Þ4 ðxÞ; ; Jð2;1ÞN ðxÞo
;
VN¼ span Jn ð1;2Þ3 ðxÞ; Jð1;2Þ4 ðxÞ; ; Jð1;2ÞN ðxÞo
: The dual Petrov–Galerkin approximation of(7) and (8)is to
find u 2 V such that
ðD3
uNðxÞ; vðxÞÞ a1ðD2
uNðxÞ; vðxÞÞ b1ðDuNðxÞ; vðxÞÞ
þ c1ðuNðxÞ; vðxÞÞ ¼ ðfðxÞ; vðxÞÞ; 8v 2 V
The choice of basis functions
We can choose suitable basis functions and their dual basis by setting
ukðxÞ ¼ Jð2;1Þkþ3 ðxÞ ¼ ð1 x2Þð1 xÞRð2;1Þk ðxÞ; k ¼ 0;1; ;N 3;
wkðxÞ ¼ Jð1;2Þkþ3 ðxÞ ¼ ð1 x2Þð1 þ xÞRð1;2Þk ðxÞ; k ¼ 0;1; ;N 3:
It is worthy noting here that the basisfukðxÞg are orthogonal
on½1; 1 in the sense that
Z 1
1
ujðxÞukðxÞ ð1 xÞ2ð1 þ xÞdx¼
0; k–j;
h2;1k ; k¼ j:
The idea behind this choice is to use trial and test functions to guarantee the satisfaction of the underlying boundary and dual boundary conditions of the third-order differential equations under investigation In contrast to other bases [1,2,24], these choices lead to linear systems with specially structured matri-ces that are well-conditioned, i.e have bounded condition numbers and therefore, can be efficiently inverted These and other items will be discussed in the section entitled ‘‘Condition numbers of the resulting matrices’’
It is clear that the two sets of orthogonal polynomials
fukðxÞg and fwkðxÞg are linearly independent, and therefore
we have
VN¼ spanfukðxÞ : k ¼ 0; 1; 2; ; N 3g;
and
VN¼ spanfwkðxÞ : k ¼ 0; 1; 2; ; N 3g:
Now the following two important lemmas will be stated and proved
Lemma 1
Proof By using Leibnitz’s rule, we have
D3Jð2;1Þkþ3 ðxÞ ¼ ð1 x2Þð1 xÞD3Rð2;1Þk ðxÞ
þ 3ð3x2 2x 1ÞD2Rð2;1Þk ðxÞ
þ 6ð3x 1ÞDRð2;1Þk ðxÞ þ 6Rð2;1Þk ðxÞ:
Making use of the relation ð1 x2Þð1 xÞD3
Rð2;1Þk ðxÞ ¼ ð1 þ 6x 7x2ÞD2
Rð2;1Þk ðxÞ
þ ðk 1Þðk þ 5Þðx 1ÞDRð2;1Þk ðxÞ;
we obtain
D3Jð2;1Þkþ3 ðxÞ ¼ 2ðx2 1ÞD2
Rð2;1Þk ðxÞ þ ½ðk 1Þðk þ 5Þðx 1Þ
þ 6ð3x 1ÞDRð2;1Þk ðxÞ þ 6Rð2;1Þk ðxÞ;
which in turn with Eq (2), and after some rather lengthy manipulation, yields
Trang 5D3Jð2;1Þkþ3 ðxÞ ¼ ðk þ 1Þðk þ 3Þ ð1 xÞDRh ð2;1Þk ðxÞ 2Rð2;1Þk ðxÞi
:
Making use of the two relations(4) and (6), we have
D3Jð2;1Þkþ3 ðxÞ ¼1
6ðk þ 1Þðk þ 3Þ kðk þ 4Þðx 1ÞRh ð3;2Þk1ðxÞ þ 12Rð2;1Þk ðxÞi
:
Finally, and in virtue of(2) and (3), and after some
manipula-tion, we get
