Journal of the Egyptian Mathematical Society (2014) xxx, xxx-xxx

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

ORIGINAL ARTICLE

On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems

E.H. Doha a, W.M. Abd-Elhameed a b *, M.M. Alsuyutic

a Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt b Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia c Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt

Received 1 December 2013; revised 26 March 2014; accepted 26 April 2014

KEYWORDS

Dual Petrov-Galerkin method;

Chebyshev polynomials of third and fourth kinds; Integrated forms; High odd-order two points boundary value problems

Abstract This article presents some spectral Petrov-Galerkin numerical algorithms based on using Chebyshev polynomials of third and fourth kinds for solving the integrated forms of high odd-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions. The principle idea behind obtaining the proposed numerical algorithms is based on constructing trial and test functions as compact combinations of shifted Chebyshev polynomials of third and fourth kinds. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Some numerical examples are illustrated for the sake of demonstrating the validity and the applicability of the proposed algorithms. The presented numerical results indicate that the proposed algorithms are reliable and very efficient.

MATHEMATICS SUBJECT CLASSIFICATION: 65M70; 65N35; 35C10; 42C10

© 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

* Corresponding author at: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

E-mail addresses: eiddoha@frcu.eun.eg (E.H. Doha), walee_9@ yahoo.com (W.M. Abd-Elhameed), muhammad.alsuyuti@gmail.com (M.M. Alsuyuti).

Peer review under responsibility of Egyptian Mathematical Society.

Chebyshev polynomials have become increasingly crucial in numerical analysis, from both theoretical and practical points of view. It is well-known that there are four kinds of Cheby-shev polynomials, and all of them are special cases of the more widest class of Jacobi polynomials. The first and second kinds are special cases of the symmetric Jacobi polynomials (i.e., ultraspherical polynomials), while the third and fourth kinds are special cases of nonsymmetric Jacobi polynomials. In literature, there is a great concentration on first and second kinds of Chebyshev polynomials Tn{x) and Un{x) and their various uses in numerous applications, (see for instance, [1-3]).

Production and hosting by Elsevier

1110-256X © 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.Org/10.1016/j.joems.2014.04.008

However, there are few articles concentrate on the other two types of Chebyshev polynomials namely, third and fourth kinds Vn(x) and Wn(x), either from theoretical or practical points of view and their uses in various applications, (see, for example, [4-6]). This motivates our interest in such polynomials. The interested readers in Chebyshev polynomials of third and fourth kinds are refereed to the excellent book of Mason and Handscomb [7].

If we were asked for ''a pecking order'' of these four Chebyshev polynomials Tn(x), Un(x), Vn(x) and Wn(x), then we would say that Tn (x) is the most important and versatile. Moreover Tn(x) generally leads to the simplest formulae, whereas results for the other polynomials may involve slight complications. However, all four polynomials have their role. For example, Un(x) is useful in numerical integration (see, Mason [8]), while Vn(x) and Wn(x) can be useful in situations in which singularities occur at one end point (+1 or -1) but not at the other (see, Mason and Handscomb [7]).

Due to their great importance in several applications, high even-order and high odd-order boundary value problems have been investigated by a large number of authors. Theorems which discuss the conditions for the existence and uniqueness of solutions of such problems are contained in a comprehensive survey in a book by Agarwal [9].

Spectral methods (see, for instance, Boyd [1] and Canuto et al. [10]) are a class of techniques used extensively in applied mathematics and scientific computing to numerically solving ordinary and partial differential equations. The numerical solution is written as an expansion in terms of certain ''basis functions'', which may be expressed in terms of various orthogonal polynomials. Spectral methods have advantage that they take on a global approach while finite element methods use a local approach, and for this reason, spectral methods have excellent error properties, and converge exponentially.

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 commonly occurring partial differential equations such as the Kortweg-de Vries equation. Also, fifth-order boundary value problems are of interest as they arise in the mathematical modelling of visco-elastic flows (see, [11,12]). Abd-Elhameed in [13] and Doha and Abd-Elhameed in [14] have constructed efficient spec-tral-Galerkin algorithms using compact combinations of ultra-spherical polynomials for solving the differentiated forms of elliptic equations of high odd-order boundary value problems. Recently, in the two papers of Abd-Elhameed et al. in [15] and Doha et al. in [16], some algorithms for solving numerically the differentiated and integrated forms of third and fifth-order differential equations based on a dual Petrov-Galerkin method using two new families of general parameters generalized Jacobi polynomials, are analyzed.

Of the important high-order differential equations are the singular and singularly perturbed problems (SPPs). These kinds of problems usually appear in quantum mechanics, optimal control, etc. The presence of small parameter in these problems prevents one to obtain satisfactory numerical solutions. It is a well-known fact that the solutions of SPPs have a multi-scale character, that is, there are thin layer(s) where the solution varies very rapidly, while away from the layer(s) the solution behaves regularly and varies slowly. The existence and uniqueness of singularly purturbed boundary value

problems was discussed by Howers [17], Kelevedjiev [18], and Roos et al. [19].

