Math Sci (2016) 10:83–93 DOI 10.1007/s40096-016-0181-1
ORIGINAL RESEARCH
Improved Jacobi matrix method for the numerical solution of Fredholm integro-differential-difference equations M. Mustafa Bahs¸ ı1
•
Ays¸ e Kurt Bahs¸ ı2 • Mehmet C ¸ evik3 • Mehmet Sezer2
Received: 1 July 2015 / Accepted: 22 May 2016 / Published online: 3 June 2016 Ó The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract This study is aimed to develop a new matrix method, which is used an alternative numerical method to the other method for the high-order linear Fredholm integro-differential-difference equation with variable coefficients. This matrix method is based on orthogonal Jacobi polynomials and using collocation points. The improved Jacobi polynomial solution is obtained by summing up the basic Jacobi polynomial solution and the error estimation function. By comparing the results, it is shown that the improved Jacobi polynomial solution gives better results than the direct Jacobi polynomial solution, and also, than some other known methods. The advantage of this method is that Jacobi polynomials comprise all of the Legendre, Chebyshev, and Gegenbauer polynomials and, therefore, is the comprehensive polynomial solution technique Keywords Orthogonal Jacobi polynomials Fredholm integro-differential-difference equation Residual error technique Matrix method & M. Mustafa Bahs¸ ı
[email protected] Ays¸ e Kurt Bahs¸ ı
[email protected] Mehmet C¸evik
[email protected] Mehmet Sezer
[email protected] 1
Department of Mechanical Engineering, Celal Bayar University, Muradiye Campus, Manisa, Turkey
2
Department of Mathematics, Celal Bayar University, Muradiye Campus, Manisa, Turkey
3
Department of Mechanical Engineering, Izmir Katip C¸elebi University, C¸ig˘li Main Campus, Izmir, Turkey
Introduction Orthogonal Jacobi polynomials The systems of polynomials remain a very active research area in mathematics, physics, engineering and other applied sciences; and the orthogonal polynomials, among others, are definitely the most thoroughly studied and widely applied systems [1–3]. The three of these systems, namely, Hermite, Laguerre, and Jacobi, are called collectively the classical orthogonal polynomials [4]. There is excessive literature on these polynomials, and the most comprehensive single account of the classical polynomials is found in the classical treatise of Szego¨ [5]. Jacobi polynomials are the common set of orthogonal polynomials defined by the formula [4] dn Pðna;bÞ ð xÞ ¼ ðn!Þ1 ð2Þn ð1 xÞa ð1 þ xÞb n dx h i nþb nþa ð1 þ xÞ ð1 xÞ
ð1Þ
Here, a and b are parameters that, for integrability purposes, are restricted to a [ 1; b [ 1. However, many of the identities and other formal properties of these polynomials remain valid under the less restrictive condition that neither a is b a negative integer. Among the many special cases, the following is the most important [4] (a) (b) (c)
The Legendre polynomials ða ¼ b ¼ 0Þ The Chebyshev polynomials ða ¼ b ¼ 1=2Þ The Gegenbouer (or ultraspherical) polynomials ða ¼ b Þ
The Jacobi polynomials Pðna;bÞ ð xÞ are defined [6, 7] with respect to the weight function xa;b ð xÞ ¼ ð1 xÞa ð1 þ xÞb
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ða [ 1; b [ 1Þ on ð1; 1Þ. It is proved that the Jacobi polynomials satisfy the following relation [7]: n X Pðna;bÞ ð xÞ ¼ Bðna;b;nÞ ðx 1Þk ; a; b [ 1 ð2Þ k¼0
where
nþaþbþk ¼2 k k ¼ 0; 1; 2; . . .; n
Bðna;b;nÞ
k
nþa ; nk
ð3Þ
These polynomials play role in rotation matrices [8], in the trigonometric Reson–Morse potential [9], and the cases of a few exact solutions in quantum mechanics [10, 11]. In very recent years, several researchers developed new numerical algorithms for some problems using Jacobi polynomials. Eslahchi et al. [12] gave a numerical solution for some nonlinear ordinary differential equations using the spectral method. Bojdi et al. [13] proposed a Jacobi matrix method for differential-difference equations with variable coefficients. Kazem [14] used the Tau method for solving fractional-order differential equations by means of Jacobi polynomials. Recently, Bharwy et al. [15–22] have used Jacobi polynomials both in operational matrix method and in spectral collocation method for solving some class of fractional differential equations; for instance, nonlinear sub-diffusion equations, delay fractional optimal control problems, time fractional Kdv equations, Caputo fractional diffusion-wave equations, fractional nonlinear cable equation, and fractional differential equations.