D3Jð2;1Þkþ3 ðxÞ ¼ 2ðk þ 1Þðk þ 3ÞRð1;2Þk ðxÞ:
By recalling the definition of Pochhammer’s symbol,
ðzÞn¼CðzþnÞCðzÞ, we have
Lemma 2
D2Jð2;1Þkþ3 ðxÞ ¼2ðk þ 3Þ2
ð2k þ 5Þ R
ð1;2Þ kþ1ðxÞ ðk þ 1Þðk þ 3Þ
kþ3 2
2
Rð1;2Þk ðxÞ
2ðkÞ2
ð2k þ 3ÞR
ð1;2Þ k1ðxÞ;
DJð2;1Þkþ3 ðxÞ ¼ ðk þ 3Þ3
2ðk þ 2Þ k þ 5
2
Rð1;2Þkþ2ðxÞ ðk þ 3Þ2
kþ3
3
Rð1;2Þkþ1ðxÞ
ðk þ 1Þðk þ 3Þ
kþ3
2
Rð1;2Þk ðxÞ þ ðkÞ2
kþ1
3
Rð1;2Þk1ðxÞ
þ ðk 1Þ3
2ðk þ 2Þ k þ 1
2
Rð1;2Þk2ðxÞ;
Jð2;1Þkþ3 ðxÞ ¼ ðk þ 4Þ3
4ðk þ 2Þ k þ 5
3
Rð1;2Þkþ3ðxÞ
3ðk þ 3Þ3
4ðk þ 2Þ k þ 3
4
Rð1;2Þkþ2ðxÞ 3ðk þ 3Þ2
4 k þ3
3
Rð1;2Þkþ1ðxÞ
þ3ðk þ 1Þðk þ 3Þ
2 k þ1
4
Rð1;2Þk ðxÞ þ 3ðkÞ2
4 k þ1
3
Rð1;2Þk1ðxÞ
3ðk 1Þ3
4ðk þ 2Þ k 1
4
Rð1;2Þk2ðxÞ
ðk 2Þ3
4ðk þ 2Þ k 1
3
Rð1;2Þk3ðxÞ:
Proof The proof of Lemma 2 is rather lengthy and it can be
accomplished by following the same procedure used in the
proof of Lemma 1 h
Now, based on the two Lemmas 1 and 2, the following
the-orem can be obtained
Theorem 2 For arbitrary constants ak, one has
D3 XN3
k¼0
akJð2;1Þkþ3 ðxÞ
k¼0
where
Moreover, if
D2 XN3 k¼0
akJð2;1Þkþ3 ðxÞ
¼XN2 k¼0
then
ek;2¼ ak1að2Þk1þ akbð2Þk þ akþ1cð2Þkþ1; ð14Þ where
að2Þk ¼2ðk þ 3Þ2
ð2k þ 5Þ; b
ð2Þ
k ¼ ðk þ 1Þðk þ 3Þ
kþ3 2
2
;
cð2Þk ¼ 2ðkÞ2
ð2k þ 3Þ: Also, if
D XN3 k¼0
akJð2;1Þkþ3 ðxÞ
¼XN1 k¼0
then
ek;1¼ ak2að1Þk2þ ak1bð1Þk1þ akcð1Þk þ akþ1dð1Þkþ1
where
að1Þk ¼ ðk þ 3Þ3
2ðk þ 2Þ k þ 5
2
; bð1Þk ¼ ðk þ 3Þ2
kþ3
3
;
cð1Þk ¼ ðk þ 1Þðk þ 3Þ
kþ3 2
2
; dð1Þk ¼ ðkÞ2
kþ1 2
3
;
lð1Þk ¼ ðk 1Þ3
2ðk þ 2Þ k þ 1
2
: Finally, if
X
N3 k¼0
akJð2;1Þkþ3 ðxÞ ¼XN
k¼0
then
ek;0¼ ak3að0Þk3þ ak2bð0Þk2þ ak1cð0Þk1þ akdð0Þk þ akþ1lð0Þkþ1
þ akþ2gð0Þkþ2þ akþ3fð0Þkþ3; ð18Þ where
að0Þk ¼ ðk þ 4Þ3
4ðk þ 2Þ k þ 5
3
; bð0Þk ¼ 3ðk þ 3Þ3
4ðk þ 2Þ k þ 3
4
;
cð0Þk ¼ 3ðk þ 3Þ2
4 kþ3 2
3
; dð0Þk ¼ 3ðk þ 1Þðk þ 3Þ
2 kþ1 2
4
;
lð0Þk ¼ 3ðkÞ2
4 k þ1
3
; gð0Þk ¼ 3ðk 1Þ3
4ðk þ 2Þ k 1
4
;
fð0Þk ¼ ðk 2Þ3
4ðk þ 2Þ k 1
3
: Now, the application of Petrov–Galerkin method on Eq
(7), yields
ðD3uNðxÞ a1D2uN b1DuNþ c1uN;wkðxÞÞ
where
uNðxÞ ¼XN3
k¼0
ak/kðxÞ; /kðxÞ ¼ Jð2;1Þkþ3 ðxÞ;
wðxÞ ¼ Jð1;2ÞðxÞ; k¼ 0; 1; ; N 3:
Trang 6Substitution of formulae (11), (13), (15) and (17) into (19)
yields
XN3
j¼0
bjRð1;2Þj ðxÞ a1
X
N2 j¼0
ej;2Rð1;2Þj ðxÞ b1XN1
j¼0
ej;1Rð1;2Þj ðxÞ
þ c1XN
j¼0
ej;0Rð1;2Þj ðxÞ; Jð1;2Þkþ3 ðxÞ
!