As an alternative approach to differentiating solution expansions is to integrate the differential equation q times, where q is the order of the equation. An advantage of this approach is that the resulted algebraic system contains a finite number of terms and hence they are cheaper in solving than those obtained by the differentiated forms. Doha et al. in [16] followed this approach for solving the integrated forms of third- and fifth-order elliptic differential equations. Moreover, Doha and Abd-Elhameed in [4] obtained new formulae for the repeated integrals of Chebyshev polynomials of third and fourth kinds and they used these formulae for solving the integrated forms of sixth-order boundary value problems.

The main objective of the present article is to develop some efficient spectral algorithms based on shifted Chebyshev third and fourth kinds-Galerkin methods for solving the integrated forms of high odd-order differential equations.

The contents of this article are organized as follows. In Section 2, some properties and relations of Chebyshev polynomials of third and fourth kinds and their shifted ones are presented. In Section 3, we discuss some algorithms for solving the integrated forms of high odd-order elliptic differential equations governed by homogeneous and nonhomogeneous boundary conditions using shifted Chebyshev third kind Petrov-Galerkin method (SC3PGM). In Section 4, we are concerned with the same equations but by using shifted Chebyshev fourth kind Petrov-Galerkin method (SC4PGM). Section 5 is concerned with discussing the condition numbers resulted from the application of the two proposed algorithms in Sections 3 and 4. In Section 6, we present three numerical examples including comparisons with some other methods aiming to exhibit the accuracy and the efficiency of our proposed algorithms. Some concluding remarks are presented in Section 7.

2. Some properties of third and fourth kinds of Chebyshev polynomials and their shifted ones

2.1. Chebyshev polynomials of third and fourth kinds

Chebyshev polynomials V(x) and Wi(x) of third and fourth kinds are polynomials in x, which can be defined by one of the two following equivalent forms (see, [7]):

, sin (i + 2)0 22i (1-1), Wi(x)= \^2 = TJJT& l)(x),

where x = cos h, and pf'b)(x) is the Jacobi polynomial of degree i. It is clear that

Wi(x) = (—1) Vi(-x). (1)

The polynomials Vi(x) and Wi(x) are orthogonal on (—1,1) with respect to the weight functions y^—f and y^J respectively, i.e.,

r1 fï+x

V(x) Vj(x) dx —

W(x) Wj(x) dx

P; i — j; 0; i-j;

and they may be generated by using the two recurrence relations

V,(x) — 2x Vi_1 (x)- Vi-2(x); i — 2; 3; ... ; with the initial values V0(x) = 1; Vi(x)=2x - 1; and

W(x) — 2x W- (x)- W-2 (x); i — 2; 3;... ;

with the initial values

W0(x) = 1; Wx (x)—2x + 1.

The following special values are of important use later:

V,(1) = 1; V,(-1) = (-1)i (2i + 1); W, (1) — 2i + 1; Wl(-1)^(-1)1. Also, for all m p 1, we have

if x — 1 ;

DmVi(x) —

■y/P (i + m)! \ r(m

TmiJ-m).\; if x = -1;

DmW(x) —

y/n(i + m)! I 2r(m+;); if x 1;

2m (i - m)!\ (-1)i+m r(m+1) '

if x 1 .

The following theorem is needed in obtaining the desired spectral numerical solutions.

Theorem 1. If the m times repeated integration of Vi(x) is denoted by

I" (x)~ J J ' J Vi(x)dx dx... dx; then

t\x) — ^ Gk;i;m Vi+m-k(x)+ ffm-l(x); k—0

(-1)k(i-k)!

Gk;i;m 2m

m! ()! (m—)! (m+i-k)!'

(-»¥(i-m)!

k even; k odd;

(I—)! (m—^))! (m+i— (¡—))\

and rm-1 (x) is a polynomial in x of degree at most (m — 1).

(For the proof of Theorem 1, see Doha and Abd-Elhameed [4]).

Theorem 1 with the aid of relation (1), enables one to get immediately the following corollary.

Corollary 1. The m times repeated integration of Wi(x) is given by

Jr\x) Gk;i;m Wi+m-k(x) + Pm-l(x);

Gk;i;m — (- 1) Gk;i;,

Pm-i(x) is a polynomial in x of degree at most (m - 1) and Gkti,m is as defined in (3).

2.2. Shifted Chebyshev polynomials of third and fourth kinds

The shifted Chebyshev polynomials of third and fourth kinds are defined on [a, b\, respectively as

VJ(.x) = V, fe^), (5)

b — a

b — a

All results of Chebyshev polynomials of third and fourth kinds, can be easily transformed to give the corresponding results for their shifted ones. The orthogonality relations of V*(x) and W*(x) on [a, b\ with respect to the weight functions

I-a and < are given by

b-x ' y x-a

'b 'x - a V(x) V*(x) dx — [b Jt—x W*(x) w;(x) dx

x- a (b - a) p ; 0;

Based on (5) and with the aid of formula (2), we have the following theorem.

