Numerical Study on the Boundary Value Problem by Using a Shooting Method

Transcription

1 Pure ad Applied Mathematics Joural 2015; 4(3: Published olie May 25, 2015 ( doi: /j.pamj SSN: (Prit; SSN: (Olie Numerical Study o the Boudary Value Problem by Usig a Shootig Method Md. Mizaur ahma 1, *, Mst. Jesmi Ara 2, Md. Nurul slam 1, Md. Shajib Ali 1 1 Dept. of Mathematics, Faculty of Applied Sciece ad Techology, slamic Uiversity, ushtia, Bagladesh 2 Departmet of Political Sciece, Natioal Uiversity, azipur, Dhaka, Bagladesh address: miza_iu@yahoo.com (M. M. ahma, jesmi_u@yahoo.com (M. J. Ara, urul_math_iu@yahoo.com (M. N. slam, shajib_301@yahoo.co.i (M. S. Ali To cite this article: Md. Mizaur ahma, Mst. Jesmi Ara, Md. Nurul slam, Md. Shajib Ali. Numerical Study o the Boudary Value Problem by Usig a Shootig Method. Pure ad Applied Mathematics Joural. Vol. 4, No. 3, 2015, pp doi: /j.pamj Abstract: the preset paper, a shootig method for the umerical solutio of oliear two-poit boudary value problems is aalyzed. Dirichlet, Neuma, ad Sturm- Liouville boudary coditios are cosidered ad umerical results are obtaied. Numerical examples to illustrate the method are preseted to verify the effectiveess of the proposed derivatios. The solutios are obtaied by the proposed method have bee compared with the aalytical solutio available i the literature ad the umerical simulatio is guaratee the desired accuracy. Fially the results have bee show i graphically. eywords: Boudary Value Problem, Shootig Method, Numerical Simulatio ad MATLAB Programmig 1. troductio There are may liear ad oliear problems i sciece ad egieerig, amely secod order differetial equatios with various types of boudary coditios, are solved either aalytically or umerically. Two-poit boudary value problems occur i a wide variety of problem such as modelig of chemical reactios, the boudary layer theory i fluid mechaics ad heat power trasmissio. The wide applicability of boudary value problems i egieerig ad scieces calls for faster ad accurate umerical methods. May authors have attempted to obtai higher accuracy rapidly by usig a umerous methods. The shootig method to compute eige-values of fourth-order two-poit boudary value problems studied by D. J. Joes [1].Wag et al [2] ivestigated applicatio of the shootig method to secod order multi poit itegral boudary value problems. wog ad Wog [3] have studied the shootig method ad o-homogeeous multipoit BVPs of secod-order ODE. Abd-Elhameed et al [4] have ivestigated a ew wavelet collectio method for solvig secod-order multipoit boudary value problems usig Chebyshev polyomials of the third ad fourth kids. See [5] studied oliear two poit boudary value problem usig two step direct method. Meade et al [] discussed about the shootig techique for the solutio two- poit boudary value problems. ahma et al [7] have studied umerical Solutios for Secod Order Boudary Value Problems usig alerki Method. Fatullayev et al [8] ivestigated umerical solutio of a boudary value problem for a secod order Fuzzy differetial equatio. raas et al [9] ivestigated the shootig method for the umerical solutio of a class of oliear boudary value problems. Cole ad Adeboye [10] studied a alterative approach to solutios of oliear two poit boudary value problems. Qiao ad Li [11] aalyzed two kids of importat umerical methods forcalculatig periodic solutios. TrugHieu [12] studied remarks o the shootig method for oliear two-poit boudary value problem. ussell ad Shampie [13] discussed umerical methods for sigular boudary value problem. Sharma et al [14] studied umerical solutio of two poit boudary value problems usig alerki-fiite elemet method. Hece the mai objective of the preset study is to solve oliear two poit boudary value problems (BVP by usig simple ad efficiet shootig method. This well-kow techique is a iterative algorithm which attempts to idetify appropriate iitial coditios for a related iitial value problem that provides the solutio to the origial boudary value problem.