¼ f; J ð1;2Þkþ3 ðxÞ
where bkand ek;2q;0 6 q 6 2 are given by(12), (14), (16) and
(18), respectively
Eq.(20)is equivalent to
XN3
j¼0
bjRð1;2Þj ðxÞ a1
X
N2 j¼0
ej;2Rð1;2Þj ðxÞ b1XN1
j¼0
ej;1Rð1;2Þj ðxÞ
þ c1XN
j¼0
ej;0Rð1;2Þj ðxÞ; Rð1;2Þk ðxÞ
!
w
¼ f; R ð1;2Þk ðxÞ
w; where w¼ ð1 x2Þð1 þ xÞ Making use of the orthogonality
relation(1), it is not difficult to show that Eq.(20)is equivalent
to
fk¼ ðbk a1ek;2 b1ek;1þ c1ek;0Þh1;2k ;
where
fk¼ f; R ð1;2Þk ðxÞ
w: This linear system may be put in the form
b1
k a1ek;2 b1ek;1þ c1ek;0
¼ f
k; k¼ 0; 1; N 3; ð22Þ where
fk¼ fk
h1;2k ; h
1;2
ðk þ 1Þðk þ 2Þðk þ 3Þ; which may be written simply in the matrix form
ðB1þ a1E2þ b1E1þ c1E0Þa ¼ f; ð23Þ
where
a¼ ða0; a1; ; aN3ÞT; f¼ f
0; f1; ; fN3
; and the nonzero elements of the matrices B1; E2; E1and E0are
given explicitly in the following theorem
Theorem 3 The nonzero elements of the matrices B1¼ ðb1kjÞ
and Ei¼ ðei
kjÞ; 0 6 i 6 2, for 0 6 k; j 6 N 3, are given as
follows:
b1kk¼ 2ðk þ 1Þðk þ 3Þ; e2
k;kþ1¼2ðkþ1Þðkþ2Þ
e2
kþ1;k¼2ðkþ3Þðkþ4Þ
kk¼ 4ðkþ1Þðkþ3Þ ð2kþ3Þð2kþ5Þ;
e1
kk¼ð2kþ3Þð2kþ5Þ4ðkþ1Þðkþ3Þ; e1
k;kþ1¼ð2kþ3Þð2kþ5Þð2kþ7Þ8ðkþ1Þðkþ2Þ ;
e1
k;kþ2¼2ðkþ1Þðkþ2Þðkþ3Þðkþ4Þð2kþ5Þð2kþ7Þ; e1
kþ1;k¼ð2kþ3Þð2kþ5Þð2kþ7Þ8ðkþ3Þðkþ4Þ ;
e1
kþ2;k¼2ðkþ3Þðkþ4Þðkþ5Þ
ðkþ2Þð2kþ5Þð2kþ7Þ; e0
kk¼3ðkþ1Þðkþ3Þ
2 kþð 1Þ4 ;
e0
k;kþ1¼ 3ðkþ1Þ2
4 kþð 3Þ3; e0k;kþ2¼ 3ðkþ1Þ3
4ðkþ4Þ kþð 3Þ4;
e0
k;kþ3¼ ðkþ1Þ3
4ðkþ5Þ kþð 5Þ3; e0kþ1;k¼3ðkþ3Þ2
4 kþð 3Þ3;
e0
kþ2;k¼3ðkþ3Þ5
0 kþ3;k¼ ðkþ4Þ3
4ðkþ2Þ kþð 5Þ :
Dual Petrov–Galerkin algorithm for fifth-order elliptic differential equations
In this section we aim to apply the GJPGM for solving the fol-lowing fifth-order elliptic linear equation
uð5ÞðxÞ þ a2uð4ÞðxÞ þ b2uð3ÞðxÞ c2uð2ÞðxÞ d2uð1ÞðxÞ
þ l2uðxÞ
governed by the homogeneous boundary conditions
We define the following two spaces
V¼ fu 2 Hð3ÞðIÞ : uð1Þ ¼ uð1Þð1Þ ¼ uð2Þð1Þ ¼ 0g;
and
V¼ fu 2 Hð3ÞðIÞ : uð1Þ ¼ uð1Þð1Þ ¼ uð2Þð1Þ ¼ 0g; where
Hð3ÞðIÞ ¼ fu : kuk3;wa;b<1g; kuk3;wa;b¼ X3
k¼0
k@kxuk2waþk;bþk
!1
:
Now, setting VN¼ V \ PN and VN¼ V\ PN We observe that:
VN¼ span Jn ð3;2Þ5 ðxÞ; Jð3;2Þ6 ðxÞ; ; Jð3;2ÞN ðxÞo
;
VN¼ span Jn ð2;3Þ5 ðxÞ; Jð2;3Þ6 ðxÞ; ; Jð2;3ÞN ðxÞo
: The dual Petrov–Galerkin approximation of(24) and (25)is to find uN2 VNsuch that
ðD5uNðxÞ; vðxÞÞ þ a2ðD4uNðxÞ; vðxÞÞ
þ b2ðD3uNðxÞ; vðxÞÞ c2ðD2uNðxÞ; vðxÞÞ
d2ðDuNðxÞ; vðxÞÞ þ l2ðuNðxÞ; vðxÞÞ
¼ ðfðxÞ; vðxÞÞ; 8v 2 V
The choice of basis functions
We can choose suitable basis functions and their dual basis – in the same way as in the previous case and for the same reasons – by setting
ukðxÞ ¼ Jð3;2Þkþ5 ðxÞ ¼ ð1 x2Þ2ð1 xÞRð3;2Þk ðxÞ; k ¼ 0;1; .; N 5;
wkðxÞ ¼ Jð2;3Þkþ5 ðxÞ ¼ ð1 x2Þ2ð1 þ xÞRð2;3Þk ðxÞ; k ¼ 0;1; .; N 5:
It is worthy noting here that the basisf/kðxÞg are orthogonal
on½1; 1 in the sense that
Z 1
1
ujðxÞukðxÞ ð1 xÞ3ð1 þ xÞ2dx¼
0; k–j;
h3;2k ; k¼ j:
It is clear that the two sets of orthogonal polynomials f/kðxÞg and fwkðxÞg are linearly independent, and therefore
we have
VN¼ spanfukðxÞ : k ¼ 0; 1; 2; ; N 5g;
and
V ¼ spanfwðxÞ : k ¼ 0; 1; 2; ; N 5g:
Trang 7The following two lemmas are needed.