Theorem 2. If the m times repeated integration of V*(x) is denoted by

m times

r\x) = j Vl(x)dx dx ...dx,

ih \ m -m W (x) = ( b—a) £ V+m—k(x) + *m—1 , \ 2 / k=0

where 0k,im is defined as in (3) and rm—1 (x) is a polynomial in x of degree (m — 1) at most.

3. Solution of high odd-order differential equations by using shifted third kind Chebyshev polynomials

In this section, we are interested in using SC3PGM to solve the integrated forms of high odd-order elliptic linear differential equations governed by the homogeneous and nonhomogene-ous boundary conditions.

3.1. High odd-order two point boundary value problems

Let us consider the following one dimensional high odd-order differential equations:

(-1)n+y2n+l)(x)+ Ylk y(k)(x) —f(x); x 2 (a; b); H P \;

m times

governed by the homogeneous boundary conditions

y(m)(a) = y(m)(b) = y(n)(a) = 0, m = 0 ,1,... ,n — 1

„M t

,1,... 1, (8)

where {ik, k — 0 ,1,..., 2n} are known constant coefficients.

It is to be noted here that the main differential operator in (7) is not symmetric, so it is convenient to use a Petrov-Galer-kin method. The difference between the Galerkin and Petrov-Galerkin methods, is that the test and trial functions in Galer-kin method are the same, however, in case of Petrov-Galerkin method, the trial functions are chosen to satisfy the boundary conditions of the differential equation, and the test functions are chosen to satisfy the dual boundary conditions.

Now, the integral equation of (7)-(8) is:

(-1Г y(x) + ZtolJ'

(2n+1—k)

y(x)(dx)

(2n+1—k)

= F(x)+YkUh Vk(x), x 2 (a,b),

y(m)(a) =y(m)(b) = y(n)(a) =0 , m = 0 , 1, F(x) = j (2n+1f(x)(dxfn+1) ,

,n- 1,

y (x)(dx)m =

y(x)dx dx... dx.

If we define the following spaces Sn = span{ V0)(x), V1(x), V2(x),..., VN(x)}, Un — {/(x) 2 SN : /(m\a) — ф(т\Ь) — ф(п\а) — 0 ,

m — 0 ,1,... ,n — 1} , WN — {W(x) 2 Sn : Wm)(a) — ф<т)(Ь) — ^(Ь) — 0 , m — 0 , 1,... ,n — 1} ,

then the spectral shifted Chebyshev third kind Galerkin procedure for solving (9) is to find ynN 2 UN such that

((—1)n+1 yN(x) , w(x))

r(2n+1-k)

yN(x)(dx

(2n+1—k)

= F(x) + J2 dk Vk(x), W(x)\ , 8 W(x) 2 W

where (y(x), W(x))wi — J^wi y(x) W(x) dx, is the scalar inner product in the weighted space L2m (a, b) and w1 — ^/frf.

3.2. The choice of trial and test functions

First we consider the case [a,b\ — [—1,1], and we aim to construct suitable basis functions and their dual basis. For this purpose, we set

/i,n(x) — Vi(x) + £PkV+k(x), x 2 [—1,1], k—1

i — 0 ,1,2,... ,N — 2n — 1, n p 1, (11)

Wi,n(x) — Vi(x) + J2qk,¡Vi+k(x), x e [—1,1\,

i — 0,1,2,..., N — 2n — 1, n P 1. (12)

The coefficients {pki} and {qki} are chosen such that /in(x) 2 &i+2n+i and Wi,n(x) 2 W,+2n+i, respectively. The (n + 1) boundary conditions (—1)— 0, m — 0 ,1,2,..., n,

and the n boundary conditions /И (1)= 0 , m = 0 ,1,2 ,...

,1, * * *,

n — 1, lead respectively to the following system of ( 2n + 1) equations:

2+ (—1)k(2i + 2k + 1)( i + k + m)! _— ( 2i + 1)( i + m)! f=j ( i + k — m)! Pki = '

i — m) !

2+ ( i + k + m)! — (i + m)!

^(i + k — m)! pki , m = 0Л2,... ,n — 1:

i — m)!

