Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2014, Article ID 456501, 11 pages http://dx.doi.org/10.1155/2014/456501
Research Article New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems W. M. Abd-Elhameed1,2 1 2
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Correspondence should be addressed to W. M. Abd-Elhameed; walee
[email protected] Received 29 April 2014; Accepted 27 August 2014; Published 14 October 2014 Academic Editor: Fazlollah Soleymani Copyright Β© 2014 W. M. Abd-Elhameed. 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. This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.
1. Introduction Classical orthogonal polynomials are successfully employed for solving ordinary and partial differential equations in spectral and pseudospectral methods (see, for instance, [1β 3]). In particular, the class of Jacobi polynomials ππ(πΌ,π½) (π₯) plays prominent roles in the applications of mathematical analysis. For example, Doha et al. in [4] used these polynomials for solving some odd-order boundary value problems (BVPs). The suggested algorithms in this paper are based on selecting certain combinations of Jacobi polynomials satisfying the boundary conditions of the given differential equation. There are some other articles in literature that have extensive theoretical and numerical studies about Jacobi polynomials (see, e.g., [5, 6]). It is well known that the classical Jacobi polynomials have two parameters. Of course, special choices of their parameters give special kinds of these polynomials. It is worthy to mention here that the classical Jacobi polynomials have six special well-known kinds of
orthogonal polynomials; they are Legendre, ultraspherical, and Chebyshev polynomials of the four kinds. All six special kinds of Jacobi polynomials have their roles and importance from both theoretical and practical points of view (see, e.g., [7β11]). The spectral methods have prominent roles in various applications such as fluid dynamics. These methods are global methods. There are three popular methods of spectral methods; they are tau, collocation, and Galerkin methods (see, e.g., [12, 13]). The choice of the suitable used spectral method suggested for solving the given equation depends certainly on the type of the differential equation and also on the type of the boundary conditions governed by it. The explicit formulae for the high-order derivatives of various orthogonal polynomials in terms of their original polynomials are useful when spectral and pseudospectral methods are used for obtaining numerical solutions of various types of differential equations. Formula for the highorder derivatives for Chebyshev polynomials in terms of
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their original polynomials is given in Karageorghis [14]. The corresponding formulae for Legendre, ultraspherical, and Jacobi polynomials are given, respectively, in Phillips [15] and Doha [16]. Due to their great importance in various applications, high even-order BVPs have been investigated by a large number of authors. For example, sixth-order BVPs are known to arise in astrophysics; the narrow convecting layers bounded by stable layers, which are believed to surround A-type stars, may be modeled by sixth-order BVPs. Sixth-order BVPs were handled by numerous numerical techniques. For example, Doha and Abd-Elhameed in [17] have developed efficient solutions of multidimensional sixth-order BVPs based on employing symmetric generalized Jacobi-Galerkin method. Also, Bhrawy et al. in [18] have developed an extension of the Legendre-Galerkin method for handling sixth-order BVPs with variable polynomial coefficients. There are other contributions concerning sixth-order BVPs; among the techniques used for handling this kind of BVPs are sinc-Galerkin method [19], nonpolynomial spline technique in [20], spline collocation method in [21], Adomian decomposition method with Greenβs function in [22], parametric quintic spline solution in [23], homotopy perturbation method in [24], and fourth order finite difference method in [25]. For extensive studies about the existence and uniqueness of solutions of such problems, the interested reader can be refered to the important book of Agarwal [26]. For other studies in high even- and high odd-order BVPs, see, for example, [27β31]. The main objective of this paper is twofold: (i) developing new formulae for the high-order derivatives of some Jacobi polynomials with certain parameters, (ii) employing the new introduced formulae in solving some sixth-order BVPs based on applying a spectral Galerkin method. The rest of the paper is as follows. In the next section, some useful properties of Jacobi polynomials are presented. In Section 3, we derive two new formulae which give explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one in terms of their original Jacobi polynomials. In Section 4, we give other two new formulae which give explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is two in terms of their original Jacobi polynomials. Some new reduction formulae for summing some terminating hypergeometric functions of the type 3 πΉ2 (1) are deduced in Section 5. Section 6 is devoted to implementing and presenting a Galerkin algorithm for numerically solving ceratin sixth-order BVPs including a numerical example aiming to illustrate the accuracy and the efficiency of the proposed algorithm. Finally, conclusions are given in Section 7.
2. Some Properties of Jacobi Polynomials The classical Jacobi polynomials associated with the real parameters (πΌ > β1, π½ > β1) and the weight function
π€(π₯) = (1 β π₯)πΌ (1 + π₯)π½ (see, e.g., [32, 33]) are a sequence (πΌ,π½) of polynomials ππ (π₯) (π = 0, 1, 2, . . .), each, respectively, of degree π. These polynomials have the following Gauss hypergeometric form: (πΌ,π½)
ππ
(πΌ + 1)π
(π₯) =
σ΅¨σ΅¨ 1 β π₯ (βπ, π + π; πΌ + 1 σ΅¨σ΅¨σ΅¨σ΅¨ ) σ΅¨ 2
2 πΉ1
π!
(β1)π (π½ + 1)π
=
π!
