Hindawi International Journal of Di๏ฌerential Equations Volume 2017, Article ID 2097317, 10 pages https://doi.org/10.1155/2017/2097317
Research Article Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations Abdulnasir Isah,1,2 Chang Phang,2 and Piau Phang3 1
Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria Department of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Malaysia 3 Faculty of Computer Science and Information Technology, Universiti Malaysia Sarawak, Sarawak, Malaysia 2
Correspondence should be addressed to Piau Phang;
[email protected] Received 2 January 2017; Revised 16 March 2017; Accepted 16 May 2017; Published 13 June 2017 Academic Editor: Patricia J. Y. Wong Copyright ยฉ 2017 Abdulnasir Isah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.
1. Introduction Fractional calculus, the calculus of derivative and integral of any order, is used as a powerful tool in science and engineering to study the behaviors of real world phenomena especially the ones that cannot be fully described by the classical methods and techniques [1]. Differential equations with proportional delays are usually referred to as pantograph equations or generalized pantograph equations. The name pantograph was originated from the study work of Ockendon and Tayler [2]. Many researchers have studied different applications of these equations in applied sciences such as biology, physics, economics, and electrodynamics [3โ5]. Solutions of pantograph equations were also studied by many authors numerically and analytically. Bhrawy et al. proposed a new generalized Laguerre-Gauss collocation method for numerical solution of generalized fractional pantograph equations [1]. Tohidi et al. in [6] proposed a new collocation scheme based on Bernoulli operational matrix for numerical solution of generalized pantograph
equation. Yusufoglu [7] proposed an efficient algorithm for solving generalized pantograph equations with linear functional argument. In [8], Yang and Huang presented a spectral-collocation method for fractional pantograph delay integrodifferential equations and in [9] Yยจuzbasi and Sezer presented an exponential approximation for solutions of generalized pantograph delay differential equations. Chebyshev and Bessel polynomials are, respectively, used in [10, 11] to obtain the solutions of generalized pantograph equations. Operational matrices of fractional derivatives and integration have become very important tool in the field of numerical solution of fractional differential equations. In this paper, a member of Appell polynomials called Genocchi polynomials is used; although this polynomial is not based on orthogonal functions, it possesses operational matrices of derivatives with high accuracy. It is very important to note that this polynomial shares some great advantages with Bernoulli and Euler polynomials for approximating an arbitrary function over some classical orthogonal polynomials; we refer the reader to [6] for these advantages. On top of that, we
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had successfully applied the operational matrix via Genocchi polynomials for solving integer-order delay differential equations [12] and fractional optimal control problems [13], and the numerical solutions obtained are comparable or even more accurate compared to some existing well-known methods. Motivated by these advantages, in this paper, we intend to extend the result for integer-order delay differential equations in [12] to fractional delay differential equations or so-called generalized fractional pantograph equations. To the best of our knowledge, this is the first time that the operational matrix based on Genocchi polynomials is applied to solve the fractional pantograph equations. On the other hand, some other types of polynomials were employed to solve some special type of fractional calculus problems; for example, Bessel polynomials were used for the solution of fractional-order logistic population model [14]; Bernstein polynomials were also used for the solution of Riccati type differential equations [15]. In this paper, we use the new operational matrix of fractional-order derivative via Genocchi polynomials to provide approximate solutions of the generalized fractional pantograph equations of the following form [1]:
2. Preliminaries 2.1. Fractional Derivative and Integration. We recall some basic definitions and properties of fractional calculus that we will use. There are various competing definitions for fractional derivatives [16, 17]. The Riemann-Liouville definition played a vital role in the development of the theory of fractional calculus. However, there are certain disadvantages of using this definition when modeling real world phenomena. To cope with these disadvantages, Caputo definition was introduced which is found to be more reliable in application. So we use this definition of fractional derivatives. We begin with the definition of Riemann-Liouville integral, in which the fractional integral operator ๐ผ of a function ๐(๐ก) is defined as follows. Definition 1. The Riemann-Liouville integral ๐ผ of fractionalorder ๐ผ of ๐(๐ก) is given by ๐ก 1 โซ (๐ก โ ๐)๐ผโ1 ๐ (๐) ๐๐, ฮ (๐ผ) 0
๐ผ๐ผ ๐ (๐ก) =
(3) +
๐ก > 0, ๐ผ โ R ,
๐ฝ ๐โ1
๐ท๐ผ ๐ฆ (๐ก) = โ โ ๐๐,๐ (๐ก) ๐ท๐ฝ๐ ๐ฆ (๐ ๐,๐ ๐ก + ๐๐,๐ ) + ๐ (๐ก) , ๐=0 ๐=0
(1)
0โค๐กโค1
where ฮ(โ
) is the Gamma function. The fractional derivative of order ๐ผ > 0 due to Riemann-Liouville is defined by (๐ท๐๐ผ ๐) (๐ก) = (
๐ ๐ ๐โ๐ผ ) (๐ผ ๐) (๐ก) , ๐๐ก
subject to the following conditions: ๐โ1
โ ๐๐,๐ ๐ฆ
๐=0
(๐)
(๐ผ > 0, ๐ โ 1 < ๐ผ < ๐) .