The determinant of the above system of equations is different from zero, hence the coefficients {pki} can be uniquely determined to give

p2k, i = '

(—l)k \k)(i + 1)k^i + 4k + 2n + 3)

( i + n + 2)k ( 2i + 2n + 3) ,

k = 1,2,..., n , 0 6 i 6 N — 2n — 1,

p2k+1 i

-1)k i + 1)k(2i + 4k — 2n + 1)

( i + n + 2)k ( 2i + 2n + 3) ,

where (z)k denotes the Pochhammer symbol, i.e., (z)k — ^pr)^. Similarly, it can be easily verified that the coefficients {qki} can be uniquely determined to give

("Dk Tk)(i+ Dk

q2 k i —-,-■ , -; k — 1,2,n, 0 6 i 6 N - 2n - 1,

( i + n + 2))

(-ir( D (i+i)k

q2 k+1,i =-, , V,-• k — 0,1,..., n, 0 6 i 6 N - 2n - 1,

( i + n + 2)k

Now, if x in (11) and (12) is replaced by (2 xb-aa-b) for x 2 [a,b], and if we define the following basis functions:

2 x - a - b\ (2 x - a - b\ „

= fan,(x), Wn 1 b_a I = wi,„(x) , x2[a,b] ,

b — a

then it is clear that the basis and dual basis functions given by

Vm(x) — V(x) + £ Pk,V+k(x), k—1

wi (x) — v (x) + j2 qk,,Kk (x), k—1

satisfy the boundary conditions (8), i.e. /*n ( x) 2 Ui+2n+1 and vin( x) 2 W,+2n+l, respectively. From now on, it would be convenient to write the basis functions /*n ( x) and their dual basis W*n ( x) in the two alternative forms:

/k» ( x) = E Pk,.vk+k (x),

k=0 2n+1

wi ( x) = Y.qk,rl+k ( x),

(13) 14)

m = 0 1 2 ... n

m times

m times

k = 0 ,1,..., n , 0 6 i 6 N — 2n — 1

where the coefficients pki and qk i are given respectively by:

(-1) 2 ( kj (i+1)i(2i+2k+2n+3) (i+n+2) k(2i+2n+3) '

k — 0 ,2 ,... ,2n ,

(-1) 2 I k-lj (i+1)I—l(2l+2k-2n-1)

(i+l+2) t—L(2l+2n+3) ; k — 1;3; ... ;2n + 1

qki—

(-1)21 njm)!

(i+n+2) k ;

(-1)" \ k-1 )(i+1)-

(i+n+2) -

k — 0; 2; ...; 2H ;

k — 1; 3; ... ; 2n + 1.

It is obvious that the basis functions /* n(x) and W*n(x) are linearly independent. Therefore, we have

Un = spanj/*.n(x) : i = 0 ,1,2 ,... ,N — 2n — 1},

Wn = spanjW*,„(x) : i = 0 ,1,2 ,... ,N — 2n — 1}.

Now, the variational formulation (10) is equivalent to

((—1 y+1 yN(x) ,wi(x) )wi

2n / (2n+1—k) )

+Y>k( yN(x) (dx) (2n+1—k) ,w:,n(x))

k=0 ' W1

= [F(x) + jrdk Vk(x),Wtn(x)) , 0 6 i 6 N — 2n — 1.

The constants Sk, 0 6 k 6 2n, would disappear if we take i P 2n + 1 in (17), then we have

-1)"+' yN(x),W*n(x))

2n / f(2n+1—k) )

+ E Ik / A (x) (dx) (2n+1—k) ,wt„(x)j

k=0 w1

= (F(x),Wln(x)^ , 2n + 1 6 i 6 N.

Let us denote

A„— a

2n+16ij6N

an — (-1 )n+1( j(x) ;Wln(x) )w

D _ i Ll2n+1-m;H

B2n+1-m;H — | b

i2n+1-m;H _ i r-(2n+1-m)

2n+16ij6N

bj+1-m;H —(f"+1-m) j(x) (dx) (2H+1-m) ; Wln(x) )t

FH — (F5; F1; ...; FI-2H-1) FH — (f(X) ; WUx)

N-2n-1

ynN(x) — E Ck/l,n(x);

— (4); c1; ... ; c"n-2h-0 :

Now Eq. (18), can be transformed into the following linear system

An + E 1m B2n+1-m;H I c' — F m—0

where the nonzero elements of the matrices An and B2n+i—m,n (0 6 m 6 2n) are given explicitly in the following theorem.

Theorem 3. If the trial and test functions /*n(x) and W*n(x) are chosen as in (13) and (14), respectively, and if

we set Oj = ( — 1)n+1(/]—2n—1,n(.x), W*n(x), and bjf+1—m,n =

(/(2n+1—m)/U—1,n(x)(dx)(2n+1—m), W*n(x))wi , 0 6 m 6 2n, then

UN+2n+1 = span{/'0,n(x), /*,n(x), ■■■ , /N,n(x) }, WN+2n+1 = spanf W0 ,n(x ), W1 ,n (x), ■■■ , WN,n(x) g,

and the nonzero elements of the matrices An and B2n+i—m,n (0 6 m 6 2n) are given explicitly by:

_ p(a — b) (i — 2n) n(2.i — 2n — ^

2(i - n + 1 )n(2i - 2n + 1)

j — (-1 )H+1 (b - a) Pkj-2H-1 q

lj-i-2n-1+k;i; j — i + s;s p 1;

, \ 2n—m+2 2n+1 2n+1

Z^ / JPk,j—2n—l ql,i 2 / k=0 1=0

0j—i+k—£—mj—2n—1+k,2n+1—m, j i + s,

m — 2n — 1 6 s 6 6n — m + 3 , 0 6 m 6 2n, (22

where 0kiim,pki and qki are as defined in (3), (15) and (16) respectively.