2 πΉ1
σ΅¨σ΅¨ 1 + π₯ (βπ, π + π; π½ + 1 σ΅¨σ΅¨σ΅¨σ΅¨ ), σ΅¨ 2 (1)
where (π§)π =
π = πΌ + π½ + 1,
Ξ (π§ + π) . Ξ (π§)
(2)
It is clear that (π½,πΌ)
(πΌ,π½)
(βπ₯) = (β1)π ππ
ππ (πΌ,π½) ππ
(1) =
(πΌ + 1)π π!
,
(πΌ,π½) ππ
(π₯) ,
(β1) =
(3)
(β1)π (π½ + 1)π π!
. (4)
The Jacobi polynomials may be generated using the recurrence relation (πΌ,π½)
2 (π + 1) (π + π) (2π + π β 1) ππ+1 (π₯) (πΌ,π½)
= (2π + π β 1)3 π₯ππ
(πΌ,π½)
(π₯) + (πΌ2 β π½2 ) (2π + π) ππ
(π₯)
(πΌ,π½)
β 2 (π + πΌ) (π + π½) (2π + π + 1) ππβ1 (π₯) , (5) (πΌ,π½)
(πΌ,π½)
starting from π0 (π₯) = 1 and π1 (π₯) = (1/2)[πΌ β π½ + (π + 1)π₯], or obtained alternatively from Rodrigueβs formula (πΌ,π½)
ππ
(π₯) =
(β1)π (1 β π₯)βπΌ (1 + π₯)βπ½ 2π π! ππ Γ π [(1 β π₯)πΌ+π (1 + π₯)π½+π ] . ππ₯
(πΌ,π½)
The polynomials ππ
(6)
(π₯) satisfy the orthogonality relation
1
(πΌ,π½)
β« (1 β π₯)πΌ (1 + π₯)π½ ππ β1
(πΌ,π½)
(π₯) ππ
(π₯) ππ₯
0, π =ΜΈ π, = { (πΌ,π½) βπ , π = π,
(7)
where (πΌ,π½)
βπ
=
2πΌ+π½+1 Ξ (π + πΌ + 1) Ξ (π + π½ + 1) . (2π + πΌ + π½ + 1) π!Ξ (π + πΌ + π½ + 1)
(8)
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The following structure formula is useful in the sequel (see Rainville [34]): (πΌ,π½)
ππ
(π₯) =
2 (π + π β 1) (2π + π β 1)3 (πΌ,π½)
Γ [(π + π β 1)2 (2π + π β 1) π·ππ+1 (π₯) (πΌ,π½)
+ (πΌ β π½) (π + π β 1) (2π + π) π·ππ
(π₯)
(πΌ,π½)
β (π + πΌ) (π + π½) (2π + π + 1) π·ππβ1 (π₯)] ,
where ππ,π =
(2π β 2π + 2πΌ + 1) Ξ (π + πΌ + 1) Ξ (π β 2π + 2πΌ + 1) , Ξ (π + 2πΌ + 2) Ξ (π β 2π + πΌ)
ππ,π =
2 (π + 1) Ξ (π + πΌ + 1) Ξ (π β 2π + 2πΌ) , Ξ (π + 2πΌ + 2) Ξ (π β 2π + πΌ β 1)
and βπ§β denotes the largest integer less than or equal to π§. Proof. Let us denote π
π β₯ 1. (9)
ππ (π₯) = β ππ ππ(πΌ,πΌ+1) (π₯) ,
(14)
πΌπ+1 (π₯) = β« ππ (π₯) ππ₯.
(15)
π=0
Also, the following theorem is of important use hereafter. Theorem 1. For all π β₯ 1, the πth derivative of the Jacobi (πΌ,π½) polynomial ππ (π₯) is given explicitly by (πΌ,π½) π·π ππ
(π₯) =
(πΌ,π½) β ππ,π,πΌ,π½,π ππ π=0
The integration of both sides of the structure formula (9) (for the case π½ = πΌ + 1) yields (πΌ,πΌ+1) β« ππ(πΌ,πΌ+1) (π₯) ππ₯ = πΌπ ππ+1 (π₯) + π½π ππ(πΌ,πΌ+1) (π₯)
πβπ
(π₯) ,
(10) +
where
(13)
(πΌ,πΌ+1) πΎπ ππβ1
(π₯) ,
(16)
π β₯ 1,
where
ππ,π,πΌ,π½,π =
(π + π)π (π + π + π)π (π + π + πΌ + 1)πβπβπ Ξ (π + π) 2π (π β π β π)!Ξ (2π + π) Γ 3 πΉ2 (βπ + π + π, π + π + π + π, π + πΌ + 1; π + π + πΌ + 1, 2π + π + 1 | 1) . (11)
π + 2πΌ + 2 , (π + πΌ + 1) (2π + 2πΌ + 3)
π½π =
β2 , (2π + 2πΌ + 1) (2π + 2πΌ + 3)
πΎπ =
β (π + πΌ) , (π + 2πΌ + 1) (2π + 2πΌ + 1)
Remark 2. Although the 3 πΉ2 (1) in formula (11) is terminating, it cannot be summed in a closed form except for certain special choices of the two parameters πΌ and π½. In case of πΌ = π½, this 3 πΉ2 (1) can be summed in a closed form with the aid of Watsonβs identity (see Doha [16]).