where Pi ð xÞ, Qj ð xÞ, K ðx; tÞ, and gð xÞ are known functions and aki , bki , cki , and lk are appropriate constants, while yð xÞ is the unknown function. Note that a g b is a given point in the spatial domain of problem. The main aim of our study, using orthogonal Jacobi polynomials, is to provide an approximate solution for the problem (4, 5), which is usually hard to find analytical solutions. We assume a solution expressed as the truncated series of orthogonal Jacobi polynomials defined by ða;bÞ
yð xÞ ffi yN
ð xÞ ¼
Fundamental matrix relations We can transform the orthogonal Jacobi polynomials Pðna;bÞ ð xÞ from algebraic form into matrix form as follow: Pða;bÞ ð xÞ ¼ Xð xÞMða;bÞ h Pða;bÞ ð xÞ ¼ Pð0a;bÞ ð xÞ
s0
ð4Þ
a
under mixed conditions n1 h i X aki yðiÞ ðaÞ þ bki yðiÞ ðbÞ þ cki yðiÞ ðgÞ ¼ lk ; i¼0
k ¼ 0; 1; . . .; n 1
123
ða;bÞ
P1
ða;bÞ
ð xÞ P 2
ð xÞ
...
ða;bÞ
PN
ð xÞ
i
ð8Þ
j¼0
K ðx; tÞyðt sÞdt;
ð7Þ
where
Fredholm integro-differential-difference equations (FIDDEs) are encountered in many model problems in biology, physics, and engineering. Also, they have been investigated using different methods by scientists [23–28]. Various numerical schemes for solving a partial integrodifferential equation are presented by Dehghan [29]. In this study, we generate a procedure to find a Jacobi polynomial solution for the nth order linear FIDDE with variable coefficients n m X X Pi ð xÞyðiÞ ð xÞ þ Qj ð xÞyðjÞ ðx sÞ
¼ gð x Þ þ
ð6Þ
where Pðna;bÞ ð xÞ, n ¼ 0; 1; . . .; N denote the orthogonal Jacobi polynomials defined by (2, 3); N is chosen N n and an ; n ¼ 0; 1; . . .; N are unknown coefficients to be determined. Note that a and b are arbitrary parameters, such that ða [ 1; b [ 1Þ.
X ð xÞ ¼ 1
Zb
an Pðna;bÞ ð xÞ
n¼0
Integro-differential-difference equations
i¼0
N X
ð x 1Þ
ðx 1Þ2
...
ðx 1ÞN
ð9Þ
and h i ða;bÞ Mða;bÞ ¼ mij ;
1 i;
jN þ 1
ð10Þ
such that 8 < 1i a þ b þ i 2 þ j aþj1 ða;bÞ 2 ; ij mij ¼ i1 ji : 0; i[j We assume the solution yð xÞ, which is defined by the truncated orthogonal Jacobi series (6) in matrix form as follow h i ða;bÞ yN ð xÞ ¼ Pða;bÞ ð xÞA ð11Þ where A ¼ ½ a0
ð5Þ
a1
...
aN T
ð12Þ
By substituting the matrix form of Jacobi polynomials (7) to (11) into (4, 5), we can obtain the fundamental
Math Sci (2016) 10:83–93
85
matrix equation of approximate solution of unknown function as h i ða;bÞ yN ð xÞ ¼ Xð xÞMða;bÞ A ð13Þ
ðiÞ
XðiÞ ð xÞ ¼ Xð xÞBi
0 2 0 0
Pi ð xÞyðiÞ ð xÞ þ
ð14Þ 3 0 07 . 7 .. . .. 7 7 N 5 N
Qj ð xÞyðjÞ ðx sÞ
j¼0 n X
Pi ð xÞXð xÞBi Mða;bÞ A þ
i¼0
m X
Qj ð xÞXð xÞBs B j Mða;bÞ A
j¼0
ð20Þ Matrix representation of integral part of problem Rb Fredholm integral part of problem is a K ðx; tÞyðt sÞdt, where K ðx; tÞ the kernel function of the Fredholm integral part of main problem is. This function can be written using the truncated Taylor Series [30] and the truncated orthogonal Jacobi series, respectively, as
K ðx; tÞ ¼
N X N X
T m n kmn x t
ð21Þ
J kmn Pðma;bÞ ð xÞPðna;bÞ ðtÞ
ð22Þ
m¼0 n¼0
ð15Þ and K ðx; tÞ ¼