Proof. The basis functions /*n(x) and their dual functions

W'l„(x) are chosen such that /*n(x) 2 &N+2n+1 and W*n(x) 2 WN+2n+1 for i = 0 , 1,... ,N. Moreover, it is clear that {^,n(x) }06,6N and fWln(x) }06,6N are linearly independent and

the dimension of both UN+2n+i and WN+2n+i is equal to (N + 1) .Hence,

UN+2n+1 = span{/**t„(x), fttn (x), ... , /N,n(x)},

WN+2n+1 = span{W0,n(x), ,n(x), ... , WN,n(x)}.

To prove (20) and (21), we make use of the two formulae (13) and (14), to get

2 n+12 n+1

aj = (—1)"+TE p^—— qJv:—2n—1+k(x), v*+t(x)) , k=0 i=0 1

which in turn, with the aid of the orthogonality relation (6), yields

p 2h+1

j — (-1 )n+l (b - a) Pkj-2n-1 q

j-i-2n-1+k i

; j — i + S;S P 1.

If we make use of relations (15) and (16), then an can be evaluated to give relations (20) and (21). To prove relation (22), we have for 0 6 m 6 2n

i2n+1—m,7 bij

/ i-(2n+1-m) \

— [J /Un-1,n (x) ( dx)( 2n+1-m\ Win ( *))

(2n+1 2n+1

H Yjkj-in-1 j+Tlk(x), Xai (x)

s k—0 1=0 and the application of Theorem 2 yields

(7 \ 2n+1-m 2 (2n+1-m)

V) £ G

,2n+1-m K+k-m-£ (X),

£J-2n-1+k,2n+1-m * j+k-m-

where Gk i m is as given in (3), which leads after making use of the orthogonality relation (6) and relation (23), to the relation

= p ( Y.Y.Pk—n-1 qt,iGj

2n-m+2 2n+12n+1

^j-i+k-l-m j-2n-1 +k , 2n+1 —m ,

k=0 '=0

6 m 6 2n ,j = i + s,m - 2n - 1 6 s 6 6n - m + 3 ,

and this completes the proof of Theorem 3. □

Now, and based on Theorem 3, it is clear that the case corresponds to im — 0 , 0 6 m 6 2n, leads to linear system with nonsingular upper triangular matrices. This result is given in the following corollary.

Corollary 2. If im — 0 , 0 6 m 6 2n, then the linear system (19) takes the matrix form An cn — Fn, where An is an upper triangular matrix whose solution can be immediately obtained by the backward substitution as:

ci = I Fn+2n+1

j=i+2n+2

i+2n+1 j Uui+2n+1 ,i+2n+1,

where the elements an and aj are given by (20) and (21) respectively.

Remark 1. If we consider the equation

(— 1)n+y2n+1) (x)^ik yik) (x) —f(x) , x 2 (a ,b) , n P 1, k—0

governed by the nonhomogeneous boundary conditions

y(m) ( a)— am , y(m) ( b) = bm , y(n) (a)=c , m = 0 ,1,... ,n - 1

then, and with the aid ofa suitable transformation, Eq. (24)-(25), can be converted to an equation similar to (7)-(8).

4. Solution of high odd-order differential equations by using fourth kind Chebyshev polynomials

In this section, we are concerned with the solution of the same problems discussed in Section 3, but by using SC4PGM. All results of this section are given without proofs.

4.1. High odd-order two point boundary value problems SC4PGM

Let us consider the same one dimensional high odd-order differential Eqs. (7) governed by the same homogeneous boundary conditions (8). If we define the following spaces

Sn — spanf W0( x) , Wi(x), Wi(x),..., WN(x)},

U n — {/ (x) 2 Sn : / (m) (a) — /(m) (b) — /(n) (a) — 0 , m — 0 , 1,... ,n — 1} ,

Wn — {W ( x) 2 Sn : W(m) ( a) — W(m) ( b) — W(n) ( b) — 0 , m — 0 , 1,... ,n — 1} ,

then, the shifted Chebyshev fourth kind spectral Petrov-Galerkin procedure for solving (7)-(8) is to find ynN 2 UN such that

^r1 fN (x), w ( x

2n / f (2n+1-k) _ \

+ Y>*( >N ( X) (dxfn+l-k\ W ( X))

k.=0 \J ' w

2n / f ( 2n+1—k)

— (fx) + £dk Wk (x) , W (xn , 8 W (x) 2 Wn.