ππ(πΌ,πΌ+1) (π₯)
πΌπ+1 (π₯) = πΌ0 β« π0(πΌ,πΌ+1) (π₯) ππ₯ π
(πΌ,πΌ+1) + β ππ [πΌπ ππ+1 (π₯) + π½π ππ(πΌ,πΌ+1) (π₯)
and
The last equation may be written alternatively in the form π+1
πΌπ+1 (π₯) = β π΄ π ππ(πΌ,πΌ+1) (π₯) ,
(π₯) =
β
π=0
where ππβ1 πΌπβ1 + ππ π½π + πΎπ+1 ππ+1 = π΄ π , π = π + 1, π, . . . , 1,
π=0
(20)
ππ+1 = ππ+2 = 0. The difference equation (20) can be solved to give (πΌ,πΌ+1) ππ,π ππβ2πβ1
βπ/2ββ1
+ β
(19)
π=0
Lemma 3. For all π β₯ 1, one has β(πβ1)/2β
(18)
π=1
(πΌ,πΌ+1) +πΎπ ππβ1 (π₯)] .
The main objective of this section is to state and prove two theorems in which the derivatives of the Jacobi polynomials ππ(πΌ,πΌ+1) (π₯) and ππ(πΌ+1,πΌ) (π₯) are given in terms of their corresponding Jacobi polynomials. We first state and prove a lemma in which the first derivative of ππ(πΌ,πΌ+1) (π₯) is expressed in terms of their original polynomials.
π·ππ(πΌ,πΌ+1)
(17)
which in turn implies that
(For a proof of Theorem 1, see Doha [16].)
3. High-Order Derivatives of ππ(πΌ+1,πΌ) (π₯)
πΌπ =
(π₯) (12)
(πΌ,πΌ+1) ππ,π ππβ2πβ2
(π₯) ,
ππ =
π+1
π+1
π=π+1 (π+π) odd
π=π+2 (π+π) even
β ππ,π π΄ π +
β
ππ,π π΄ π ,
π = 0, 1, . . . , π,
(21)
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where
where
ππ,π =
(π + π + 2πΌ + 2) Ξ (π + πΌ + 1) Ξ (π + 2πΌ + 2) , (22) Ξ (π + 2πΌ + 2) Ξ (π + πΌ + 1)
ππ,π =
(2π β 2π + 2πΌ + 1) Ξ (π + πΌ + 1) Ξ (π β 2π + 2πΌ + 1) , Ξ (π + 2πΌ + 2) Ξ (π β 2π + πΌ)
ππ,π =
(π β π) Ξ (π + πΌ + 1) Ξ (π + 2πΌ + 2) . Ξ (π + 2πΌ + 2) Ξ (π + πΌ + 1)
ππ,π =
2 (π + 1) Ξ (π + πΌ + 1) Ξ (π β 2π + 2πΌ) , Ξ (π + 2πΌ + 2) Ξ (π β 2π + πΌ β 1)
(23)
If we substitute (21) into (14), we get
and this completes the proof of of Lemma 3.
ππ (π₯) π { π+1 } { π+1 } = β { β ππ,π π΄ π + β ππ,π π΄ π , } ππ(πΌ,πΌ+1) (π₯) . { π=π+1 } π=0 π=π+2 (π+π) even {(π+π) odd } (24)
On the other hand, if we differentiate (19) with respect to π₯, we have ππ (π₯) =
π+1
β π΄ π π·ππ(πΌ,πΌ+1) π=0
(π₯) .
(25)
Now, we extend the result of Lemma 3 to give the πthderivative of the Jacobi polynomials ππ(πΌ,πΌ+1) (π₯) in terms of their original polynomials. The general formula is stated in the following theorem. Theorem 4. For all π β₯ 1, the πth-derivative of the classical Jacobi polynomials ππ(πΌ,πΌ+1) (π₯) of any degree and for any order in terms of their corresponding Jacobi polynomials is given explicitly by π·π ππ(πΌ,πΌ+1) (π₯) =
π { π+1 } { π+1 } β { β ππ,π π΄ π + β ππ,π π΄ π , } ππ(πΌ,πΌ+1) (π₯) { π=π+1 } π=0 π=π+2 (π+π) even {(π+π) odd }
(26)
π=0
+
β π=0
(31) (πΌ,πΌ+1) π΅π,π,π ππβ2πβπβ1 (π₯) ,
3 π΄ π,π,π = (2π (π + π β 1)!Ξ (π + πΌ + 1) Ξ (π β π + πΌ + ) 2 Γ Ξ (π β π β 2π + 2πΌ + 2))
π+1 π=0
Expanding the left-hand side of (26) and collecting the similar terms, (26) may be writtenβafter some rather manipulationβin the form β π΄ π π΅π (π₯) =
(πΌ,πΌ+1) β π΄ π,π,π ππβ2πβπ (π₯)
where
= β π΄ π π·ππ(πΌ,π+1) (π₯) .