Then, using (13) and (14), we may write h i ða;bÞ ðiÞ ðiÞ y ð xÞ ffi yN ð xÞ ¼ XðiÞ ð xÞMða;bÞ A
N X N X m¼0 n¼0
where
¼ Xð xÞBi Mða;bÞ A
ð16Þ
Similarly, the relation between the matrix form of unknown function and matrix form of its delay forms’ derivatives yðjÞ ðx sÞ can be expressed as yð jÞ ðx sÞ ¼ Xð jÞ ðx sÞMða;bÞ A ¼ Xðx sÞB j Mða;bÞ A
ð17Þ
¼ Xð xÞBs B j Mða;bÞ A
T kmn ¼
1 omþn K ð0; 0Þ ; m!n! oxm otn
m; n ¼ 0; 1; . . .; N
J is the Taylor coefficient and kmn is the Jacobi coefficient. The expressions (21) and (22) can be written using matrix forms of the Jacobi polynomials, respectively, as T ð23Þ K ðx; tÞ ¼ Xð xÞB1 KT ðXðtÞB1 ÞT ; KT ¼ kmn T J K ðx; tÞ ¼ Pða;bÞ ð xÞKJ Pða;bÞ ðtÞ ; KJ ¼ kmn ð24Þ
The following relation can be obtained from Eqs. (7), (23), and (24),
where
2 0 0 6 0 ðsÞ 6 6 6 6 0 6 Bs ¼ 6 6 6 6 6 4 0
m X
i¼0
Differential-difference part of problem is i¼0 Pi ð xÞy ð xÞ P ðjÞ þ m j¼0 Qj ð xÞy ðx sÞ:First, to explain the relation between the matrix form of the unknown function and the matrix form of its derivative yðiÞ ð xÞ, we introduce the relation between Xð xÞ and its derivatives XðiÞ ð xÞ can be expressed as
ð19Þ
Using (16) and (19), the matrix form of differential-difference part of Eq. (4) becomes
¼
Pn
0 1 60 0 6 .. B¼6 . 6 40 0 0 0
yðjÞ ðx sÞ ¼ Xð xÞBs Bi Mða;bÞ A
n X
Matrix representation of differential-difference part of problem
where 2
Thus, it is seen that
1 0 1 1
ðsÞ
1
ðsÞ0
2 0 2 1
ðsÞ1 ..
.. . 0
ðsÞ
2
0
.
3 N ðsÞ 7 0 7 7 7 N ðsÞN1 7 7 1 7 7 .. 7 7 . 7 5 N ðsÞ0 N N
ð18Þ
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T Xð xÞB1 KT ðB1 ÞT XT ðtÞ ¼ Pða;bÞ ð xÞKJ Pða;bÞ ðtÞ T ¼ Xð xÞMða;bÞ KJ Mða;bÞ XT ðtÞ T ) B1 KT ðB1 ÞT ¼ Mða;bÞ KJ Mða;bÞ 1 T 1 ) KJ ¼ Mða;bÞ B1 KT ðB1 ÞT Mða;bÞ
n1 h i X aki yðiÞ ðaÞ þ bki yðiÞ ðbÞ þ cki yðiÞ ðgÞ i¼0
¼
n1 X
½aki XðaÞ þ bki XðbÞ þ cki XðgÞBi Mða;bÞ A ¼ lk ;
i¼0
k ¼ 0; 1; . . .; n 1
ð28Þ
or
T 1 KT ¼ ðB1 Þ1 Mða;bÞ KJ Mða;bÞ ðB1 ÞT By substituting the Eqs. (19) and (25) into ðt sÞdt, we derive the matrix relation Zb
K ðx; tÞyðt sÞdt ¼
a
Zb
ð25Þ Rb a
K ðx; tÞy
T Pða;bÞ ð xÞKJ Pða;bÞ ðtÞ
Method of solution We substitute obtained matrix relations in the previous subsections given in Eqs. (20) and (27) into fundamental problem to build the fundamental matrix equation of the problem. For this purpose, we can define collocation points as follow: xs ¼ a þ
a
XðtÞBs Mða;bÞ Adt ¼P
ða;bÞ
ð xÞKJ QA
ð26Þ
such that Zb T Pða;bÞ ðtÞ XðtÞBs Mða;bÞ dt Q¼
ba s; N
s ¼ 0; 1; 2; . . .; N
As can be observed, standard collocation points dividing the domain interval ½a; b of the problem into N equal parts are employed. Accordingly, we obtain the system of matrix equations n m X X Pi ðxs ÞXðxs ÞBi Mða;bÞ A þ Qj ðxs ÞXðxs ÞBs B j Mða;bÞ A i¼0
j¼0 ða;bÞ
a
¼ gðxs Þ þ Xðxs ÞM
Zb T ¼ Mða;bÞ XT ðtÞXðtÞBs Mða;bÞ dt a
¼ Mða;bÞ
T
KJ QA
The fundamental matrix equation becomes (
HBs Mða;bÞ
n X
i
Pi XB M
ða;bÞ
i¼0
þ
m X
) j
Qj XBs B M
ða;bÞ
XM
ða;bÞ
KJ Q A ¼ G
j¼0
ð29Þ
where H¼
Zb a
hij ¼
where
X ðtÞXðtÞdt ¼ hij ;