\ k—0 J w2

In such case, the basis functions and their dual basis functions are selected to take the following forms:

<t>"i,n (x) — Pk, -K+k (x) ,

Vin(x) — £ qk, iW+k(x) ,

where the constants pki and qk, t can be computed to give:

q> k i —

qk i \ -qk,i ,

pk i -pk i

k — 0 ,2 ,... ,2n,

k — 1,3 ,..., 2n + 1,

k — 1,3 ,..., 2n + 1,

(26) 27)

(28) (29)

and the coefficients pki and qk, t are given in (15) and (16), respectively.

Now, the main result of this section is given in the following theorem.

Theorem 4. If the trial and test functions /* it„ (x) and W*i,n ( x) are chosen as given in (26) and (27), respectively, and if fN(x) — Ylk=(0n-lc'k />*k,n( x), is the Petrov-Galerkin approximation to (9), then the expansion coefficients {ck,k — 0 , 1 ... n - 2n - 1 } satisfy the matrix system:

An + Zlm ^^-mn I c" — F" ,

where the nonzero elements of the matrices An and B2n+1— ( 0 6 m 6 2n) are given explicitly as follows:

r — 0,1,... ,N - 2n - 1,

_n{b - a) {i - 2n) n = 2{i - n + 1)n '

= {-1 )n+1 {b - a) (p) E Pkj-2n-1 qJ-

1 qj-i-2n-1+k,i

_ \ 2n-m+2 2n+12n+1

r2n+1-m,n „/ b a\ -

' ' k=0 1=0

qi,t G,

j — i + s, m — 2n — 1 6 s 6 6n — m + 3, 0 6 m 6 2n,

where Gkiim ,pk,i and qk,j are as defined in (4), (28) and (29) respectively.

We note that the case corresponds to im — 0 , 0 6 m 6 2n, leads to linear system with nonsingular upper triangular matrices. The result for such case is summarized in the following corollary.

Corollary 3. If im — 0 , 0 6 m 6 2n, then the system (30) reduces to the matrix equation An cn — Fn, where An is an upper triangular matrix whose solution can directly obtained by the backward substitution

cn = I F+2n+1

'i+2n+1 j c I / 'i+2n+1, i+2n+1>

j=i+2n+2

i = 0 ,1,... ,N - 2n - 1, where an and an. are given by (31) and (32) respectively.

5. Condition numbers of the two resulting matrices in systems (19) and (30)

In this section, we discuss the condition numbers of the two resulting matrices that appear in the two linear systems (19) and (30). If we apply the two spectral methods namely, SC3PGM and SC4PGM, then the two linear systems resulted from solving the integral equation of the equation

((—1 )"+1 y'2"+1' (x) = f{x)j are given respectively by Ancn —

Fn and Ancn — Fn. Thus, 8n P 1, we note that the condition number of the matrix An behaves like O(N) for large values of N, while the condition number of the matrix An is independent of N. Moreover, it has been noted that the combined matrices Dn — An + J2m=o B2n+i—m,n in (30) and Dn — An+ Ylm=0B2n+i—m,n in (19) have the same conditions numbers of the matrices An and An, respectively. Hence the propagation of roundoff errors should not be very significant. To ascertain these observations, Table 1 illustrates the condition numbers

for the matrices An and Dn = An + Y^m=0B

m=0B2n+1—m,n in (19) for some values of the parameter N and (a,b) — (—1,1),n — 1,2, while Table 2 illustrates the condition numbers for the matrices An and Dn — An + Yim=0B2n+i—m,n in (30), n — 1, 2, for the same values of N and in the same interval.

Remark 2. It should be noted here that the main advantage of following the integrated form approach over the differentiated one is that the condition numbers in the integrated approach are smaller than those obtained via the differentiated approach. It can be shown that if the two algorithms SC3PGM and SC4PGM are applied but on the differentiated forms (7) and (8), then the condition numbers of the matrices appear in

Table 1 Condition number of the matrices An and Dn.

N n j{An) j{Dn) n k{A„ ) j{Dn)

'• = i+s,s P 1, {32) 16 4.093 8.611 8.472 7.434

24 1 4.389 8.849 2 10.179 9.034

32 4.539 8.880 11.084 9.819

'-i+k-i-mJ-2n-l+k,2n+1-m ; 40 4.629 8.881 11.643 10.279

N n n KD

16 3.031 2.293 0.663 0.395

24 1 4.561 3.544 2 0.951 0.600

32 6.087 4.797 1.241 0.809

40 7.613 6.052 1.531 1.021

Table 3 Error E for N = 8 , 10 , 12 , 14 , 16 , 18 for Example 1.