π=0
β(πβπ)/2β
β(πβπβ1)/2β
Equating the two right-hand sides of the two relations (24) and (25) yields
π+1
(30)
π+1
β π΄ π π·ππ(πΌ,π+1) π=0
(π₯) ,
(27)
where
Γ ((π β 1)!π!Ξ (π + 2πΌ + 2) Ξ (π β π β 2π + πΌ + 1) 3 β1 Γ Ξ (π β π β π + πΌ + )) , 2
(32)
1 π΅π,π,π = (2π (π + π)!Ξ (π + πΌ + 1) Ξ (π β π + πΌ + ) 2 Γ Ξ (π β π β 2π + 2πΌ + 1))
π΅π (π₯) =
β(πβ1)/2β
(33)
(πΌ,πΌ+1) β ππ,πβ2πβ1 ππβ2πβ1 (π₯)
π=0
Γ ((π β 1)!π!Ξ (π + 2πΌ + 2) Ξ (π β π β 2π + πΌ) (28)
βπ/2ββ1
(πΌ,πΌ+1) + β ππ,πβ2πβ2 ππβ2πβ2 (π₯) , π=0
and ππ,π , ππ,π are given by (22) and (23), respectively. This immediately yields π·ππ(πΌ,πΌ+1) (π₯) =
β(πβ1)/2β
βπ/2ββ1
(πΌ,πΌ+1) + β ππ,π ππβ2πβ2 (π₯) , π=0
Proof. We proceed by induction on π. For π = 1, Theorem 4 reduces identically to Lemma 3. Assume that relation (31) holds, and we have to show that π·π+1 ππ(πΌ,πΌ+1) (π₯) =
(πΌ,πΌ+1) β ππ,π ππβ2πβ1 (π₯)
π=0
3 β1 Γ Ξ (π β π β π + πΌ + )) . 2
β(πβπβ1)/2β
β π=0
(29)
(πΌ,πΌ+1) π΄ π,π,π+1 ππβ2πβπβ1 (π₯)
β(πβπ)/2ββ1
+
β π=0
(34) (πΌ,πΌ+1) π΅π,π,π+1 ππβ2πβπβ2 (π₯) .
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5 π
If we differentiate relation (31), then we get π·π+1 ππ(πΌ,πΌ+1) (π₯) =
Γ β (((π + π β 1)! (2π β 4π β 2π + 2πΌ + 1) π=0
β(πβπ)/2β
(πΌ,πΌ+1) β π΄ π,π,π π·ππβ2πβπ (π₯)
π=0
β(πβπβ1)/2β
+
β π=0
1 Γ Ξ (π β π + πΌ + )) 2
(35)
3 β1 Γ (π!Ξ (π β π β π + πΌ + )) ) , 2
(πΌ,πΌ+1) π΅π,π,π π·ππβ2πβπβ1 (π₯) ,
π΅π,π,π = (2π (π + π + 1) Ξ (π + πΌ + 1)
and in virtue of Lemma 3, one can write π·π+1 ππ(πΌ,πΌ+1) (π₯) = β + β, 1
Γ Ξ (π β π β 2π + 2πΌ) )
(36)
2
Γ ((π β 1)!Ξ(π + 2πΌ + 2)Ξ(π β π β 2π + πΌ β 1))
where
β1
π
β
Γ β (((π + π β 1)! (2π β 4π β 2π + 2πΌ + 1)
1
π=0 β(πβπ)/2β
=
β(πβπβ1)/2ββπ
β π΄ π,π,π
β
π=0
π =0
β(πβπβ1)/2β
β(πβπβ1)/2ββπβ1
+
β π=0
π΅π,π,π
1 Γ Ξ (π β π + πΌ + )) 2
(πΌ,πΌ+1) ππβπβ2π,π ππβπβ2πβ2π β1 (π₯)
β π =0
(πΌ,πΌ+1) ππβπβ2πβ1,π ππβπβ2πβ2π β3 (π₯) ,
β β(πβπ)/2β
β(πβπ)/2ββπβ1
β π΄ π,π,π
+
β
(πΌ,πΌ+1) ππβπβ2π,π ππβπβ2πβ2π β2 (π₯)
π=0
π =0
β(πβπβ1)/2β
β(πβπ)/2ββπβ1
β π=0
π΅π,π,π
β π =0
Lemma 5. For every nonnegative integer π, one has
(πΌ,πΌ+1) ππβπβ2πβ1,π ππβπβ2πβ2π β2 (π₯) .
(37) If we expand β1 and β2 and collecting similar terms and after some lengthy manipulations, we get π·π+1 ππ(πΌ,πΌ+1) (π₯) =
β(πβπβ1)/2β
β π=0
+
β π=0
π
(π + π β 1)! (2π β 4π β 2π + 2πΌ + 1) Ξ (π β π + πΌ + 1/2) π!Ξ (π β π β π + πΌ + 3/2) π=0 β
=
2 (π + π)!Ξ (π β π + πΌ + 1/2) . ππ!Ξ (π β π β π + πΌ + 1/2) (41)
(πΌ,πΌ+1) π΄π,π,π ππβ2πβπβ1 (π₯)
β(πβπ)/2ββ1
(38) (πΌ,πΌ+1) π΅π,π,π ππβ2πβπβ2 (π₯) ,
Now, the application of Lemma 5 in (40) yields 3 π΄π,π,π = (2π+1 (π + π)!Ξ (π + πΌ + 1) Ξ (π β π + πΌ + ) 2
where
Γ Ξ (π β π β 2π + 2πΌ + 1))
π
π΄π,π,π = β {π΄ π,π,π ππβπβ2π,πβπ + π΅π,π,π ππβπβ2πβ1,πβπβ1 } , π=0
(39)
π
π΅π,π,π = β {π΄ π,π,π ππβπβ2π,πβπ + π΅π,π,π ππβπβ2πβ1,πβπ } .