2
T
0 P i ð x0 Þ 6 0 P ð x1 Þ i 6 Pi ¼ 6 .. 6 4 .
1 ðb 1Þi1þj ða 1Þi1þj ; i1þj i; j ¼ 1; 2; 3. . .; N þ 1
2
Finally, substituting the form (7) into expression (26) yields the matrix relation Zb
K ðx; tÞyðt sÞdt ¼ Xð xÞMða;bÞ KJ QA
0 0
0
Qj ðx1 Þ
6 6 Qj ¼ 6 6 4
Matrix representation of conditions In this section, we write to the matrix form of mixed conditions of the problem given Eq. (5), using the matrix relation (16), as
.. . 0
ð27Þ
a
123
0 Qj ð x 0 Þ
0
3 0 0 7 7 7 .. 7; .. . 5 . P i ð xN Þ 3 0 0 7 7 7 .. 7; .. . 5 . Qj ðxN Þ
2
gðx0 Þ 6 gðx1 Þ 6 G¼6 6 .. 4 . gðxN Þ 2 Xðx0 Þ 6 Xðx1 Þ 6 X¼6 6 .. 4 . XðxN Þ
3 7 7 7 7 5 3 7 7 7 7 5
Equation (29), which is matrix representation of the Eq. (4), corresponds to a system of N þ 1 algebraic equations. This system indicates N þ 1 unknown coefficients, such that a0 ; a1 ; a2 ; . . .; aN . Briefly, if we define W¼
n X
Pi XBi Mða;bÞ þ
i¼0
XMða;bÞ KJ Q
m X j¼0
Qj XBs B j Mða;bÞ
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87
under last definition W matrix Eq. (29) transforms into the augmented matrix form ½W; G:
ð30Þ
Similarly, from (28), the matrix form of mixed conditions can be obtained briefly as Uk A ¼ lk or ½Uk ; lk ;
k ¼ 0; 1; 2; . . .; n 1
For the purpose of calculating the corrected solution, we now define the residual function using the Jacobi polynomial solution by obtained the our method as n ðiÞ X ða;bÞ RN ð x Þ ¼ Pi ð x Þ y N ð xÞ i¼0
ð31Þ
þ
m X
ða;bÞ ðjÞ Qj ð xÞ yN ðx sÞ gð xÞ
j¼0
such that
n1 X ½aki XðaÞ þ bki XðbÞ þ cki XðgÞBi Mða;bÞ Uk ¼
Zb
ða;bÞ
K ðx; tÞyN
ðt sÞdt
ð33Þ
a
i¼0 ða;bÞ
Consequently, to find the Jacobi polynomial solution of Eq. (4) under the mixed conditions (5), we replace the row matrix (31) by last n rows of the augmented matrix (30), which yields the new matrix equation form written as follow ~ : ~ G W; ð32Þ ~ ¼ N þ 1, then we can find the ~ ¼ rank W; ~ G If rank W matrix of unknown coefficient of Jacobi series via
1 ~ Note that the matrix A (thereby, the coef~ A¼ W G. ficients a0 ; a1 ; a2 ; . . .; aN ) is uniquely determined [22]. Equation (4) has also a unique solution under the conditions (5). Thus, we get the Jacobi polynomial solution for arbitrary parameters a and b: yð xÞ ffi
ða;bÞ yN ð xÞ
¼
N X
where yN
ða;bÞ
for arbitrary parameters a and b. Hence, yN ð xÞ satisfies the problem n m ðiÞ ð jÞ X X ða;bÞ ða;bÞ Pi ð x Þ y N ð xÞ þ Qj ð xÞ yN ð x sÞ i¼0
n¼0
Error analysis In this part of study, it is given to a useful error estimation procedure for orthogonal Jacobi polynomial solution of the problem. Also, this procedure is used to obtain the improved solution of the problem (4, 5) according to the direct Jacobi polynomial solution. For this purpose, we use the residual correction technique [31, 32] and error estimation by the known Tau method [33, 34]. Recently, Yu¨zbas¸ ı and Sezer [35] solved a class of the Lane–Emden equations using the improved BCM with residual error function. Yu¨zbas¸ ı et al. [36] proposed an improved Legendre method for to obtain the approximate solutions of a class of the integro-differential equations. Wei and Chen [37] presented a numerical method called spectral methods for classes Volterra type integro-differential equations with weakly singular kernel and smooth solutions.