N e SC3PGM SC4PGM

1/16 1.3 > < 10-6 1.3 > < 10-6

8 1/32 5.5 > < 10-7 5.3 > < 10-7

1/64 2.2 > < 10-7 2.1 > < 10-7

1/16 1.4 > < 10-8 1.1 > s 10-8

10 1/32 5.8 > < 10-9 4.7 > c 10-9

1/64 2.4 > < 10-9 1.9 > c 10-9

1/16 9.8 > c 10-11 6.6 > < 10-11

12 1/32 4.2 > c 10-11 2.8 > c 10-11

1/64 1.7 > c 10-11 1.1 > c 10-11

1/16 4.9 > c 10-13 2.7 > c 10-13

14 1/32 2.1 > c 10-13 1.2 > c 10-13

1/64 8.4 > c 10-14 4.6 > < 10-14

1/16 6.5 > c 10-16 1.6 > c 10-15

16 1/32 7.4 > c 10-16 2.5 > c 10-16

1/64 2.2 > c 10-16 1.7 > c 10-16

Table 4 Comparison between different methods for Example 1.

e Methods in [20] SC3PGM SC4PGM

1/16 5.4 x 10-6 6.5 > 10-16 1.6 > 10-15

1/32 2.8 > 10-6 7.4 > 10-16 2.5 > 10-16

1/64 1.4 > 10-7 2.2 > 10-16 1.7 > 10-16

Table 5 Error E for N = 8 , 10 , 12 , 14 , 16 for Example 2.

N SC3PGM SC4PGM

8 1.2 > < 10-6 7.6 > < 10-7

10 3.4 > c 10-9 1.8 > c 10-9

12 5.1 > c 10-12 2.3 > < 10-12

14 4.8 > c 10-15 1.9 > c 10-15

16 1.1 > c 10-16 9.0 > c 10-17

Table 2 Condition number of the matrices An and Dn

Table 6 Comparison between different methods for Example 2.

x Methods in [21,22] Method in [23] Method in [24]

0.1 1.0 > < 10-9 7.0 > < 10-4 2.3 > < 10-7

0.2 2.0 > < 10-9 7.2 > < 10-4 1.6 > < 10~6

0.3 1.0 > < 10-8 4.1 > < 10-4 4.6 > < 10~6

0.4 2.0 > < 10~8 4.6 > < 10-4 8.9 > < 10~6

0.5 3.1 > < 10-8 4.7 > < 10-4 1.3 > < 10~5

0.6 3.7 > < 10~8 4.8 > < 10-4 1.6 > < 10-5

0.7 4.1 > < 10-8 3.9 > < 10-4 1.6 > < 10~5

0.8 3.1 > < 10~8 3.1 > < 10-4 1.2 > < 10-5

0.9 1.4 > < 10~8 1.6 > < 10-4 5.1 > < 10~6

Method in [25] Method in [26] SC3PGM SC4PGM

0.0 3.38 > < 10-15 5.6 > < 10-17 5.6 > < 10-17

1.0 X 10-5 4.47 > < 10-15 1.1 > c 10-16 1.1 > c 10-16

1.0 > 10"5 3.44 > < 10-15 0.0 5.6 > c 10-17

1.0 > 10"4 6.88 > < 10-16 5.6 > c 10-17 0.0

3.2 > 10"4 3.17 > < 10-15 0.0 5.6 > c 10-17

3.6 > 10~4 7.28 > < 10-15 0.0 0.0

1.4 > 10"4 1.05 > < 10-14 5.6 > c 10-17 5.6 > c 10-17

3.1 > 10"4 1.15 > < 10-14 0.0 5.6 > c 10-17

5.8 > 10"4 8.68 > < 10-15 5.6 > c 10-17 1.1 > < 10-16

the resulting two linear systems are both of O(N2n+1), for large values of N.

6. Numerical results

In this section, we give two numerical results for the sake of testing the efficiency and the applicability of the proposed algorithms of Sections 3 and 4. We consider the following two examples.

Example 1. Consider the following singulary perturbed linear third-order boundary value problem (see, [20]):

-e y(3) ( x)+y( x) — 81 e2 cos ( 3x) + 3 e sin ( 3x), 0 6 x 6 1,

subject to the boundary conditions

y (0)=0, y (1) = 3 e sin ( 3) , y( 1) (0) = 9 e ,

with the analytic solution y( x) — 3 e sin (3x).

In Table 3, we list the maximum pointwise error E — jy - yNj using (SC3PGM and SC4PGM) with various values of N and e, while in Table 4, we introduce a comparison between the best absolute errors obtained by (SC3PGM and SC4PGM) in case of N — 18 with the best absolute errors obtained by using method in [20]. This table shows that our two methods are more accurate if compared with the method developed in [20].

Example 2. Consider the following linear fifth-order boundary value problem (see, [21-26]):

u(5) (x) — u(x)- 15 ex - 10 x ex , 0 6 x 6 1, governed by the boundary conditions u (0)—0 , u(1) (0) —1, u(2) (0)—0 , u (1)—0 , u(1) (1)

— —e.

The exact solution of this problem is u(x) — x (1 — x)ex.