(0) = ๐๐ , ๐ = 0, 1, . . . , ๐ โ 1,
(2)
The following are important properties of Riemann-Liouville fractional integral ๐ผ๐ผ : ๐ผ๐ผ ๐ผ๐ฝ ๐ (๐ก) = ๐ผ๐ผ+๐ฝ ๐ (๐ก) ,
where ๐๐,๐ , ๐ ๐,๐ , and ๐๐,๐ are real or complex coefficients; ๐ โ 1 < ๐ผ < ๐, 0 < ๐ฝ0 < ๐ฝ1 < โ
โ
โ
< ๐ฝ๐โ1 < ๐ผ, while ๐๐,๐ (๐ก) and ๐(๐ก) are given continuous functions in the interval [0, 1]. The rest of the paper is organized as follows: Section 2 introduces some mathematical preliminaries of fractional calculus. In Section 3, we discuss some important properties of Genocchi polynomials. In Section 4, we derive the Genocchi delay operational matrix and we apply the collocation method for solving fractional pantograph equation (1) using the Genocchi operational matrix of fractional derivative and the delay operational matrix in Section 5. In Section 6, the proposed method is applied to several examples and conclusion is given in Section 7.
๐ท๐ผ ๐ถ = 0,
๐ผ๐ผ ๐ก๐ฝ =
๐ผ > 0, ๐ฝ > 0,
ฮ (๐ฝ + 1) ๐ฝ+๐ผ ๐ก . ฮ (๐ฝ + ๐ผ + 1)
(5)
Definition 2. The Caputo fractional derivative ๐ท๐ผ of a function ๐(๐ก) is defined as ๐ท๐ผ ๐ (๐ก) =
๐ก ๐(๐) (๐) 1 ๐๐, โซ ฮ (๐ โ ๐ผ) 0 (๐ก โ ๐)๐ผโ๐+1
(6)
๐ โ 1 < ๐ผ โค ๐, ๐ โ N. Some properties of Caputo fractional derivatives are as follows:
(๐ถ is constant) ,
0, { { ๐ท ๐ก = { ฮ (๐ฝ + 1) ๐ฝโ๐ผ { ๐ก , { ฮ (๐ฝ + 1 โ ๐ผ) ๐ผ ๐ฝ
(4)
๐ฝ โ N โช {0} , ๐ฝ < โ๐ผโ ๐ฝ โ N โช {0} , ๐ฝ โฅ โ๐ผโ or ๐ฝ โ N, ๐ฝ > โ๐ผโ ,
(7)
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where โ๐ผโ denotes the smallest integer greater than or equal to ๐ผ and โ๐ผโ denotes the largest integer less than or equal to ๐ผ. Similar to the integer-order differentiation, the Caputo fractional differential operator is a linear operator; that is, ๐ผ
๐ผ
๐ผ
๐ท (๐๐ (๐ก) + ๐๐ (๐ก)) = ๐๐ท ๐ (๐ก) + ๐๐ท ๐ (๐ก)
(8)
for ๐ and ๐ constants.
3. Genocchi Polynomials and Some Properties Genocchi polynomials and numbers have been extensively studied in many different contexts in branches of mathematics such as elementary number theory, complex analytic number theory, homotopy theory (stable homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), and quantum physics (quantum groups). The classical Genocchi polynomial ๐บ๐ (๐ฅ) is usually defined by means of the exponential generating functions [18โ20]. โ ๐ก๐ 2๐ก๐๐ฅ๐ก ๐บ = , โ (๐ฅ) ๐ ๐๐ก + 1 ๐=0 ๐!
(|๐ก| < ๐) ,
(9)
where ๐บ๐ (๐ฅ) is the Genocchi polynomial of degree ๐ and is given by ๐ ๐ ๐บ๐ (๐ฅ) = โ ( ) ๐บ๐โ๐ ๐ฅ๐ . ๐=0 ๐
(10)
๐บ๐โ๐ here is the Genocchi number. Some of the important properties of these polynomials include 1
โซ ๐บ๐ (๐ฅ) ๐บ๐ (๐ฅ) ๐๐ฅ = 0
๐
2 (โ1) ๐!๐! ๐บ๐+๐ (๐ + ๐)!
๐๐บ๐ (๐ฅ) = ๐๐บ๐โ1 (๐ฅ) , ๐ โฅ 1, ๐๐ฅ ๐บ๐ (1) + ๐บ๐ (0) = 0,
๐ > 1.
๐, ๐ โฅ 1,
(11)
where Gram (๐บ1 , ๐บ2 , . . . , ๐บ๐) ๓ตจ๓ตจโจ๐บ , ๐บ โฉ โจ๐บ , ๐บ โฉ 1 2 ๓ตจ๓ตจ 1 1 ๓ตจ๓ตจ ๓ตจ๓ตจโจ๐บ , ๐บ โฉ โจ๐บ , ๐บ โฉ 2 2 ๓ตจ๓ตจ 2 1 = ๓ตจ๓ตจ๓ตจ๓ตจ . . ๓ตจ๓ตจ .. .. ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจโจ๐บ , ๐บ โฉ โจ๐บ , ๐บ โฉ ๓ตจ ๐ 1 ๐ 2
โ
โ
โ
โจ๐บ1 , ๐บ๐โฉ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ โ
โ
โ
โจ๐บ2 , ๐บ๐โฉ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ . .. ๓ตจ๓ตจ ๓ตจ๓ตจ โ
โ
โ
. ๓ตจ๓ตจ ๓ตจ โ
โ
โ
โจ๐บ๐ , ๐บ๐โฉ๓ตจ๓ตจ๓ตจ
Now, to prove that this determinant is not equal to zero, we first reduce the Gram matrix to an upper triangular matrix by Gaussian elimination and it is not difficult to see that the elements of the diagonal of the reduced matrix are given by ๐ (๐) =
(๐! (๐ + 1)!)2 , (2๐!) (2๐ + 1)!
๐ โ N.