j¼0
Zb
n1 X
ða;bÞ
K ðx; tÞyN
ðt sÞdt ¼ gð xÞ þ RN ð xÞ
a
ðiÞ ðiÞ ðiÞ ða;bÞ ða;bÞ ða;bÞ aki yN ðaÞ þ bki yN ðbÞ þ cki yN ðgÞ
i¼0
¼ lk ;
k ¼ 0; 1; . . .; n 1
ð34Þ
The error function eN ð xÞ can also be defined as ða;bÞ
an Pðna;bÞ ð xÞ
ð xÞ is the approximate solution of Eqs. (4, 5)
eN
ða;bÞ
ð xÞ ¼ yð xÞ yN
ð xÞ
ð35Þ
where yð xÞ is the exact solution of the Eqs. (4, 5). Substituting (35) into (4, 5) and also using (33) and (34), we derive the error differential equation with homogenous conditions: n m X X ða;bÞ ðiÞ ða;bÞ ðjÞ P i ð xÞ eN ð xÞ þ Q j ð xÞ eN ð x sÞ i¼0
j¼0
Zb
ða;bÞ
K ðx; tÞeN
ðt sÞdt ¼ RN ð xÞ
a n1 X
ðiÞ ð i Þ ð i Þ ða;bÞ ða;bÞ ða;bÞ aki eN ðaÞ þ bki eN ðbÞ þ cki eN ðgÞ ¼ 0
i¼0
ð36Þ By solving the problem (36) using the present method given in the previous section, we get the error estimation function ða;bÞ
ða;bÞ
eN;M ð xÞ to eN ð xÞ. Note that M must be bigger than N and error estimation is found using the residual function RN ð xÞ. Consequently, by means of the orthogonal Jacobi polynomials ða;bÞ
yN
ða;bÞ
ð xÞ and eN;M ð xÞ, we obtain the corrected Jacobi solution
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ða;bÞ
ða;bÞ
yN;M ð xÞ ¼ yN
ða;bÞ
ð xÞ þ eN;M ð xÞ
ð37Þ ða;bÞ
Finally, we construct the Jacobi error function eN
ð xÞ
ða;bÞ
and the corrected Jacobi error function EN;M ð xÞ ða;bÞ
eN
ða;bÞ
ð xÞ ¼ yð xÞ yN
ða;bÞ EN;M ð xÞ
ð xÞ;
ð38Þ
¼ yð xÞ yN;M ð xÞ:
ð39Þ
Illustrative examples We apply the Jacobi matrix method to four examples via the symbolic computation program Maple [38]. In these ða;bÞ examples, the term eN ð xÞ represent absolute error ða;bÞ function and also EN;M ð xÞ represent the absolute error function of the corrected Jacobi polynomial solution. Example 1 As the first example, we consider the FIDDE [23, 24] 0
0
y ð xÞ þ xy ðx 1Þ yð xÞ þ yðx 1Þ Z1 ¼ gðxÞ þ Kðx; tÞyðt 1Þdt
ð40Þ
1
with initial condition yð1Þ 2yð0Þ þ yð1Þ ¼ 0 Here, m ¼ 1; s ¼ 1; P1 ð xÞ ¼ 1; P0 ð xÞ ¼ 1; gð xÞ ¼ x 2; Q1 ð xÞ ¼ x; Q0 ð xÞ ¼ 1; K ðx; tÞ ¼ x þ t; a ¼ 1; b ¼ 1; g ¼ 0; a00 ¼ 1; b00 ¼ 2; c00 ¼ 1; l0 ¼ 0. We assume that the Eq. (40) has a Jacobi polynomial solution in the following form, ða;bÞ
y ð xÞ ¼ a0 P 0
ða;bÞ
ð x Þ þ a 1 P1
ða;bÞ
ð x Þ þ a2 P2
ð xÞ
where N ¼ 2 and ða; bÞ ¼ ð0:5; 0:5Þ, which are chosen arbitrary; then, according to (8) h i Pða;bÞ ð xÞ ¼ Pð0a;bÞ ð xÞ Pð1a;bÞ ð xÞ Pð2a;bÞ ð xÞ 2 1 15 15 3 ð x 1 Þ ¼ 1 : þx þ xþ 2 8 4 2 The collocation points are computed as fx0 ¼ 1; x1 ¼ 0; x2 ¼ 1g and from Eq. (30), the matrix equation of the Eq. (40) is fP1 XBM þ P0 XM þ Q1 XB1 BM Q0 XB1 M XMKJ Qg A¼G
123
where 2
1
6 P1 ¼ Q 1 ¼ 4 0 0 2 1 0 6 P0 ¼ 4 0 1 0 0 2 1 0 6 0 Q1 ¼ 4 0 0 0 2 1 2 6 X ¼ 41 1 1 2 1 6 B1 ¼ 4 0 0 2
1
6 6 6 M ¼ 60 6 4 0 2 2 6 6 6 Q ¼ 61 6 4 1 4
0
3 2 1 0 1 1 6 3 8
0
1 0 0
0
3
7 0 5; 1 3
7 0 5; 1 3
7 0 5; 1 2 3 0 4 6 7 1 5; B ¼ 4 0
1 0
0 2
0 3
0
1
0
0
1 1 0
0
3
0 7 2 5; 0 15 3
3 0 7 2 5;
3 6 7 G ¼ 4 2 5 1
2 8 7 1 7 15 7 6 7; KJ ¼ 4 1 4 7 5 0 3 2 7 3 4 7 7 5 7 7 8 7 5 81 160
0 0
3
7 0 5; 0
Hence, we obtain the matrix W as follows 3 2 8 2 10 7 6 3 7 6 2 7 6 W¼6 0 3 7 7 6 3 5 4 4 2 2 3 Using Eq. (31), we can write the matrix equation of the condition of the problem as ½U; k ¼ ½ 0
0
3;
0