Lemma 3
D5Jð3;2Þkþ5 ðxÞ ¼ 3ðk þ 1Þðk þ 2Þðk þ 4Þðk þ 5ÞRð2;3Þk ðxÞ:
Proof Setting a¼ 2; b ¼ 1 in relation(5), we get
ð1 x2ÞRð3;2Þk ðxÞ ¼ 12
ð2k þ 5Þ3 ðk þ 2Þð2k þ 7ÞR
ð2;1Þ
h
þ 2ðk þ 3ÞRð2;1Þkþ1ðxÞ ðk þ 4Þð2k þ 5ÞRð2;1Þkþ2ðxÞi
:
Making use of this relation and with the aid of the two
relations(6)(for q¼ 2) and(10), we obtain
D5Jð3;2Þkþ5 ðxÞ ¼ 1
ð2k þ 5Þ3hð2k þ 7Þðk 1Þ7Rð3;4Þk2ðxÞ
þ 2ðkÞ7Rð3;4Þk1ðxÞ ð2k þ 5Þðk þ 1Þ7Rð3;4Þk ðxÞi
: Finally, from the two relations (2) and (3), and after some
lengthy manipulation, we get
D5Jð3;2Þkþ5 ðxÞ ¼ 3ðk þ 1Þðk þ 2Þðk þ 4Þðk þ 5ÞRð2;3Þk ðxÞ:
Lemma 4
D4Jð3;2Þkþ5 ðxÞ ¼ 3ðk þ 2Þðk þ 4Þ3
2kþ 7 R
ð2;3Þ kþ1ðxÞ þ3ðk þ 1Þ2ðk þ 4Þ2
2 k þ5
2
Rð2;3Þk ðxÞ
þ3ðkÞ3ðk þ 4Þ
ð2k þ 5Þ R
ð2;3Þ
D3Jð3;2Þkþ5 ðxÞ ¼ 3ðkþ 4Þ4
4 kþ 7
2
Rð2;3Þkþ2ðxÞþ3ðk þ 2Þðkþ 4Þ3
2 k þ5
3
Rð2;3Þkþ1ðxÞ
þ3ðk þ 1Þ2ðk þ4Þ2
2 kþ5 2
2
Rð2;3Þk ðxÞ 3ðkÞ3ðk þ 4Þ
2 kþ3 2
3
Rð2;3Þk1ðxÞ
3ðk 1Þ4
4 kþ3
2
2
D2Jð3;2Þkþ5 ðxÞ ¼ 3ðk þ 4Þ5
8ðk þ 3Þ k þ 7
3
Rð2;3Þkþ3ðxÞ þ9ðk þ 4Þ4
8 k þ5
4
Rð2;3Þkþ2ðxÞ
þ9ðk þ 2Þðk þ 4Þ3
8 k þ5
3
Rð2;3Þkþ1ðxÞ 9ðk þ 1Þ2ðk þ 4Þ2
4 k þ3
4
Rð2;3Þk ðxÞ
9ðkÞ3ðk þ 4Þ
8 k þ3
3
Rð2;3Þk1ðxÞ þ9ðk 1Þ4
8 k þ1
4
Rð2;3Þk2ðxÞ
þ 3ðk 2Þ5
8ðk þ 3Þ k þ 1
3
DJð3;2Þkþ5 ðxÞ ¼ 3ðk þ 5Þ5
16ðk þ 3Þ k þ 7
4
Rð2;3Þkþ4ðxÞ
þ 3ðk þ 4Þ5
4ðk þ 3Þ k þ 5
5
Rð2;3Þkþ3ðxÞ þ3ðk þ 4Þ4
4 k þ 5
4
Rð2;3Þkþ2ðxÞ
9ðk þ 2Þðk þ 4Þ3
4 k þ 3
5
Rð2;3Þkþ1ðxÞ 9ðk þ 1Þ2 ðk þ 4Þ2
8 k þ 3
4
Rð2;3Þk ðxÞ
þ9ðkÞ3 ðk þ 4Þ
4 k þ 1
5
Rð2;3Þk1ðxÞ þ3ðk 1Þ4
4 k þ 1
4
Rð2;3Þk2ðxÞ
3ðk 2Þ5
4ðk þ 3Þ k 1
5
Rð2;3Þk3ðxÞ 3ðk 3Þ5
16ðk þ 3Þðk 1 Þ4R
ð2;3Þ k4 ðxÞ;
ð30Þ
and
Jð3;2Þkþ5 ðxÞ ¼ 3ðk þ 6Þ5
32ðk þ 3Þ k þ 7
5
Rð2;3Þkþ5ðxÞ
þ 15ðk þ 5Þ5
32ðk þ 3Þ k þ 5
6
Rð2;3Þkþ4ðxÞ þ 15ðk þ 4Þ5
32ðk þ 3Þ k þ 5
5
Rð2;3Þkþ3ðxÞ
15ðk þ 4Þ4
8 k þ3
6
Rð2;3Þkþ2ðxÞ 15ðk þ 2Þðk þ 4Þ3
16 k þ3
5
Rð2;3Þkþ1ðxÞ
þ45ðk þ 1Þ2ðk þ 4Þ2
16 k þ1
6
Rð2;3Þk ðxÞ þ15ðkÞ3ðk þ 4Þ
16 k þ1
5
Rð2;3Þk1ðxÞ
15ðk 1Þ4
8 k 1
6
Rð2;3Þk2ðxÞ 15ðk 2Þ5
32ðk þ 3Þ k 1
5
Rð2;3Þk3ðxÞ
þ 15ðk 3Þ5
32ðk þ 3Þ k 3
6
Rð2;3Þk4ðxÞ þ 3ðk 4Þ5
32ðk þ 3Þ k 3
5
Rð2;3Þk5ðxÞ: ð31Þ
Applying Petrov–Galerkin method to(24) and (25) and if
we make use of the two Lemmas 3 and 4, and after performing some lengthy manipulation, then the numerical solution of(24) and (25)can be obtained This solution is given in the follow-ing Theorem
Theorem 4 If uNðxÞ ¼PN5
0 akJð3;2Þkþ5 ðxÞ is the Petrov– Galerkin approximation to(24) and (25), then the expansion coefficientsfak: k¼ 0; 1; ; N 5g satisfy the matrix system
ðB2þ a2G4þ b2G3þ c2G2þ d2G1þ l2G0Þa ¼ f; ð32Þ where the nonzero elements of the matrices B2¼ b2
k;j