๐
โ๐ (๐)
(17)
๐=1
is not equal to zero. Therefore, the set ๐ด is the set of linearly independent sets. 3.1. Function Approximation. Assume that {๐บ1 (๐ก), ๐บ2 (๐ก), . . . , ๐บ๐(๐ก)} โ ๐ฟ2 [0, 1] is the set of Genocchi polynomials and ๐ = Span{๐บ1 (๐ก), ๐บ2 (๐ก), . . . , ๐บ๐(๐ก)}. Let ๐(๐ก) be arbitrary element of ๐ฟ2 [0, 1]; since ๐ is a finite dimensional subspace of ๐ฟ2 [0, 1] space, ๐(๐ก) has a unique best approximation in ๐, say ๐โ (๐ก), such that ๓ตฉ๓ตฉ ๓ตฉ ๓ตฉ ๓ตฉ โ (18) ๓ตฉ๓ตฉ๐ (๐ก) โ ๐ (๐ก)๓ตฉ๓ตฉ๓ตฉ2 โค ๓ตฉ๓ตฉ๓ตฉ๐ (๐ก) โ ๐ฆ (๐ก)๓ตฉ๓ตฉ๓ตฉ2 , โ๐ฆ (๐ก) โ ๐. This implies that, โ๐ฆ(๐ก) โ ๐, โจ๐ (๐ก) โ ๐โ (๐ก) , ๐ฆ (๐ก)โฉ = 0,
(19)
โ
where โจโ
โฉ denotes inner product. Since ๐ (๐ก) โ ๐, there exist the unique coefficients ๐1 , ๐2 , . . . , ๐๐ such that ๐
Lemma 3. The set ๐ด = {๐บ1 (๐ก), ๐บ2 (๐ก), . . . , ๐บ๐(๐ก)} โ ๐ฟ2 [0, 1] is a linearly independent set in ๐ฟ2 [0, 1].
(20)
๐=1
(12)
Before we move to the next level, we need the following linear independence on which the rest of theoretical results are based.
(16)
Clearly, one can see that, for any โ N, ๐(๐) =ฬธ 0. Consequently, the determinant given by
๐ (๐ก) โ ๐โ (๐ก) = โ ๐๐ ๐บ๐ (๐ก) = C๐ G (๐ก) ,
(13)
(15)
where C = [๐1 , ๐2 , . . . , ๐๐]๐ , G(๐ก) ๐บ๐(๐ก)]๐ . Using (19), we have
=
[๐บ1 (๐ก), ๐บ2 (๐ก), . . . ,
โจ๐ (๐ก) โ C๐ G (๐ก) , ๐บ๐ (๐ก)โฉ = 0 ๐ = 1, 2, . . . , ๐;
(21)
for simplicity, we write C๐ โจG (๐ก) , G (๐ก)โฉ = โจ๐ (๐ก) , G (๐ก)โฉ ,
(22)
Proof. To show that ๐ด is the set of linearly independent elements of ๐ฟ2 [0, 1], it is enough to show that the Gram determinant is not zero. That is,
where โจG(๐ก), G(๐ก)โฉ is an ๐ ร ๐ matrix. 1 Let ๐ = โจG(๐ก), G(๐ก)โฉ = โซ0 G(๐ก)G๐ (๐ก)๐๐ก. The entries of the matrix ๐ can be calculated from (11). Therefore, any function ๐(๐ก) โ ๐ฟ2 [0, 1] can be expanded by Genocchi polynomials as ๐(๐ก) = C๐ G(๐ก), where
Gram (๐บ1 , ๐บ2 , . . . , ๐บ๐) =ฬธ 0,
C = ๐โ1 โจ๐ (๐ก) , G (๐ก)โฉ .
(14)
(23)
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4. Genocchi Operational Matrix
where 0 [ [2 [ [ [0 [ [0 ๐=[ [ [. [ .. [ [ [0 [ [0
In this section, we derive the operational matrices for the delay and that of fractional derivative based on Genocchi polynomials for the solution of fractional pantograph equations. 4.1. Genocchi Delay Operational Matrix. The Genocchi delay vector G(๐ก โ ๐) can be expressed as G (๐ก โ ๐) = RG (๐ก) ,
(24)
R = ๐1 ๐โ1 0
0
[ 1 0 [ โ2๐ [ [ 3๐2 โ3๐ 1 [ [ [ โ4๐3 6๐2 โ4๐ =[ [ [ 5๐4 โ10๐3 10๐2 [ [ [ . .. .. [ .. . . [
0
0
0
0
0
0
1
0
โ5๐
1
โ
โ
โ
.. .
(25)
0 0 โ
โ
โ
0 0 โ
โ
โ
(28)
(29)
๐ท๐ผ G (๐ก)๐ = ๐๐ผ G (๐ก)๐ ,
(30)
where ๐๐ผ is ๐ ร ๐ operational matrix of fractional derivative of order ๐ผ in Caputo sense and is defined as follows:
1
๐ (๐ก โ ๐) = โ๐๐ ๐บ๐ (๐ก โ ๐) = C RG (๐ก) ,
.. .. . . โ
โ
โ
Theorem 5 (see [21]). Suppose that G(๐ก) is the Genocchi vector given in (20) and let ๐ผ > 0. Then,
where ๐1 = โซ0 G(๐กโ๐)G๐ (๐ก)๐๐ก and ๐๐ (๐) = (โ1)๐โ๐ (( ๐๐ )) ๐๐โ๐ , ๐ = 1, 2, . . . , ๐. Also, for any delay function ๐(๐ก โ ๐), we can express it in terms of Genocchi polynomials as shown in (26):
๐
0 4 โ
โ
โ
In the following theorem, the operational matrix of fractional-order derivative for the Genocchi polynomials is given.