Consequently, to find the Jacobi polynomial solution of the problem under the mixed conditions, we replace the row matrix ½U; k by last row of the matrix ½W; G and obtain ~ as ~ G the matrix W;
Math Sci (2016) 10:83–93
2 2 6 6 ~ ~ W; G ¼ 6 40 0
8 3 2 3 0
89
3 3 7 7 7: 3; 2 5 3; 0
37651331e 1ðx 1Þ3 þ 0:4006530189e 1ðx 1Þ4 þ 0; 2888171229e 2ðx 1Þ5 0:8352418987e 3ðx 1Þ6
10;
ð0:4;0:5Þ
and the estimated error function is e6;7 ð xÞ ¼ 0:58 5645023e 3 þ 0:460274411e 2x 0:1546818100 e 1ðx 1Þ2 0:2185948038e 1ðx 1Þ3 0:47 16294 762e 2ðx 1Þ4 þ 0:2241198695e 2ðx 1Þ5 0:167994 29 91 e 3ðx 1Þ6 0:1485433260e 3ðx 1Þ7 . Then, we calculate the corrected solution function simply as the sum of the approximate solution and the estimated error function
~ , we obtain the ~ G Solving the new augmented matrix W; Jacobi polynomial coefficient matrix T 5 A¼ 3 0 : 2 From Eq. (11), the Jacobi polynomial solution of the ð0:5;0:5Þ
ð0:4;0:5Þ
problem is y2 ð xÞ ¼ 3x þ 4, which is the exact solution of the problem. Furthermore, we can obtain the exact solution of the problem for any value of N and corresponding suitable values of ða; bÞ. Example 2 We consider a third-order FIDDE with variable coefficients 000
00
y6;7
ð xÞ
ða;bÞ
Now, we determine the maximum error for yN ða;bÞ EN
ð41Þ 0
¼ ¼
1
yð xÞ ;
axb
ð xÞ as,
o
The maximum errors EN for different values of N are given in Table 2, and it is seen that the error decreases continually as N increases. The maximum error for the corrected Jacobi polynomial solution (37) is calculated in a similar way, ða;bÞ
ða;bÞ
EN;M ¼ yN;M ð xÞ yð xÞ1 n ða;bÞ ¼ max yN;M ð xÞ yð xÞ ;
ð0:4;0:5Þ y6 ð xÞ
derived by employing these values, as ¼ 0:299 3926085 þ 0:5377164192x 0:4041110104ðx 1Þ2 0:68
Present method N¼6
ða;bÞ yN nð x Þ yð xÞ1 ða;bÞ max yN ð xÞ
ða;bÞ
00
with the initial conditionsyð0Þ ¼ 0; y ð0Þ ¼ 1; y ð0Þ ¼ 0. where gð xÞ ¼ ðx þ 1Þðsinðx 1Þþ cosð xÞÞ cosð2Þ þ 1. The exact solution of problem is yð xÞ ¼ sinð xÞ: After several trials, it has been determined that ða; bÞ ¼ ð0:4; 0:5Þ gives the most accurate result; therefore, the approximate solution of the third-order FIDDE has been
xi
ð0:4;0:5Þ
ð xÞ þ e6;7
the relative absolute error function Table 1 shows ð0:4;0:5Þ ð xÞ and the corrected absolute error function e6 ð0:4;0:5Þ ð xÞ for this example. E6;7
0
y ð xÞ þ y ðx 1Þ xy ð xÞ xyðx 1Þ Z1 ¼ gðxÞ þ yðt 1Þdt
Table 1 Relative absolute error ð0:4;0:5Þ ð xÞ and the function e6
ð0:4;0:5Þ
ð xÞ ¼ y6
N; M ¼ 6; 7
axb
o
Taylor method [23]
Tau method [24]
N¼6
N¼6
N¼7
N¼7
corrected absolute error ð0:4;0:5Þ function E6;7 ð xÞ for
1:0
2:88e02
1:04e03
8:58e02
6:03e02
3:84e02
5:05e03
Example 2
0:8
1:36e02
5:72e04
3:93e02
2:28e02
1:83e02
2:38e03
0:6
5:22e03
2:53e04
1:50e02
6:63e03
7:00e03
9:14e04
Table 2 Maximum error ð0:4;0:5Þ ) for Example 2 (EN
0:4
1:38e03
7:70e05
4:12e03
1:20e03
1:86e03
2:42e04
0:2
1:52e04
9:68e06
4:85e04
6:90e05
2:04e04
2:65e05
0:0
0
0
0
0
0
0
0:2
1:10e04
9:22e06
4:59e04
5:30e05
1:48e04
1:91e05
0:4
7:20e04
6:97e05
3:69e03
8:09e04
9:67e04
1:25e04
0:6
1:90e03
2:15e04
1:28e02
3:82e03
2:55e03
3:30e04
0:8
3:34e03
4:71e04
3:17e02
1:14e02
4:44e03
5:78e04
1:0
4:36e03
8:26e04
6:57e02
2:73e02
5:76e03
7:53e04
N ða;bÞ
EN
4
6
8
10
12
1:6321e01
2:8883e02
6:2300e03
2:9748e03
2:9748e04