and
Gi¼ gi k;j
;ð0 6 i 6 4Þ, for 0 6 k; j 6 N 5, are given as follows:
b2kk¼ rk; g4
2 kþ5 2
2
; g4 k;kþ1¼3ðk þ 1Þ3ðk þ 5Þ
2kþ 7 ;
g4 kþ1;k¼ 3ðk þ 2Þ5
ðk þ 3Þð2k þ 7Þ; g
3
2 k þ5
2
;
g3 k;kþ1¼3ðk þ 1Þ3ðk þ 5Þ
2 k þ5
3
;
g3 k;kþ2¼3ðk þ 1Þ4
4 kþ7 2
2
; g3 kþ1;k¼3ðk þ 2Þðk þ 4Þ3
2 kþ5 2
3
;
g3 kþ2;k¼3ðk þ 4Þ4
4 k þ7
2
;
g2
4 kþ3 2
4
; g2 k;kþ1¼9ðk þ 1Þ3ðk þ 5Þ
8 kþ5 2
3
;
g2 k;kþ2¼9ðk þ 1Þ4
8 k þ5
4
;
g2k;kþ3¼ 3ðk þ 1Þ5
8ðk þ 6Þ k þ 7
3
; g2kþ1;k¼9ðk þ 2Þðk þ 4Þ3
8 k þ5
3
;
g2 kþ2;k¼9ðk þ 4Þ4
8 kþ5 2
4
;
g2
8ðk þ 3Þ k þ7
2
3
; g1
8 kþ3 2
4
;
g1k;kþ1¼9ðk þ 1Þ3ðk þ 5Þ
4 k þ3 ;
Trang 8k;kþ2¼3ðk þ 1Þ4
4 k þ5
4
; g1
4ðk þ 6Þ k þ 5
5
;
g1
16ðk þ 7Þ k þ7
2
4
;
g1
kþ1;k¼9ðk þ 2Þðk þ 4Þ3
4 kþ3
2
5
; g1 kþ2;k¼3ðk þ 4Þ4
4 kþ5 2
4
;
g1kþ3;k¼ 3ðk þ 4Þ5
4ðk þ 3Þ k þ 5
5
;
g1kþ4;k¼ 3ðk þ 5Þ5
16ðk þ 3Þ k þ 7
4
; g0kk¼ 15rk
16 k þ1
6
;
g0
k;kþ1¼15ðk þ 1Þ3ðk þ 5Þ
16 k þ3
5
;
g0
k;kþ2¼15ðk þ 1Þ4
8 k þ3
6
; g0
32ðk þ 6Þ k þ 5
5
;
g0
32ðk þ 7Þ k þ5
2
6
;
g0
32ðk þ 8Þ k þ7
2
5
; g0 kþ1;k¼15ðk þ 2Þðk þ 4Þ3
16 kþ3 2
5
;
g0
kþ2;k¼15ðk þ 4Þ4
8 kþ3
2
6
;
g0
32ðk þ 3Þ k þ5
2
5
; g0
32ðk þ 3Þ k þ5
2
6
;
g0kþ5;k¼ 3ðk þ 6Þ5
32ðk þ 3Þ k þ 7
5
; where rk¼ 3ðk þ 1Þðk þ 2Þðk þ 4Þðk þ 5Þ
Structure of the coefficient matrices in the linear systems(23)
and (32)
This section is concerned with discussing the structure of the
coefficient matrices B1 and E3q ð1 6 q 6 3Þ which appear in
the linear system (23), and the coefficient matrices B2 and
G5qð1 6 q 6 5Þ in the linear system(32) Hence, we will
dis-cuss the structure of the two combined matrices
D1¼ B1þ a1E2þ b1E1þ c1E0; and D2
¼ B2þ a2G4þ b2G3þ c2G2þ d2G1þ l2G0:
Also, the influence of these structures on the efficiency for
solv-ing the two systems(23) and (32)will be discussed
Since the two matrices B1 and B2 are diagonal, the two
cases correspond to a1¼ b1¼ c1¼ 0 in (23) and
a2¼ b2¼ c2¼ d2¼ l2¼ 0 in (32) lead to two diagonal
sys-tems The results for these two cases are summarized in the
fol-lowing two important corollaries
Corollary 1 If uNðxÞ ¼PN3
k¼0akJð2;1Þkþ3 ðxÞ is the Galerkin approximation to problem (7) and (8), for a1¼ b1¼ c1¼ 0,
then the expansion coefficients fak: k¼ 0; 1; ; N 3g are
given explicitly by:
ak¼kþ 2
16 fk; k¼ 0; 1; ; N 3;
where fk¼R1
1ð1 x2Þð1 þ xÞfðxÞRð1;2Þk ðxÞdx
Corollary 2 If uNðxÞ ¼PN5
k¼0akJð3;2Þkþ5 ðxÞ is the Petrov–Galer-kin approximation to problem (24) and (25), for
a2¼ b2¼ c2¼ d2¼ l2¼ 0, then the expansion coefficients
fak: k¼ 0; 1; ; N 5g are given explicitly by
ak¼kþ 3
384 fk; k¼ 0; 1; ; N 5;
where
fk¼
Z 1
1