[๐๐ (1) ๐๐ (2) ๐๐ (3) ๐๐ (4) ๐๐ (5) โ
โ
โ
1]
๐
3 0 โ
โ
โ
0 0 ] 0 0 0] ] ] 0 0 0] ] 0 0 0] ]. ] .. .. .. ] . . .] ] ] ๐ โ 1 0 0] ] 0 ๐ 0]
๐ ๐๐ G (๐ก)๐ = G (๐ก) (๐๐ ) . ๐๐ก๐
โ
โ
โ
0
] โ
โ
โ
0] ] โ
โ
โ
0] ] ] โ
โ
โ
0] ], ] โ
โ
โ
0] ] ] .. .. ] . .] ]
0 0 โ
โ
โ
0
Thus, ๐ is ๐ ร ๐ operational matrix of derivative. It is not difficult to show inductively that the ๐th derivative of G(๐ก) can be given by
where R is the ๐ ร ๐ operational delay matrix given by
1
0 0 โ
โ
โ
(26)
๐=1
(๐ผ)
๐
where C is given in (23). The following lemma is also of great importance. Lemma 4. Let ๐บ๐ (๐ก) be the Genocchi polynomials; then ๐ท๐ผ ๐บ๐ (๐ก) = 0, for ๐ = 1, ..., โ๐ผโ โ 1, ๐ผ > 0. The proof of this lemma is obvious; one can use (7) and (8) on (10).
0 [ .. [ [ . [ [ 0 [ [ [ โ๐ผโ [ [โ๐ โ๐ผโ,๐,1 [ [๐=โ๐ผโ [ .. [ =[ . [ [ ๐ [ [ โ๐ [ ๐,๐,1 [ ๐=โ๐ผโ [ [ .. [ . [ [ ๐ [ [ โ ๐๐,๐,1 [ ๐=โ๐ผโ
0
โ
โ
โ
.. .
โ
โ
โ
0
โ
โ
โ
โ๐ผโ
โ ๐โ๐ผโ,๐,2 โ
โ
โ
๐=โ๐ผโ
.. .
โ
โ
โ
โ ๐๐,๐,2
โ
โ
โ
.. .
โ
โ
โ
๐
๐=โ๐ผโ
๐
โ ๐๐,๐,2 โ
โ
โ
๐=โ๐ผโ
0
] ] ] ] ] 0 ] ] ] โ๐ผโ ] โ ๐โ๐ผโ,๐,๐] ] ] ๐=โ๐ผโ ] .. ] ] , (31) . ] ] ๐ ] โ ๐๐,๐,๐ ] ] ] ๐=โ๐ผโ ] ] .. ] . ] ] ๐ ] ] โ ๐๐,๐,๐ ๐=โ๐ผโ ] .. .
where ๐๐,๐,๐ is given by 4.2. Genocchi Operational Matrix of Fractional Derivative. If we consider the Genocchi vector G(๐ก) given by G(๐ก) = [๐บ1 (๐ก), ๐บ2 (๐ก), . . . , ๐บ๐(๐ก)], then the derivative of G(๐ก) with the aid of (12) can be expressed in the matrix form by ๐G (๐ก)๐ = ๐G (๐ก)๐ , ๐๐ก
(27)
๐๐,๐,๐ =
๐!๐บ๐โ๐ ๐. (๐ โ ๐)!ฮ (๐ + 1 โ ๐ผ) ๐
(32)
๐บ๐โ๐ is the Genocchi number and ๐๐ can be obtained from (23). Proof. For the proof, see [21].
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4.3. Upper Bound of the Error for the Operational Matrix of Fractional Derivative ๐๐ผ . We begin here by proving the upper bound of the error of arbitrary function approximation by Genocchi polynomials in the following Lemma.
Theorem 8. Suppose that ๐(๐ก) โ ๐ฟ2 [0, 1] is approximated by ๐๐ (๐ก) as ๐
๐๐ (๐ก) = โ๐๐ ๐บ๐ (๐ก) = C๐ G (๐ก) ;
(38)
๓ตฉ ๓ตฉ lim ๓ตฉ๓ตฉ๐ (๐ก) โ ๐๐ (๐ก)๓ตฉ๓ตฉ๓ตฉ = 0.
(39)
๐=1
๐+1
Lemma 6. Suppose that ๐(๐ก) โ ๐ถ [0, 1] and ๐ = Span{๐บ1 (๐ก), ๐บ2 (๐ก), . . . , ๐บ๐(๐ก)}; if C๐ G(๐ก) is the best approximation of ๐(๐ก) out of ๐, then ๐
๓ตฉ๓ตฉ ๓ตฉ ๓ตฉ๓ตฉ๐ (๐ก) โ C๐ G (๐ก)๓ตฉ๓ตฉ๓ตฉ โค ๓ตฉ ๓ตฉ (๐ + 1)!โ2๐ + 3 ,
(33)
where ๐
= max๐กโ[0,1] |๐(๐+1) (๐ก)|. To see this, we set {1, ๐ก, . . . , ๐ก๐ } as a basis for the polynomial space of degree ๐. Define ๐ฆ1 (๐ก) = ๐(0) + ๐ก๐๓ธ (0) + (๐ก2 /2!)๐๓ธ ๓ธ (0) + โ
โ
โ
+ ๐ (๐ก /๐!)๐(๐) (0). From Taylorโs expansion, one has |๐(๐ก)โ๐ฆ1 (๐ก)| = |(๐ก๐+1 /(๐+ 1)!)๐(๐+1) (๐๐ก )|, where ๐๐ก โ (0, 1). Since C๐ G(๐ก) is the best approximation of ๐(๐ก) out of ๐ and ๐ฆ1 (๐ก) โ ๐, from (18), one has
then, ๐โโ ๓ตฉ
The proof of this theorem obviously follows from Lemma 6. The operational matrix error vector ๐ธ๐ผ is given by ๐ธ๐ผ = ๐๐ผ G (๐ก) โ ๐ท๐ผ G (๐ก) , where ๐1๐ผ
[๐๐ผ ] [ 2] [ ] ๐ธ = [ . ]; [.] [.] ๐ผ
๓ตฉ๓ตฉ ๓ตฉ2 ๓ตฉ๓ตฉ๐ (๐ก) โ C๐ G (๐ก)๓ตฉ๓ตฉ๓ตฉ โค ๓ตฉ๓ตฉ๓ตฉ๓ตฉ๐ (๐ก) โ ๐ฆ1 (๐ก)๓ตฉ๓ตฉ๓ตฉ๓ตฉ22 ๓ตฉ ๓ตฉ2
๐ผ
1
1
2
0
โค
๐ก๐+1 ) (๐ + 1)!