Γ (π!π!Ξ (π + 2πΌ + 2) Ξ (π β π β 2π + πΌ) 1 β1 Γ Ξ (π β π β π + πΌ + )) 2
π=0
It is not difficult to show that
= π΄ π,π,π+1 ,
π΄π,π,π = (2πβ1 (2π β 2π + 2πΌ + 1) Ξ (π + πΌ + 1)
1 π΅π,π,π = (2π+1 (π + π + 1)!Ξ (π + πΌ + 1) Ξ (π β π + πΌ + ) 2
Γ Ξ (π β π β 2π + 2πΌ + 1)) Γ ((π β 1)!Ξ (π + 2πΌ + 2) Ξ (π β π β 2π + πΌ))
(40)
To complete the proof of Theorem 4, the following lemmaβwhich can be easily proved by inductionβis required.
2
=
3 β1 Γ (π!Ξ (π β π β π + πΌ + )) ) . 2
β1
Γ Ξ (π β π β 2π + 2πΌ))
6
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Γ (π!π!Ξ (π + 2πΌ + 2) Ξ (π β π β 2π + πΌ β 1) 1 β1 Γ Ξ (π β π β π + πΌ + )) 2 = π΅π,π,π+1 . (42)
π·π ππ (π₯) =
β(πβπ)/2β
2π (π β 1)!
This proves formula (34) and hence completes the proof of Theorem 4.
+
Remark 6. It is worthy to note here that relation (31) may be written alternatively in the equivalent form
Γ
β π=0
2π (π β 1)! β(πβπβ1)/2β
β π=0
πβπ
π·π ππ(πΌ,πΌ+1) (π₯) = β πΈπ,π,π ππ(πΌ,πΌ+1) (π₯) ,
(π β π)! (π + π β 1)! ππβ2πβπ (π₯) π! (π β π β π)!
(π β π β 1)! (π + π)! ππβ2πβπβ1 (π₯) . π! (π β π β π)! (47)
(43)
π=0
Corollary 9. For all π β₯ 1, the πth-order derivative of ππ (π₯) of any degree and for any order in terms of their corresponding polynomials is given explicitly by
where πΈπ,π,π =
2π Ξ (π + πΌ + 1) Ξ (π + 2πΌ + 2) (π β 1)!Ξ (π + πΌ + 1) Ξ (π + 2πΌ + 2)
π·π ππ (π₯) =
((π β π + π β 2) /2)!Ξ ((π + π + π + 2πΌ + 3) /2) { , { { { ((π β π β π) /2)!Ξ ((π + π β π + 2πΌ + 3) /2) { { { { { (π + π + π) even, { { Γ{ { ((π β π + π β 1) /2)!Ξ ((π + π + π + 2πΌ + 2) /2) { { { , { { { ((π β π β π β 1) /2)!Ξ ((π + π β π + 2πΌ + 4) /2) { { { (π + π + π) odd. {
β(πβπ)/2β
2π (π β 1)! β
β π=0
2π (π β 1)! β(πβπβ1)/2β
Γ
(π β π)! (π + π β 1)! ππβ2πβπ (π₯) π! (π β π β π)!
β π=0
(π β π β 1)! (π + π)! ππβ2πβπβ1 (π₯) . π! (π β π β π)! (48)
(44) Remark 7. As a direct consequence of relation (3), the πthderivative of ππ(πΌ+1,πΌ) (π₯) is given by π·π ππ(πΌ+1,πΌ) (π₯) =
β(πβπ)/2β
(πΌ+1,πΌ) β π΄ π,π,π ππβ2πβπ (π₯)
π=0
(45) β(πβπβ1)/2β
β
β π=0
(πΌ+1,πΌ) π΅π,π,π ππβ2πβπβ1 (π₯) ,
where π΄ π,π,π and π΅π,π,π are given by (32) and (33), respectively, or alternatively in the form πβπ
π·π ππ(πΌ+1,πΌ) (π₯) = β (β1)π+π+π πΈπ,π,π ππ(πΌ+1,πΌ) (π₯) ,
(46)
4. High-Order Derivatives of ππ(πΌ,πΌ+2) (π₯) and ππ(πΌ+2,πΌ) (π₯) In this section, and following similar procedures to those followed in Section 3, we can obtain new derivatives formulae for the high-order derivatives of the Jacobi polynomials ππ(πΌ,πΌ+2) (π₯) and ππ(πΌ+2,πΌ) (π₯) in terms of their corresponding Jacobi polynomials. The details of the required computations are lengthy and will not be given here. Theorem 10. For all π β₯ 1, the πth-derivative of the classical Jacobi polynomials ππ(πΌ,πΌ+2) (π₯) of any degree and for any order in terms of their corresponding Jacobi polynomials is given explicitly by
π=0
where πΈπ,π,π is given by (44). As a special case of the two formulae (31) and (45) and if we set πΌ = β1/2, the derivatives of Chebyshev polynomials of third kind (ππ (π₯)) and of fourth kind (ππ (π₯)) can be easily deduced. These two results are given in the following two corollaries.