ð1 x2Þð1 þ xÞfðxÞRð2;3Þk ðxÞdx:
Now, each of the matrices E3q ð1 6 q 6 3Þ and
G5qð1 6 q 6 5Þ is a band matrix and the total number of non-zero diagonals upper or lower the main diagonal for each matrix
is q Thus the coefficient matrices D1 and D2 are at most four-band and six-four-band matrices, respectively These special struc-tures of D1and D2simplify greatly the solution of the two linear systems(23) and (32) These two systems can be decomposed by LU-factorization Moreover, the operations required for con-structing these factorizations are of order 21ðN 2Þ and 55ðN 4Þ, respectively Also, the number of operations required for solving the two decomposable triangular systems are of order 13ðN 2Þ and 21ðN 4Þ respectively
Note 2 The total number of operations mentioned in the pre-vious discussion includes the number of all subtractions, addi-tions, divisions and multiplications[35]
Treatment of nonhomogeneous boundary conditions
This section is devoted to describe the way of how third- and fifth-order BVPs governed by nonhomogeneous boundary conditions can be converted to BVPs governed by homoge-neous boundary conditions
Now, Let us consider the one-dimensional third-order equation
uð3ÞðxÞ a1uð2ÞðxÞ b1uð1ÞðxÞ þ c1uðxÞ ¼ fðxÞ;
x2 I ¼ ð1; 1Þ;
governed by the nonhomogeneous boundary conditions:
Now, and if we make use of the transformation VðxÞ ¼ uðxÞ þ a0þ a1xþ a2x2; ð34Þ where
a0¼a 3aþþ 2a
1
4 ; a1¼a aþ
a2¼aþ aþ 2a
1
then, the transformation (34) turns the nonhomogeneous boundary conditions (33) into the homogeneous boundary conditions:
Trang 9Hence, it is sufficient to solve the following modified
one-dimensional third-order equation:
Vð3ÞðxÞ a1Vð2ÞðxÞ b1Vð1ÞðxÞ þ c1VðxÞ ¼ fðxÞ;
governed by the homogeneous boundary conditions (35),
where VðxÞ is given by(34), and
fðxÞ ¼ fðxÞ þ ð2a1a2 b1a1þ c1a0Þ þ ð2b1a2þ c1a0Þx
þ c1a2x2:
Now, the application of the GJPGM to the modified Eq
(36), leads to the following equivalent system of equations
ðB1þ a1E2þ b1E1þ c1E0Þa ¼ f;
B1; E2; E1 and E0 are the matrices defined in Theorem 3, and
f¼ f
0; f
1; ; f
N3
, where
fk¼
2a1a2 b1a1þ c1a0; k¼ 0;
6
5ð2b1a2þ c1a0Þ; k¼ 1;
10
8
>
>
<
>
>
:
and fk¼R1
1ð1 þ x2Þð1 þ xÞRð1;2Þk ðxÞfðxÞdx
The same procedure can be applied to solve the following
fifth-order BVP:
uð5ÞðxÞ þ a2uð4ÞðxÞ þ b2uð3ÞðxÞ c2uð2ÞðxÞ d2uð1ÞðxÞ
þ l2uðxÞ
governed by the nonhomogeneous boundary conditions
uð1Þ ¼ a; uð1Þð1Þ ¼ a1
; uð2Þð1Þ ¼ b: ð38Þ
In such case,(37) and (38)can be turned into
Vð5ÞðxÞ þ a2Vð4ÞðxÞ þ b2Vð3ÞðxÞ c2Vð2ÞðxÞ d2Vð1ÞðxÞ