from Theorem 7, we get
๓ตฉ2 ๓ตฉ๓ตฉ (๐+1) ๓ตฉ๓ตฉ๐ (๐๐ก )๓ตฉ๓ตฉ๓ตฉ๓ตฉ ๐๐ก ๓ตฉ
(34)
(35)
๓ตจ๓ตจ ๓ตฉ๓ตฉ ๐ผ ๓ตฉ๓ตฉ ๓ตจ๓ตจ๓ตจ๓ตจ ๐ผ ๓ตฉ๓ตฉ๐๐ ๓ตฉ๓ตฉ = ๓ตจ๓ตจ๐ท ๐บ๐ (๐ก) ๓ตจ๓ตจ ๓ตจ
๓ตจ๓ตจ ๓ตจ๓ตจ ๐!๐บ๐โ๐ ๐๐ ) ๐บ๐ (๐ก)๓ตจ๓ตจ๓ตจ๓ตจ โโ( โ ๓ตจ๓ตจ ๐=1 ๐=โ๐ผโ (๐ โ ๐)!ฮ (๐ + 1 โ ๐ผ) ๓ตจ ๓ตจ ๓ตจ๓ตจ ๐ ๐!๐บ๐โ๐ ๓ตจ โค โ ๐ก๐โ๐ผ ๓ตจ๓ตจ๓ตจ๓ตจ๐0 (๐ก) โ ๐)!ฮ + 1 โ ๐ผ) (๐ (๐ ๓ตจ๓ตจ ๐=โ๐ผโ ๓ตจ ๓ตจ ๐ ๐ ๓ตจ๓ตจ๓ตจ ๐!๐บ๐โ๐ ๐ก๐โ๐ผ โ โ ๐๐ ๐บ๐ (๐ก)๓ตจ๓ตจ๓ตจ๓ตจ โค โ โ ๐)!ฮ + 1 โ ๐ผ) (๐ (๐ ๓ตจ ๓ตจ๓ตจ ๐=โ๐ผโ ๐=1 ๐
We use the following theorem from [22]. Theorem 7 (see [22]). Suppose that ๐ป is a Hilbert space and ๐ is a closed subspace of ๐ป such that dim ๐ < โ and ๐ฆ1 , ๐ฆ2 , . . . , ๐ฆ๐ is a basis for ๐. Let ๐ be an arbitrary element in ๐ป and let ๐ฆ0 be the unique best approximation of ๐ out of ๐. Then, (36)
๐
(43)
1/2
Gram (๐ (๐ก) , ๐บ1 (๐ก) , . . . , ๐บ๐ (๐ก)) ) โ
( Gram (๐บ1 (๐ก) , . . . , ๐บ๐ (๐ก))
where Gram (๐ฆ1 , ๐ฆ2 , . . . , ๐ฆ๐ ) โ
โ
โ
โจ๐ฆ1 , ๐ฆ๐ โฉ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ โ
โ
โ
โจ๐ฆ2 , ๐ฆ๐ โฉ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ .. ๓ตจ๓ตจ๓ตจ๓ตจ . โ
โ
โ
. ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ โ
โ
โ
โจ๐ฆ๐ , ๐ฆ๐ โฉ๓ตจ๓ตจ๓ตจ
.
Thus, according to equations (29) and (30) in [21], one has
which is the desired error bound.
๓ตฉ2 Gram (๐, ๐ฆ1 , ๐ฆ2 , . . . , ๐ฆ๐ ) ๓ตฉ๓ตฉ , ๓ตฉ๓ตฉ๐ โ ๐ฆ0 ๓ตฉ๓ตฉ๓ตฉ = Gram (๐ฆ1 , ๐ฆ2 , . . . , ๐ฆ๐ )
(42)
1/2
Taking the square root of both sides, one has ๐
๓ตฉ๓ตฉ ๓ตฉ ๓ตฉ๓ตฉ๐ (๐ก) โ C๐ G (๐ก)๓ตฉ๓ตฉ๓ตฉ โค ๓ตฉ ๓ตฉ (๐ + 1)!โ2๐ + 3
๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๐ ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐ (๐ก) โ โ๐ ๐บ (๐ก)๓ตฉ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ 0 ๐ ๐ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๐=1 ๓ตฉ ๓ตฉ Gram (๐ (๐ก) , ๐บ1 (๐ก) , . . . , ๐บ๐ (๐ก)) =( ) Gram (๐บ1 (๐ก) , . . . , ๐บ๐ (๐ก))
1 ๐
2 ๐
2 . โซ ๐ก2๐+2 ๐๐ก = 2 ((๐ + 1)!) 0 ((๐ + 1)!)2 (2๐ + 3)
๓ตจ๓ตจโจ๐ฆ , ๐ฆ โฉ โจ๐ฆ , ๐ฆ โฉ 1 2 ๓ตจ๓ตจ 1 1 ๓ตจ๓ตจ ๓ตจ๓ตจโจ๐ฆ , ๐ฆ โฉ โจ๐ฆ , ๐ฆ โฉ 2 2 ๓ตจ๓ตจ 2 1 = ๓ตจ๓ตจ๓ตจ๓ตจ . .. ๓ตจ๓ตจ .. . ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจโจ๐ฆ , ๐ฆ โฉ โจ๐ฆ , ๐ฆ โฉ ๓ตจ ๐ 1 ๐ 2
(41)
[๐๐ ]
๓ตจ2 ๓ตจ โค โซ ๓ตจ๓ตจ๓ตจ๐ (๐ก) โ ๐ฆ1 (๐ก)๓ตจ๓ตจ๓ตจ ๐๐ก 0 =โซ (
(40)
(37)
.