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and the results are shown in Table 3 for miscellaneous values of N, M. The decrease in maximum error, as M increases, is indisputable. Finally, the third-order FIDDE has also been solved using Legendre, Gegenbauer (also Chebyshev), and Jacobi polynomials, for comparison purposes. The maximum error values are given in Table 4, and it is seen that Jacobi-based solution gives slightly better results. Table 3 Maximum error N; M ða;bÞ
EN;M
ð0:4;0:5Þ (EN;M )
Example 3 The third example is a second-order FIDE [24–26] with variable coefficients 00
0
y ð xÞ þ 4xy ð xÞ ¼
8x4
2 ðx2 þ 1Þ3
Z1 0
t2 þ 1 ð x 2 þ 1Þ 2
yðtÞdt;
ð42Þ
0x1 under the boundary conditions yð0Þ ¼ 1; yð1Þ ¼ 1=2:
for Example 2
Here, P2 ð xÞ ¼ 1; P1 ð xÞ ¼ 4x; P0 ð xÞ ¼ 0; Q0 ð xÞ ¼ 0; gð xÞ 3
6; 7
6; 8
10; 11
10; 15
¼ 8x4 =ðx2 þ 1Þ ; a ¼ 0; b ¼ 0; K ðx; tÞ ¼ 2ðt2 þ 1Þ=
1:0480e03
6:7241e04
3:5602e04
1:7160e05
ðx2 þ 1Þ ; s ¼ 0; a ¼ 0.
2
1
The exact solution of this problem is ðx2 þ 1Þ . Figure 1 shows a comparison of the Jacobi polynomial Table 4 Comparison of different polynomial bases for maximum error values for Example 2 ða; bÞ
E4
E6
Legendre base
ð0; 0Þ
0:1632109624
0:02888311052
Gegenbauer base
ð0:5; 0:5Þ
0:1632109623
0:0288831090
Jacobi base
ð0:4; 0:5Þ
0:1632109622
0:0288831072
ða;bÞ
Fig. 1 Comparison of the Jacobi polynomial solution ða;bÞ y5 ð xÞ and the corrected Jacobi polynomial solution ða;bÞ y5;6 ð xÞ with the exact solution for Example 3
ða;bÞ
ð0;0Þ
solution yN
ð xÞ, and the corrected Jacobi polynomial
ð0;0Þ yN;M ð xÞ,
solution is for ðN; M Þ ¼ ð5; 6Þ and ða ¼ b ¼ 0Þ, with the exact solution yðxÞ. It is apparently seen that the corrected Jacobi polynomial solution almost coincides with the exact solution. Table 5 and Fig. 2 show a comparison of the absolute error with the corrected absolute errors, for N ¼ 5; 8 and
Exact Soluon Jacobi Polinomial Soluon Improved Jacobi Polynomial Soluon
1
0.9
0.8
0.7
0.6
0.5 0
Table 5 Comparison of the absolute error with the corrected absolute errors for Example 3
123
0.2
0.4
0.6
0.8
1
Absolute error ð0;0Þ ð0;0Þ e5 ð xÞ e8 ð xÞ
Corrected absolute errors ð0;0Þ ð0;0Þ E5;6 ð xÞ E5;7 ð xÞ
ð0;0Þ E5;7;8 ð xÞ
ð0;0Þ E5;7;9 ð xÞ
0:125
2:097e2
1:087e2
1:298e3
2:627e4
2:723e4
6:754e5
0:250
3:883e2
2:010e2
2:347e3
4:903e4
5:024e4
1:252e4
0:375
5:295e2
2:737e2
3:197e3
6:574e4
6:821e4
1:703e4
0:500
6:296e2
3:264e2
3:799e3
7:817e4
8:114e4
2:027e4
0:625
6:815e2
3:596e2
4:173e3
8:862e4
8:957e4
2:239e4
0:750 0:875
6:581e2 4:838e2
3:656e2 2:992e2
4:257e3 3:403e3
1:128e3 1:501e3
9:439e4 9:147e4
2:345e4 2:260e4
xi
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91
Fig. 2 Comparison of the absolute error functions of the solution Jacobi polynomial ða;bÞ e5 ð xÞ with the corrected
0.08 Absolute Error Corrected Absolute Error 0.06
solution Jacobi polynomial ða;bÞ E5;6 ð xÞ for Example 3 0.04
0.02
0
Table 6 Comparison of the absolute errors of Jacobi polynomial solution, improved Jacobi polynomial solution, wavelet collocation, wavelet Galerkin, and ChFD methods for Example 4
0
0.2
00
0
0.6
0.8
Wavelet collocation [26]
0:125
8:5e10
1:1e11
2:6e02
2:7e04
1:8e10
0:250
6:3e10
5:0e11
1:3e02
3:0e05
4:4e10
0:375
1:0e09
5:8e11
9:3e03
2:6e04
1:4e09
0:500
1:3e09
7:6e11
5:1e03
4:3e04
2:4e10
0:625
1:6e09
7:5e11
2:5e03
5:6e04
1:7e09
0:750 0:875