π·π ππ(πΌ,πΌ+2) (π₯) =
β(πβπ)/2β
(πΌ,πΌ+2) β πΊπ,π,π ππβ2πβπ (π₯)
π=0
(49) β(πβπβ1)/2β
+
β π=0
(πΌ,πΌ+2) π»π,π,π ππβ2πβπβ1 (π₯) ,
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7
where πΊπ,π,π = (2βπβ2π (π + π β 1)!Ξ (π + πΌ + 1) Ξ (2π β 2π + 2πΌ + 3)) Γ ((π β 1)!π!Ξ (π + 2πΌ + 3) (π β π β 2π + πΌ + 2) β1
Γ Ξ(π β π + πΌ + 2)) Γ
Ξ (π β π β 2π + 2πΌ + 3) Ξ (π β π β 2π + πΌ + 5/2) Ξ (2π β 2π β 4π + 2πΌ + 3) Ξ (π β π β π + πΌ + 5/2) 2
Γ (π β π (π + 2π β 2πΌ β 3) β πΌ (π + 2π) (50) π»π,π,π = (2βπβ2π (π + π)!Ξ (π + πΌ + 1) Ξ (2 (π β π + πΌ + 1))) Γ ((π β 1)!π!Ξ (π + 2πΌ + 3) (π β π β 2π + πΌ + 1) β1
Γ Ξ(π β π + πΌ + 1))
Ξ (π β π β 2π + πΌ + 3/2) Ξ (π β π β 2π + 2πΌ + 2) . Ξ (π β π β π + πΌ + 3/2) Ξ (2π β 2π β 4π + 2πΌ + 1) (51)
Remark 11. As a direct consequence of relation (49), the πthderivative of ππ(πΌ+2,πΌ) (π₯) is given by π·π ππ(πΌ+2,πΌ)
β(πβπ)/2β
(π₯) =
β π=0
(πΌ+2,πΌ) πΊπ,π,π ππβ2πβπ
β(πβπβ1)/2β
β
β π=0
3 πΉ2
(π₯) (52)
(πΌ+2,πΌ) π»π,π,π ππβ2πβπβ1 (π₯) ,
where πΊπ,π,π and π»π,π,π are given by (50) and (51), respectively.
Ξ (π + πΌ + 1/2) Ξ (π + π + πΌ + 1) βπ (π β 1)! (π + πΌ)
((π β π + π β 2) /2)!Ξ ((π β π β π + 1) /2) { { , { { Ξ ((π + π + π + 2πΌ)/2)Ξ((π + π β π + 2πΌ + 1)/2) { { { { { (π + π + π) πVππ, { Γ{ { β ((π β π + π β 1) /2)!Ξ ((π β π β π + 2) /2) { { { , { {Ξ ((π + π + π + 2πΌ + 1)/2)Ξ((π + π β π + 2πΌ + 2)/2) { { { (π + π + π) πππ. { (53) Corollary 13. For all π, π, π β Zβ₯0 and π β€ π β π, one has 3 πΉ2
=
2β2(π+πΌ+2) Ξ (2π + 2πΌ + 4) Ξ (π + π + πΌ + 1) (π β 1)!Ξ (π + πΌ + 3)
((π β π + π β 2) /2)!Ξ ((π β π β π + 1) /2) { { { { Ξ ((π + π β π + 2πΌ + 5)/2)Ξ((π + π + π + 2πΌ + 4)/2) { { { { { { Γ ππ,π,πΌ , { { { { (π + π + π) πVππ, Γ{ { { { { 4 ((π β π + π β 1) /2)!Ξ ((π β π β π + 2) /2) { { { , { { Ξ ((π + π β π + 2πΌ + 4)/2)Ξ((π + π + π + 2πΌ + 3)/2) { { { { (π + π + π) πππ, { 3 πΉ2
In this section, four new reduction formulae for the 3 πΉ2 (1) that appears in relation (11) are deduced for certain choices of the two parameters πΌ and π½. These formulae are given in the following four corollaries.
=
3 πΉ2
(βπ + π + π, π + π + π + 2πΌ + 2, π + πΌ + 1; π + π + πΌ + 1, 2π + 2πΌ + 3 | 1)
=
Ξ (π + πΌ + 3/2) Ξ (π + π + πΌ + 1) βπ (π β 1)!