By considering Theorem 8 and (43), we can conclude that by increasing the number of the Genocchi bases the vector ๐๐๐ผ tends to zero. For comparison purpose in Table 1, we show below the errors of operational matrix of fractional derivative based on Genocchi polynomials and shifted Legendre polynomials derived in [23, 24] when ๐ = 10 and ๐ผ = 0.75 at different
6
International Journal of Differential Equations Table 1: Comparison of the operational matrix errors for the GPOMFD and SLPOMFD.
๐ธ๐ผ ๐1 ๐2 ๐3 ๐4 ๐5 ๐6 ๐7 ๐8 ๐9 ๐10
๐ฅ=1 GPOMFD 0.00000 0.01288 0.02204 0.00004 0.03944 0.00292 0.15747 0.00748 1.21090 0.04902
๐ฅ=0 SLPOMFD 0.00000 0.01343 0.03462 0.02916 0.17666 0.33223 2.61748 11.8527 38.04476 806.14232
๐ฅ = 0.5
GPOMFD 0.00000 0.51657 0.77767 0.01241 1.28048 0.02450 5.38767 0.15766 39.19581 1.32181
points on [0, 1]. From this table, it is clear that the accuracy of Genocchi polynomials operational matrix of fractional derivative (GPOMFD) is better than the shifted Legendre polynomials operational matrix of fractional derivatives (SLPOMFD). We believe that this is the case for any value of ๐ because the Genocchi polynomials have smaller coefficients of individual terms compared to shifted Legendre polynomials.
5. Collocation Method Based on Genocchi Operational Matrices
SLPOMFD 0.00000 0.51648 1.56518 3.17608 5.36557 8.00571 15.06874 11.80719 54.74287 629.08480
GPOMFD 0.00000 0.00353 0.00531 0.00008 0.00911 0.00001 0.03839 0.00060 0.27897 0.00519
Substituting (44) and (46) in (1), we have ๐ผ
๐บ (๐ก) (๐๐ ) ๐ถ ๐ฝ ๐โ1
๐ฝ๐
= โ โ ๐๐,๐ (๐ก) ๐บ (๐ ๐,๐ ๐ก + ๐๐,๐ ) (๐๐ ) ๐ถ + ๐ (๐ก) ,
where ๐บ(๐ ๐,๐ ๐ก + ๐๐,๐ ) = [๐บ1 (๐ ๐,๐ ๐ก + ๐๐,๐ ), ๐บ2 (๐ ๐,๐ ๐ก + ๐๐,๐ ), . . . , ๐บ๐(๐ ๐,๐ ๐ก + ๐๐,๐ )]. Also the initial condition will produce ๐ other equations: ๐
โ ๐๐,๐ ๐บ (0) (๐๐ ) ๐ถ = ๐๐ , ๐ = 0, 1, . . . , ๐ โ 1.
To find the solution ๐ฆ๐(๐ก) we collocate (47) at the collocation points ๐ก๐ = ๐/(๐ โ ๐), ๐ = 1, 2, . . . , ๐ โ ๐, to obtain ๐ผ
๐บ (๐ก๐ ) (๐๐ ) ๐ถ ๐ฝ ๐โ1
๐ฆ๐ (๐ก) โ โ ๐๐ ๐บ๐ (๐ก) = ๐บ (๐ก) ๐ถ,
(44)
๐=1
๐บ (๐ก) = [๐บ1 (๐ก) , ๐บ2 (๐ก) , . . . , ๐บ๐ (๐ก)] ;
(45)
for ๐ = 1, 2, . . . , ๐ โ ๐. Additionally, one can also use both the operational matrix of fractional derivative and delay operational matrix to solve problem (1). According to (44), we can approximate the delay function ๐ฆ(๐ ๐,๐ ๐ก๐ + ๐๐,๐ ) and its fractional derivative using the operational matrices P and R as follows:
๐ท๐ฝ๐ ๐ฆ๐ (๐ ๐,๐ ๐ก๐ + ๐๐,๐ ) = R๐ถ๐ ๐๐ฝ๐ ๐บ (๐ก) .
(50)
Putting this approximation together with (44) in (1), we have
๐ผ
๐ท๐ผ ๐ฆ๐ (๐ก) = ๐บ (๐ก) (๐๐ ) ๐ถ, ๐ฝ๐
(49)
๐=0 ๐=0
๐ฆ๐ (๐ ๐,๐ ๐ก๐ + ๐๐,๐ ) = R๐ถ๐ ๐บ (๐ก) ,
thus, ๐ท๐ผ ๐ฆ๐(๐ก) and ๐ท๐ฝ๐ ๐ฆ๐(๐ก), ๐ = 0, 1, . . . , ๐ โ 1, can be expressed, respectively, as follows:
๐ท๐ฝ๐ ๐ฆ๐ (๐ก) = ๐บ (๐ก) (๐๐ ) ๐ถ, ๐ = 0, 1, . . . , ๐ โ 1.