4:0e09 3:8e09
1:1e10 1:5e10
1:0e03 2:6e04
6:5e04 7:2e04
7:7e10 1:3e09
1 exþ1 2 þ x þ 50x 35 ex xþ1 Z1 þ ext yðtÞdt 0
with conditions yð0Þ ¼ 1; yð1Þ ¼ e The exact solution of problem is Taking ða ¼ 0:4; b ¼ 0:5Þ, the absolute errors of Jacobi polynomial solution for N ¼ 7 and the absolute errors of the improved Jacobi polynomial solution for N ¼ 7; M ¼ 8 are compared with those of the wavelet Galerkin, the wavelet collocation, and the Chebyshev finite difference (ChFD) methods [25, 26], in Table 6. Considering the errors of the different methods, it is observed that the smallest errors are obtained using the improved Jacobi polynomial solution.
Wavelet Galerkin [26]
1
Present method ð:4;:3Þ ð:4;:3Þ ð xÞ e7 E7:8 ð xÞ
xi
M ¼ 6; 7; 8; 9. The parameters are taken as ða ¼ b ¼ 0Þ. The corrected absolute errors are corrected, once more and the last two columns show these values. It is noticed that sequential corrections tend to decrease the absolute error. Example 4 [25, 26] The last example is the second-order Fredholm integro-differential equation x2 y þ 50xy 35y ¼
0.4
ChFD [17] N¼7
Example 5 Consider the first-order linear FIDDE [39] Zx yðtÞ 4 3=2
x 24 x2 ; 0 x 1: 1 dt ¼ 105 ðx tÞ2 0
We assume that the problem has a Jacobi polynomial solution in the form ða;bÞ
y ð xÞ ¼ a0 P 0
ða;bÞ
ð x Þ þ a 1 P1
ða;bÞ
ð x Þ þ a2 P2
ða;bÞ
ð xÞ þ a3 P3
ð xÞ
where N ¼ 3 and ða; bÞ ¼ ð0:2; 0:3Þ, which are chosen arbitrary. Using the mentioned methods, the Jacobi polynomial solution of the problem is obtained by ð0:2;0:3Þ
y3 ð xÞ ¼ x x3 , which is the exact solution of the problem [39]. Furthermore, we can obtain the exact solution of the problem for any value of N 3 and corresponding suitable values of ða; bÞ.
Conclusions A new matrix method based on Jacobi polynomials and collocation points has been introduced to solve high-order linear FIDDE with variable coefficients. Jacobi polynomials are the common set of orthogonal polynomials,
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which are the most extensively studied and widely applied systems. The solution of the FIDDE is expressed as a truncated series of orthogonal Jacobi polynomials, which is then transformed from algebraic form into matrix form. The problem and the mixed conditions are also represented in matrix form. Finally, the solution is obtained as a truncated Jacobi series written in matrix form using collocation points. A new error estimation procedure for polynomial solution and a technique to find a high accuracy solution are developed. Most of the previous studies dealt with solutions using Legendre, Chebyshev, and Gegenbauer polynomials. In this study, however, we have proposed a Jacobi polynomial solution that comprises all of these polynomial solutions. The new Jacobi matrix method has been applied to four illustrative examples. It is well seen from these examples that the method yields either the exact solution or a high accuracy approximate solution for delay integro-differential equation problems. The accuracy of the approximate solution can be increased using the proposed error analysis technique depending on residual function. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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