(βπ + π + π, π + π + π + 2πΌ + 3, π + πΌ + 1; π + π + πΌ + 1, 2π + 2πΌ + 4 | 1)
5. Reduction Formulae for Some Terminating Hypergeometric Functions of the Type 3 πΉ2 (1)
Corollary 12. For all π, π, π β Zβ₯0 and π β€ π β π, one has
(βπ + π + π, π + π + π + 2πΌ, π + πΌ + 1; π + π + πΌ + 1, 2π + 2πΌ + 1 | 1)
=
+2π (π + π) β π β 3π + πΌ2 + 3πΌ + 2) ,
Γ
((π β π + π β 2) /2)!Ξ ((π β π β π + 1) /2) { , { { { Ξ ((π + π + π + 2πΌ + 2)/2)Ξ((π + π β π + 2πΌ + 3)/2) { { { { { (π + π + π) πVππ, { Γ{ { ((π β π + π β 1) /2)!Ξ ((π β π β π + 2) /2) { { , { { { Ξ ((π + π + π + 2πΌ + 3)/2)Ξ((π + π β π + 2πΌ + 4)/2) { { { (π + π + π) πππ, {
(βπ + π + π, π + π + π + 2πΌ β 1, π + πΌ + 1; π + π + πΌ + 1, 2π + 2πΌ | 1)
Ξ (π + πΌ + 1/2) Ξ (π + π + πΌ + 1) 2βπ (π β 1)! (π + πΌ) (π + πΌ β 1) (π + πΌ)
((π β π + π β 2) /2)!Ξ ((π β π β π + 1) /2) { { { { Ξ ((π + π β π + 2πΌ + 1)/2)Ξ((π + π + π + 2πΌ)/2) { { { { Γ π { π,π,πΌ , { { { (π + π + π) πVππ, Γ{ { { { β4 ((π β π + π β 1) /2)!Ξ ((π β π β π + 2) /2) { { , { { { Ξ ((π + π β π + 2πΌ)/2)Ξ((π + π + π + 2πΌ β 1)/2) { { { (π + π + π) πππ, { (54)
8
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where ππ,π,πΌ = π2 + 2πΌ (π + π + 3) + 3π + π2 + 3π β π2 + π + 2πΌ2 + 4, ππ,π,πΌ = π2 + 2πΌ (π + π β 1) β π + π2 β π β π2 + π + 2πΌ2 .
6.1. Basis Functions in Terms of Certain Parameters Jacobi Polynomials. In this section, we consider four kinds of basis functions satisfying the boundary conditions (59). Assume that these basis functions can be expressed in the following forms:
(55) 6
(πΌ,πΌ+1) π1,π = ππ(πΌ,πΌ+1) (π₯) + β π1,π,π ππ+π (π₯) ,
Proof. The proofs of the four reduction formulae in Corollaries 12 and 13 can be deduced immediately, by comparing the four relations (43), (46), (49), and (52) with the corresponding results obtained from the general relation in (10).
π=1 6
(πΌ+1,πΌ) π2,π = ππ(πΌ+1,πΌ) (π₯) + β π2,π,π ππ+π (π₯) , π=1
6. Jacobi Galerkin Algorithms for Sixth-Order Two-Point Boundary Value Problems
π3,π =
In this section, and as an application, we are interested in applying the introduced derivatives formulae which were obtained in Sections 3 and 4, for the sake of solving the following sixth-order two-point BVPs: βπ¦(6) (π₯) + πΎπ¦ (π₯) = π (π₯) ,
π₯ β (β1, 1) ,
(56)
governed by the nonhomogeneous boundary conditions π¦(π) (Β±1) = Β±πΌπ ,
π = 0, 1, 2,
(57)
where πΎ is a real constant. We draw the readerβs attention that (56) governed by the nonhomogeneous boundary conditions (57) can be easily transformed to the equation (see [17]) βπ’(6) (π₯) + πΎπ’ (π₯) = π (π₯) ,
π₯ β (β1, 1) ,
(58)
governed by the homogeneous boundary conditions π’(π) (Β±1) = 0,
π = 0, 1, 2,
(59)
where 5
π (π₯) = π (π₯) + βππ π₯π ,
(πΌ,π½)
(π₯) , π1
(πΌ,π½)
(π₯) , π2
(πΌ,πΌ+2) β π3,π,π ππ+π π=1
(π₯) ,
6
(πΌ+2,πΌ) π4,π = ππ(πΌ+2,πΌ) (π₯) + β π4,π,π ππ+π (π₯) , π=1
and π = 0, 1, 2, . . . , π β 6. The coefficients {ππ,π,π }, 1 β€ π β€ 4, are uniquely determined such that each member of ππ,π (π₯), 1 β€ π β€ 4, satisfies the boundary conditions (57). For example, the coefficients {π1,π,π } are given explicitly by π1,π,π 3 ) (π + π)!(π + πΌ + 3/2)π/2 (β1)π/2 Ξ (πΌ + 1) ( π/2 { { , { { Ξ (π + π + πΌ + 1) (π + πΌ + 9/2)π/2 { { { { { π = 2, 4, 6, { { { { { 3 { {((β1)(πβ1)/2 (π β 7) Ξ (πΌ + 1) ( ) ={ (π β 1) /2 { { { { { { Γ Ξ (π + π + 1) (π + πΌ + 3/2)(πβ1)/2 ) { { { { { β1 { { Γ (2Ξ(π + π + πΌ + 1)(π + πΌ + 9/2)(π+1)/2 ) , { { π = 1, 3, 5. {
(64)
Now, the four variational formulations corresponding to the four choices of the basis functions can be written as
and ππ , 0 β€ π β€ 2π β 1, are some constants. Now, we define the following spaces: (πΌ,π½)
(π₯) +
(63)
6
(60)
π=0
ππ = span {π0
ππ(πΌ,πΌ+2)
(πΌ,π½)
(π₯) , . . . , ππ
(π₯)} ,
(βπ·6 π’π,π, ππ,π )π€ + πΎ(π’π,π, ππ,π )π€ = (π, ππ,π )π€ , π
π
π
1 β€ π β€ 4, (65)
π
ππ = {V β ππ : π· V (Β±1) = 0, π = 0, 1, 2} . (61) The Jacobi-Galerkin procedure for solving (56) subject to the boundary conditions (57) is to find π’π β ππ such that 6
(βπ· π’π, V)π€ + πΎ(π’π, V)π€ = (π, V)π€ ,
βV β ππ, 1
(62)
where π€ = (1 β π₯)πΌ (1 + π₯)π½ , and (π’, V)π€ = β«β1 π€π’V ππ₯ is the scalar inner product in the weighted space πΏ2π€ (β1, 1).