๐ฝ๐
+ ๐ (๐ก๐ )
where the Genocchi coefficient vector ๐ถ and the Genocchi vector ๐บ(๐ก) are given by ๐ถ๐ = [๐1 , ๐2 , . . . , ๐๐] ,
(48)
๐=0
= โ โ ๐๐,๐ (๐ก๐ ) ๐บ (๐ ๐,๐ ๐ก๐ + ๐๐,๐ ) (๐๐ ) ๐ถ
๐
(47)
๐=0 ๐=0
๐โ1
In this section, we use the collocation method based on Genocchi operational matrix of fractional derivatives and Genocchi delay operational matrix to solve numerically the generalized fractional pantograph equation. We now derive an algorithm for solving (1). To do this, let the solution of (1) be approximated by the first ๐ terms Genocchi polynomials. Thus, we write
SLPOMFD 0.00000 0.00345 0.01047 0.02487 0.02570 0.05647 0.04226 0.92194 15.64398 20.30772
(46)
๐ผ
๐ฝ ๐โ1
๐บ (๐ก) (๐๐ ) ๐ถ = โ โ ๐๐,๐ (๐ก) R๐ถ๐ ๐๐ฝ๐ ๐บ (๐ก) + ๐ (๐ก) . ๐=0 ๐=0
(51)
International Journal of Differential Equations
7
Table 2: Comparison errors obtained by the present method and those obtained in [25] when ๐ผ = 0.6 and ๐ = 0.3 for Example 1. ๐ก 0.2 0.4 0.6 0.8 1.0
Error (new method) [25] 0.0781197 0.129928 0.190687 0.248601 0.307649
Error (FAM) [25] 0.078155 0.129978 0.19076 0.248694 0.307763
Thus, collocating (51) at the same collocation point as that in (47), we get ๐ ๐ผ
๐ฝ ๐โ1
subject to ๐ฆ (0) = 0, ๐ฆ๓ธ (0) = 0,
๐ ๐ฝ๐
๐บ (๐ก๐ ) (๐ ) ๐ถ = โ โ ๐๐,๐ (๐ก๐ ) R๐ถ ๐ ๐บ (๐ก๐ ) ๐=0 ๐=0
(56)
๐ฆ๓ธ ๓ธ (0) = 0,
(52)
+ ๐ (๐ก๐ ) .
where
Hence, (49) or (52) is ๐ โ ๐ nonlinear algebraic equation. Any of these equations together with (48) makes ๐ algebraic equations which can be solved using Newtonโs iterative method. Consequently, ๐ฆ๐(๐ฅ) given in (44) can be calculated.
6. Numerical Examples
๐ (๐ก) =
(57)
5/2
๐ถ = โ๐บ (๐ก) ๐ถ + ๐บ (๐ก โ 0.5) ๐ถ + ๐ (๐ก) .
(58)
Also from the initial conditions we have ๐บ (0) ๐ถ = 0, ๐บ (0) (๐๐ ) ๐ถ = 0,
Example 1. Consider the following example solved in [25]: 2 ๐ฆ1โ๐ผ/2 (๐ก) + ๐ฆ (๐ก โ ๐) โ ๐ฆ (๐ก) ฮ (3 โ ๐ผ)
ฮ (4) 1/2 3 ๐ก + ๐ก + (๐ก โ 0.5)3 . ฮ (3/2)
The exact solution of this problem is known to be ๐ฆ(๐ก) = ๐ก3 . This problem is solved in [1] using generalized LaguerreGauss collocation scheme. We apply our technique with ๐ = 4. Approximating (55) with Genocchi polynomials, we have ๐บ (๐ก) (๐๐ )
In this section, some numerical examples are given to illustrate the applicability and accuracy of the proposed method. All the numerical computations have been done using Maple 18.
๐ท๐ผ ๐ฆ (๐ก) =
Error (present method) 3.37154๐ธ โ 03 6.52102๐ธ โ 03 9.77309๐ธ โ 03 1.30349๐ธ โ 02 1.64103๐ธ โ 02
(59)
2
๐บ (0) (๐๐ ) ๐ถ = 0. Thus, collocating (58) at ๐ก = 0.267339, we get
(53)
+ 2๐โ๐ฆ (๐ก) โ ๐2
โ 3.507078326 + 15.27479180๐4 + 2๐1 + 0.2727646328๐3 โ 1.930640400๐2 = 0,
(60)
and (59) gives
subject to ๐ฆ (๐ก) = 0,
๐ก โค 0.
๐1 โ ๐2 + ๐4 = 0, (54)
2๐1 โ 3๐2 = 0,
The exact solution for this example is given by ๐ฆ(๐ก) = ๐ก . We solve the example when ๐ผ = 0.6 and ๐ = 0.3. In Table 2, we compare the errors obtained by our method with those obtained using FAM and new approach in [25]. As reported in [25], the time required for the new method is 104.343750 seconds and for the FAM the time taken is 215.031250 seconds for completing the same task, whereas in our method we only need 38.080 seconds to complete the computations.
6๐3 โ 12๐4 = 0.