þ l2VðxÞ
governed by the homogenous boundary conditions
Vð1Þ ¼ Vð1Þð1Þ ¼ Vð2Þð1Þ ¼ 0;
where
VðxÞ ¼ uðxÞ þ a0þ a1xþ a2x2þ a3x3þ a4x4;
with
a0¼ 1
16 2a1
þ 8a1
þ 2b 5a 11aþ
;
a1¼1
4 a
1
þ a1
þþ 3a 3aþ
;
a2¼1
8 6a1
þþ 2b 3aþ 3aþ
;
a3¼1
4 a1
a1
þ aþ aþ
;
a4¼ 1
16 4a
1
þþ 3a 3aþ
; and
fðxÞ ¼ ðl2a0 d2a1 2c2a2þ 6ba3þ 24a2a4Þ þ ðl2a1 2d2a2
6c2a3þ 24b2a4Þx þ ðl2a2 3d2a3 12c2a4Þx2
þ ðl2a3 4d2a4Þx3þ l2a4x4þ fðxÞ:
If the GJPGM is applied to Eq.(39), then the following equiv-alent system of equations is obtained
ðB2þ aG4þ bG3þ cG2þ dG1þ lG0Þa ¼ f; where B2; Gi ð0 6 i 6 4Þ are the matrices defined in Theorem 4, and
f¼ f
0; f1; ; fN5
;
f
k¼
l2a0 d2a1 2c2a2þ 6b2a3þ 24a2a4; k¼ 0;
8
7ðl2a1 2d2a2 6c2a3þ 24b2a4Þ; k¼ 1;
4
3ðl2a2 3d2a3 12c2a4Þ; k¼ 2;
50
8
>
>
>
>
>
>
and fk¼R1
1ð1 þ x2Þð1 þ xÞRð2;3Þk ðxÞfðxÞdx
Condition numbers of the resulting matrices
In the direct collocation method, the condition numbers be-have like OðN6Þ and OðN10Þ for third- and fifth-order BVPs, respectively, (N: maximal degree of polynomials) In this arti-cle, improved condition numbers with OðN4Þ and OðN6Þ are obtained, respectively, for third- and fifth-order BVPs The advantage with respect to propagation of rounding errors is demonstrated
For GJPGM, the resulting systems obtained for the two differential equations uð3ÞðxÞ ¼ fðxÞ and uð5ÞðxÞ ¼ fðxÞ are
B1a1¼ f and B2a2¼ f, where B1 and B2 are two diagonal matrices their elements are given by b1kkand b2kk, where
b1kk¼ 2ðk þ 1Þðk þ 3Þ; b2kk¼ 3ðk þ 1Þðk þ 2Þðk þ 4Þðk þ 5Þ: Thus we note that the condition numbers of the matrices B1 and B2 behave like Oðk2Þ and Oðk4Þ for large values of k, respectively The evaluation of the condition numbers for the matrices B1and B2are easy because of their special structures, since B1and B2are diagonal matrices, so their eigenvalues are their diagonal elements In such case, the condition number can be defined as:
Condition number of the matrix
¼Maxðeigenvalue of the matrixÞ Minðeigenvalue of the matrixÞ:
InTable 1, we list the values of the conditions numbers of the matrices B1 and B2, respectively, for different values of N Remark 1 If we add P3
q¼1E3qð1 6 q 6 3Þ and P5
q¼1G5q ð1 6 q 6 5Þ, where the matrices E3qand G5qare the matrices their nonzero elements are given explicitly in Theorems 3 and
4, to the matrices B1and B2, respectively, then we find that the eigenvalues of the matrices D1¼ B1þP3
q¼1E3q; D2¼ B2þ P5