Table 5 lists the maximum pointwise error E —ju — uNj using (SC3PGM and SC4PGM) with various values of N. In Table 6, we introduce a comparison between the best absolute errors obtained by (SC3PGM and SC4PGM) at N — 16 with the best error obtained by using the methods developed in [21-26] is displayed in Table 6. This table shows that our two methods are more accurate if compared with the methods obtained in all of these papers.

7. Concluding remarks

We have presented some efficient direct solvers for the integrated forms of high odd-order differential equations using shifted Chebyshev polynomials of third and fourth kinds based on applying Petrov-Galerkin spectral method. We have found that for some particular differential equations, the resulting systems of linear equations are upper triangular, and this is certainly reduces the computational cost for the numerical solutions for these special cases. The presented numerical examples exhibit the high accuracy and efficiency of the proposed algorithms.

Acknowledgment

The authors are very grateful to the anonymous referees for carefully reading the paper and for their comments and suggestions which have greatly improved the manuscript.

References

[1] J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, Mineola, 2001.

[2] E.H. Doha, W.M. Abd-Elhameed, Y.H. Youssri, Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type, New Astron. 23-24 (2013) 113117.

[3] K. Julien, M. Watson, Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods, J. Comput. Phys. 228 (2009) 1480-1503.

[4] E.H. Doha, W.M. Abd-Elhameed, On the coefficients of integrated expansions and integrals of Chebyshev polynomials of the third and fourth kinds, B. Malays. Math. Sci. Soc. 37 (2) (2014) 383-398.

[5] E.H. Doha, W.M. Abd-Elhameed, M.A. Bassuony, New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods, J. Comput. Phys. 236 (2013) 563-579.

[6] M.R. Eslahchi, M. Dehghan, S. Amani, The third and fourth kinds of Chebyshev polynomials and best uniform approximation, Math. Comput. Model. 55 (2012) 1746-1762.

[7] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman and Hall, New York, NY, CRC, Boca Raton, 2003.

[8] J.C. Mason, Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms, J. Comput. Appl. Math. 49 (1993) 169-178.

[9] R.P. Agarwal, Boundary Value Problems for High Ordinary Differential Equations, World Scientific, Singapore, 1986.

[10] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1989.

[11] A.R. Davies, A. Karageorghis, T.N. Phillips, Spectral Galerkin methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 647-662.

[12] A. Karageorghis, T.N. Phillips, A.R. Davies, Spectral collocation methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Int. J. Numer. Methods Eng. 26 (1988) 805-813.

[13] W.M. Abd-Elhameed, Efficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2n + 1)th-order linear differential equations, J. Egypt Math. Soc. 17 (2009) 189211.

[14] E.H. Doha, W.M. Abd-Elhameed, Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of (2n + 1)th-order linear differential equations, Math. Comput. Simulat. 79 (2009) 3221-3242.

[15] W.M. Abd-Elhameed, E.H. Doha, Y.H. Youssri, Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials, Quaest. Math. 36 (2013) 15-38.

[16] E.H. Doha, W.M. Abd-Elhameed, Y.H. Youssri, Efficient spectral-Petrov-Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials, Appl. Math. Comput. 218 (2012) 7727-7740.

[17] F.A. Howers, Singular Perturbation and Differential Inequalities, vol. 168, Memories of the American mathematical Society, Providence, Rhode, Island, 1976.

[18] P. Kelevedjiev, Existence of positive solutions to a singular second order boundary value problem, Nonlinear Anal.: Theory Methods Appl. 50 (8) (2002) 1107-1118.

[19] H.G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, vol. 24, Springer Series in Computational Mathematics, SpringerVerlag, Berlin, Heidelberg, 2008.

[20] G. Akram, Quartic spline solution of a third order singularly perturbed boundary value problem, Anziam. J. 53 (2012) 44-58.

[21] A. Wazwaz, The numerical solution of fifth-order boundary-value problems by Adomian decomposition method, J. Comput. Appl. Math. 136 (2001) 259-270.

[22] M.A. Noor, S.T. Mohyud-Din, Variational iteration method for fifth-order boundary value problems using he's polynomials, Math. Probl. Eng. 2008 (2008) (12 pages), Article ID 954794. http://dx.doi.org/10.1155/2008/954794.

[23] H.N. Ccglar, S.H. Caglar, E.H. Twizell, The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions, Appl. Math. Lett. 12 (1999) 20-30.

[24] M.A. Noor, S.T. Mohyud-Din, A new approach to fifth-order boundary value problems, Int. J. Nonlinear Sci. 7 (2) (2009) 143-148.

[25] J. Zhang, The numerical solution of fifth-order boundary value problems by the variational iteration method, J. Comput. Math. Appl. 58 (2009) 2347-2350.

[26] J. Rashidinia, M. Ghasemi, R. Jalilian, An O(h6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algor. 55 (2010) 403-428.