2
Example 2 (see [1]). Consider the following generalized fractional pantograph equation: ๐ท5/2 ๐ฆ (๐ก) = โ๐ฆ (๐ก) โ ๐ฆ (๐ก โ 0.5) + ๐ (๐ก) , ๐ก โ [0, 1]
(55)
(61)
Solving these equations, we have ๐1 = 0.4999969148, ๐2 = 0.7499953722, ๐3 = 0.4999969148,
(62)
๐4 = 0.2499984574. Thus, ๐ฆ(๐ก) = ๐บ(๐ฅ)๐ถ is calculated and we have 0.9999938296๐ก3 which is almost the exact solution. In Table 3, we compare the absolute errors obtained by our method (with only few terms ๐ = 4) and the absolute errors obtained in [1] when ๐ = 22 with different Laguerre parameters ๐ฝ.
8
International Journal of Differential Equations Table 3: Comparison of the absolute errors obtained by the present method and those obtained in [1] for Example 2.
๐ก
๐ = 22 [1] ๐ฝ=3 1.019๐ธ โ 05 6.051๐ธ โ 05 1.495๐ธ โ 04 2.559๐ธ โ 04 3.546๐ธ โ 04 4.261๐ธ โ 04 4.592๐ธ โ 04 4.510๐ธ โ 04 4.055๐ธ โ 04 3.311๐ธ โ 04
๐ฝ=2 1.030๐ธ โ 04 6.510๐ธ โ 04 1.740๐ธ โ 03 3.283๐ธ โ 03 5.138๐ธ โ 03 7.175๐ธ โ 03 9.303๐ธ โ 03 1.147๐ธ โ 02 1.367๐ธ โ 02 1.589๐ธ โ 02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
๐ฝ=5 6.273๐ธ โ 06 3.892๐ธ โ 05 1.023๐ธ โ 04 1.901๐ธ โ 04 2.944๐ธ โ 04 4.088๐ธ โ 04 5.306๐ธ โ 04 6.597๐ธ โ 04 7.977๐ธ โ 04 9.468๐ธ โ 04
๐=4 Present method 6.17040๐ธ โ 09 4.93630๐ธ โ 08 1.66600๐ธ โ 07 3.94910๐ธ โ 07 7.71300๐ธ โ 07 1.33280๐ธ โ 06 2.11640๐ธ โ 06 3.15920๐ธ โ 06 4.49820๐ธ โ 06 6.17040๐ธ โ 06
Table 4: Comparison of the absolute errors obtained by the present method and those in [26] for Example 4. ๐ก 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Absolute error [26] ๐ = 4 0.100๐ธ โ 07 0.115๐ธ โ 07 0.115๐ธ โ 07 0.107๐ธ โ 07 0.967๐ธ โ 08 0.811๐ธ โ 08 0.641๐ธ โ 08 0.440๐ธ โ 08 0.223๐ธ โ 08 0.372๐ธ โ 09
Example 3. Consider the following fractional pantograph equation: ๐ท1/2 ๐ฆ (๐ก) = 2๐ฆ (
3๐ก 8๐ก3/2 9๐ก2 โ )+ , ๐ก โ [0, 1] 2 2 3โ๐
(63)
Absolute error (present method) ๐ = 3 1.15000๐ธ โ 09 2.00000๐ธ โ 09 2.55000๐ธ โ 09 2.80000๐ธ โ 09 2.70000๐ธ โ 09 2.40000๐ธ โ 09 1.70000๐ธ โ 09 8.00000๐ธ โ 10 5.00000๐ธ โ 10 2.00000๐ธ โ 09
Example 4. Consider the following fractional pantograph equation solved in [26]: ๐ท2 ๐ฆ (๐ก) + ๐ท3/2 ๐ฆ (๐ก) + ๐ฆ (๐ก) ๐ก 3๐ก2 ๐ก + 4โ + 2, ๐ก โ [0, 1] = ๐ฆ( ) + 2 4 ๐
subject to ๐ฆ (0) = 0, ๐ฆ (1) = 1.
subject to (64)
The exact solution of this problem is known to be ๐ฆ(๐ก) = ๐ก2 . We solve (63) using our technique with ๐ = 3 only. As in Example 2, we obtained the values of the coefficients to be 1 ๐1 = , 2 1 ๐2 = , 2
(66)
๐ฆ (0) = 0, ๐ฆ (1) = 1.
(67)
The exact solution of this problem is known to be ๐ฆ(๐ก) = ๐ก2 . As in Example 3, we solve (66) using our technique with ๐ = 3 and the values of the coefficients obtained are ๐1 = 0.5000000011,
(65)
1 ๐3 = . 3 Thus, ๐ฆ๐(๐ก) = ๐บ(๐ก)๐ถ is calculated to be ๐ก2 which is the exact solution and so there is nothing to compare for the error is zero.
๐2 = 0.5000000011,
(68)
๐3 = 0.3333333384. Thus, ๐ฆ๐(๐ก) = ๐บ(๐ก)๐ถ is calculated and compared with the exact solution. This problem is solved using Taylor collocation method in [26] when ๐ = 4, 5, and 6. In Table 4, we compare the absolute errors obtained by present method when ๐ = 3 with the errors obtained when ๐ = 4 in [26].
International Journal of Differential Equations
9
7. Conclusion In this paper, a collocation method based on the Genocchi delay operational matrix and the operational matrix of fractional derivative for solving generalized fractional pantograph equations is presented. The comparison of the results shows that the present method is an excellent mathematical tool for finding the numerical solutions delay equation. The advantage of the method over others is that only few terms are needed and every operational matrix involves more numbers of zeroes; as such the method has less computational complexity and provides the solution at high accuracy.
[9]
[10]
[11]
Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this manuscript.
Acknowledgments This work was supported in part by MOE-UTHM FRGS Grant Vot 1433. The authors also acknowledge financial support from UTHM through GIPS U060. The third author would like to acknowledge Faculty of Computer Science and Information Technology, UNIMAS, for providing continuous support.
[12]
[13]
[14]
[15]
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