q¼1G5q are all real and positive Moreover, the effect of these additions does not significantly change the values of the condition numbers for the systems This means that matrices
Trang 10B1and B2, which resulted from the highest derivatives of the
differential equations under investigation, play the most
important role in the propagation of the roundoff errors
The numerical results ofTable 2illustrate this remark
Convergence analysis
In this section, we state and prove two theorems to ascertain
that the nonsymmetric generalized Jacobi polynomials
expan-sion of a function uðxÞ 2 Hð2Þð1; 1Þ, converges uniformly to
uðxÞ For k 1; ak: bk and ak bk mean that
limk!1a k
bk¼ 1 and limk!1a k
bk<1, respectively The following theorem is needed in the sequel
Theorem 5 (Bernstein type Inequality [36]) The well-known
Legendre polynomials LkðxÞ; k ¼ 0; 1; 2; , satisfy the
follow-ing inequality
ffiffiffiffiffiffiffiffiffi
sin h
p
Lkðcos hÞ <
ffiffiffiffiffiffi 2 pk
r
; 0 < h < p:
Theorem 6 A function uðxÞ ¼ ð1 xÞ2ð1 þ xÞfðxÞ 2
Hð2Þð1; 1Þ, with jfð2ÞðxÞj 6 L, can be expanded as an infinite
sum of nonsymmetric generalized Jacobi polynomials
Jð2;1Þkþ3 ðxÞ : k ¼ 0; 1; 2;
, and the series converges uni-formly to uðxÞ Explicitly, the expansion coefficients in
uðxÞ ¼P1
k¼0akJð2;1Þkþ3 ðxÞ, satisfy the following inequality:
jakj <9L
pk3; 8k P 0:
Proof Since nJð2;1Þkþ3 ðxÞ : k ¼ 0; 1; 2; o
are orthogonal basis of Hð2Þð1; 1Þ, then
ak¼ 1
h2;1k
Z 1
1
Jð2;1Þkþ3 ðxÞuðxÞ ð1 xÞ2ð1 þ xÞdx;
where h2;1k is as defined in(1) With the aid of the relation
ðk 1Þð2k 3Þ Lk3ðxÞ
2k 3 2k 1Lk2ðxÞ
Lk1ðxÞ þ2k 3
2k 1LkðxÞ
; and after integration by parts two times, we get,
ak¼ðk þ 1Þðk þ 3Þ 2ð2k þ 3Þ
Z 1
1
IkðxÞfð2ÞðxÞdx;
where
IkðxÞ ¼ Lk2ðxÞ
4 k1 2
2
Lk1ðxÞ ð2k þ 1Þð2k þ 5Þ
3LkðxÞ ð2k 1Þð2k þ 5Þ
þ3ð2k þ 3ÞLkþ1ðxÞ 4ð2k þ 1Þ k þ5
2
2
þ 3Lkþ2ðxÞ ð2k þ 1Þð2k þ 7Þ
3Lkþ3ðxÞ ð2k þ 5Þð2k þ 9Þ
Lkþ4ðxÞ ð2k þ 5Þð2k þ 7Þ
þð2k þ 3ÞLkþ5ðxÞ
8 k þ5
3
¼X7 m¼0
bm;kLkþm2ðxÞ; say:
Now, making use of the substitution x¼ cos h, yields
ak¼ðk þ 1Þðk þ 3Þ 2ð2k þ 3Þ
X7 m¼0
bm;k
Z p 0
Lkþm2ðcoshÞfð2ÞðcoshÞsinhdh
;
Therefore, we have
jakj 6ðk þ 1Þðk þ 3ÞL
2ð2k þ 3Þ
X7
jbm;kj
Z p 0
jLkþm2ðcos hÞj sin h dh
:
Table 1 Condition number for the matrices Bn; n¼ 1; 2
Table 2 Condition number for the matrices Dn; n¼ 1; 2
N Cond ðD 1 Þ CondðD1 Þ
N 4
16 55.287 2:159 101 827.262 1:262 102
20 88.679 2:217 101 2278.4 1:424 102
24 129.929 2:256 101 5104.45 1:539 102
28 179.037 2:284 101 9980.18 1:624 102
32 236.003 2:305 10 1 17715.3 1:689 10 2
36 300.826 2:321 10 1 2925.4 1:742 10 2
40 373.507 2:334 10 1 45677.4 1:784 10 2