边值问题

tion of sixth-order boundary value problems using modified decomposition method. EL-Gamel et al. [3] used
Sinc–Galerkin method for the solution of sixth-order boundary value problems.
9 >>>=
yð1ÞðaÞ ¼ b0; yð2ÞðaÞ ¼ c0;
yð1ÞðbÞ ¼ b1; >>>; yð2ÞðbÞ ¼ c1;
ð1:1Þ
where ai; bi and ci; i ¼ 0; 1 are finite real constants and functions f and g are continuous in the interval ½a; b. The paper is organized into four sections. In Section 2, quintic spline is defined. Using derivative continu-
Applied Mathematics and Computation 189 (2007) 887–892
/locate/amc
Quintic spline solution of linear sixth-order boundary value problems
In this paper, the following sixth-order linear special case boundary value problem is considered
yð6ÞðxÞ þ f ðxÞyðxÞ ¼ gðxÞ; yðaÞ ¼ a0;
a 6 x 6 b; yðbÞ ¼ a1;
ð2:3Þ ð2:4Þ
S.S. Siddiqi et al. / Applied Mathematics and Computation 189 (2007) 887–892
889
120 ½N iÀ2 þ 26N iÀ1 þ 66N i þ 26N iþ1 þ N iþ2 ¼ h4 ½yiÀ2 À 4yiÀ1 þ 6yi À 4yiþ1 þ yiþ2;
Qði3ÞðxiÞ ¼ ni; >;
ð2:2Þ
and yðxÞ be the exact solution of the system (1.1).
The derivatives of spline Q at knots lead ai; bi; ci; di; ei and li to be calculated as
ð2:1Þ
where xi ¼ a þ ih;
i ¼ 0; 1; 2; . . . ; k; and
h
¼
bÀa k
:
Let, for i ¼ 0; 1; 2; . . . ; k,
QiðxiÞ ¼ yi;
Qði 1Þ ðxi Þ
¼
mi;
9 >=
Qði2ÞðxiÞ ¼ M i; Qði4ÞðxiÞ ¼ N i
* Corresponding author. E-mail addresses: shahidsiddiqiprof@ (S.S. Siddiqi), toghazala2003@ (G. Akram), saimanazeer123@yahoo.
com (S. Nazeer).
ities at knots the consistency relations between the values of spline and its derivatives at knots are determined in this section as well. The end conditions for the solutions of sixth-order boundary value problems are also derived in the same section. In Section 3, quintic spline solution of (1.1) is determined. In Section 4, two examples are considered for the usefulness of the problem developed and are compared with those developed by EL-Gamel et al. [3] and Wazwaz [8].
2. Quintic spline
Let Q be the quintic spline defined on ½a; b as
QiðxÞ ¼ aiðx À xiÞ5 þ biðx À xiÞ4 þ ciðx À xiÞ3 þ diðx À xiÞ2 þ eiðx À xiÞ þ li; x 2 ½xiÀ1; xi; i ¼ 1; 2; . . . ; k;
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.178
888
S.S. Siddiqi et al. / Applied Mathematics and Computation 189 (2007) 887–892
ai
¼
N iþ1 À N i 120h
;
bi
¼
Ni ; 24
1
h2
ci ¼ 6ຫໍສະໝຸດ M iþ1 À M i þ 6 ðÀ2N i À N iþ1Þ ;
di
¼
Mi 2
;
1
h4

ei ¼ h yiþ1 À yi þ 360 ð7N iþ1 þ 8N iÞ ;
li ¼ yi:
Applying the first and third-order derivatives continuities at the knots the following consistency relations are
Shahid S. Siddiqi *, Ghazala Akram, Saima Nazeer
Department of Mathematics, University of the Punjab, Lahore, Postcode 54590, Pakistan
Abstract
There are few techniques available to numerically solve sixth-order boundary value problems with two point boundary conditions. In this paper, quintic spline method is developed for the numerical solutions of linear special case sixth-order boundary value problems. Two examples are considered for the numerical illustration of the method developed. The method is also compared with those developed by EL-Gamel et al. [Mohamed EL-Gamel, John R. Cannon, Ahmed I. Zayed, Sinc–Galerkin method for solving sixth-order BVPs, Math. Comput. 73 (247) (2003)] and Wazwaz [A. Wazwaz, The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. Math. Comput. 118 (2001) 311–325]. The comparison shows that the quintic spline method is more efficient and effective tool and yields better results. Ó 2006 Elsevier Inc. All rights reserved.
obtained:
h2
1
M i ¼ À 120 ½N iÀ1 þ 8N i þ N iþ1 þ h2 ½yiÀ1 À 2yi þ yiþ1; i ¼ 1; 2; . . . ; k À 1;
1
1
N i ¼ À h4 ½30yiþ1 À 60yi þ 30yiÀ1 À 2h2 ½54M i þ 3M iÀ1 þ 3M iþ1; i ¼ 1; 2; . . . ; k À 1;
Keywords: Quintic spline; Boundary value problems; End conditions
1. Introduction
Scott and Watts [9] described the numerical solution of linear BVP using a combination of superposition and orthonormalization. In this paper difficult examples of BVP of orders 2, 4 and 8 are considered. Scott and Watts [10] described several computer codes that were developed using the superposition and orthonormalization technique and invariant imbedding. Sixth-order boundary value problems are known to arise in astrophysics, the narrow convecting layers bounded by stable layers, which are believed to surround A type stars, may be modelled by sixth-order boundary value problems, see [2,7]. Boutayeb and Twizell [2] developed a family of numerical methods for the solution of special and general nonlinear sixth-order boundary value problems. Twizell and Boutayeb [7] developed a family of numerical methods for the solution of special and general sixth-order boundary value problems with application to benard layer eigenvalue problems. Siddiqi and Twizell [5] presented the solution of sixth-order boundary value problem using the sextic spline.