where π€1 = (1 β π₯)πΌ (1 + π₯)πΌ+1 ,
π€2 = (1 β π₯)πΌ (1 + π₯)πΌβ1 ,
π€3 = (1 β π₯)πΌ (1 + π₯)πΌ+2 ,
π€4 = (1 β π₯)πΌ (1 + π₯)πΌβ2 , πβ6
π’π,π = β ππ,π ππ,π . π=0
(66)
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9
For 1 β€ π β€ 4, let us denote π
ππ,π = (π, ππ,π )π€ ,
π
gi = (ππ,0 , ππ,1 , . . . , ππ,πβ6 ) ,
π
π π΄ π = (πππ )
Table 1: Maximum absolute error πΈ for π = 8, 12, 16.
0β€π,πβ€πβ6
,
π π΅π = (πππ )
0β€π,πβ€πβ6
,
(67)
π
ci = (ππ,0 , ππ,1 , . . . , ππ,πβ6 ) , and then (65) is equivalent to the following four matrix systems: (π΄ π + πΎπ΅π ) ci = gi ,
8
(68)
where the nonzero elements of the matrices π΄ π and π΅π , 1 β€ π β€ 4, can be given explicitly. Now, we give a numerical example to show the applicability and the efficiency of the proposed algorithm. Example 14. Consider the following sixth-order linear boundary value problem (see El-Gamel et al. [19], Akram and Siddiqi [20], and Lamnii et al. [21]): π¦(6) (π‘) β π¦ (π‘) = β6ππ‘ , π¦ (0) = 1, π¦ (1) = 0,
σΈ
π‘ β [0, 1] ,
12
σΈ σΈ
π¦ (0) = 0,
π¦ (0) = β1,
π¦σΈ (1) = βπ,
π¦σΈ σΈ (1) = β2π.
(69)
π‘
The analytic solution to (69) is (1 β π‘)π . The transformation π‘ = (1 + π₯)/2 turns (69) into π’(6) (π₯) β π’ (π₯) = β6π(π₯+1)/2 ,
π₯ β [β1, 1] ,
π’ (β1) = 1,
π’σΈ (β1) = 0,
π’σΈ σΈ (β1) = β1,
π’ (1) = 0,
π’σΈ (1) = βπ,
π’σΈ σΈ (1) = β2π,
(70)
(1+π₯)/2
with the analytic solution π’(π₯) = (1/2)(1 β π₯)π . In Table 1, we denote πΈ = |π’ β π’π| by the absolute errors resulting from the application of the Jacobi Galekin method (JGM) for problem (70) for various values of the parameters πΌ, π½, and π. Moreover, Table 2 displays a comparison between the best errors resulting from the application of JGM with the results obtained by applying the following three methods: (i) sinc-Galerkin method developed in [19], (ii) nonpolynomial spline technique developed in [20], (iii) spline collocation method developed in [21]. The results in Table 2 indicate that our method is more accurate if compared with all of the above-mentioned methods.
7. Conclusions This paper is concerned with introducing four new analytical formulae for the πth-derivative of certain parameters Jacobi polynomials in terms of their corresponding Jacobi polynomials. Moreover, some new reduction formulae for
16
πΌ
π½
1 β 2 1 2
1 2 1 β 2
πΈ
0
1
2.72189 β
10β8
1 1 β 2 3 2
0 3 2 1 β 2
2.71475 β
10β8
0
2
9.35247 β
10β8
2 1 β 2 1 2
0 1 2 1 β 2
8.87656 β
10β8 1.03683 β
10β14
0
1
2.56597 β
10β14
1 1 β 2 3 2
0 3 2 1 β 2
2.56661 β
10β14
0
2
1.43156 β
10β13
2 1 β 2 1 2
0 1 2 1 β 2
1.37795 β
10β13
0
1
2.57166 β
10β16
1 1 β 2 3 2
0 3 2 1 β 2
2.83627 β
10β16
0
2
5.22802 β
10β16
2
0
3.47378 β
10β16
1.5451 β
10β8 1.497 β
10β8
8.1876 β
10β8 7.6352 β
10β8
1.03251 β
10β14
9.67494 β
10β14 9.25758 β
10β14
3.15286 β
10β16 3.1637 β
10β16
2.4723 β
10β16 3.38054 β
10β16
Table 2: Comparison between the best errors of different methods. Error JGM Method in [19] Method in [20] Method in [21] πΈ 2.4723 β
10β16 1.00 β
10β4 1.70 β
10β7 2.370 β
10β9
summing some terminating hypergeometric functions of unit argument are deduced. As an application, and with the aid of the new introduced derivatives formulae, some spectral solutions of a special sixth-order boundary value problem are presented. To the best of our knowledge, all the presented theoretical formulae in this paper are completely
10 new. Moreover, we do believe that these new introduced formulae can be applied for solving other kinds of high-order boundary value problems.
Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 212-965-1434. The author, therefore, acknowledges with thanks DSR technical and financial support.
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