2 97 Md. Mizaur ahma et al.: Numerical Study o the Boudary Value Problem by Usig a Shootig Method 2. Mathematical Formulatio For a geeral boudary value problem for a secod-order ordiary differetial equatio, the simple shootig method is stated as follows: Let, x ''( t f ( t, xt (, x '( t, t a, b = [ ] xa ( = α, xb ( = β (2.1 be the BVP i questio ad let x(t,sdeote the solutio of the VP x ''( t f ( t, xt (, x '( t, = t [ a, b] xa ( = α, x'( a = s (2.2 where is a parameter that ca be varied. The VP (2.2 is solved with differet values of s with, e.g., ug utta-4 method util the boudary coditio o the right side xb ( = β becomes fulfilled. As metioed above, the solutio xt (, s of (2.2 depeds o the parameters. Let us defie a fuctio F( s : = xbs (, β f the BVP (2.1 has a solutio, the the fuctio F(s has a root, which is just the value of the slope x'( a givig the solutio x(tof the BVP i questio. The zeros of F(s ca be foud with, e.g., Newto s method. Newto s method is probably the best kow method for fidig umerical approximatios to the zeroes of a real-valued fuctio. The idea of the method is to use the first few terms of the Taylor series of a fuctio F(s i the viciity of a suspected root, i.e., F s h F s F s h Q h 2 ( + = ( + '( + ( Wheres is the thapproximatio of the root. Now if oe iserts h = s s, oe obtais F( s = F( s + F '( s ( s s As the ext approximatio to the root we choose the zero of this fuctio, i.e, F( s = F( s + F '( s ( s s = F( s s+ 1 = s F '( s (2.3 The derivative F '( s ca be calculated usig the forward differece formula F( s + δs F( s F '( s = δs where δ s is small. Notice that this procedure ca be ustable ear a horizotal asymptote. 3. Method of Solutio Techique Cosider the boudary value problem for the secod-order differetial equatio of the form,=,= (3.1 The the two iitial value problems is give by,=, =0 (3.2,=0, =1 (3.3 The if is the solutio to (3.2 ad is the solutio to equatio (3.3 the solutio to equatio (3.1 is = +, 0 (3.4 For the oliear case, the techique remais the same as that used to obtai a solutio to equatio (3.1 except that a sequece of iitial value problems of the form; =!,,,,=, =" # (3.5 where " # are real umber are ow required. Let," # be solutio of the iitial value problem3.5. We wat to have a sequece ' # (so that Oe of the choices for. is; lim # -," # = (3.. = = / / = 0 / Choosig the parameter " # for 1 1 to satisfy (3. is ot easy ad ca be complicated by the fact that; is a oliear equatio;," # =0 =!4,,",,"5,,,"=,,"=" # (3.7 The subscript 1 is dropped iside the fuctioal otatio for coveiece differetiatig equatio (3.7 with respect to t ad assumig that the order of differetiatio of x ad t is reversible gives 7,"=!4,,",,"5,"+ 7!4,,",,"5," 7,"=0, 7 7,"=1 Simplificatio usig 9," to represet results i 7,"

3 Pure ad Applied Mathematics Joural 2015; 4(3: =!,, 9+!,, 9,,9= 0,9 =1 =! 9+! 9,,9=0,9 =1 Fially, the Secat formula i geeral form is as follows: " # =" # :7 ;< 7 ;<7 ; :7 ;< :7 ;< We ca update " # usig the iformatio from 9," as follows: " # =" #,7 ;< =,7 ;< For a give accuracy >, the algorithm termiated if;," # <>. 4. esults ad Discussio this sectio, we explai five umerical examples of BVP which are available i the literature. The computatios programmig laguage, associated with the examples, are performed by MATLAB [15, 1, 17]. Example 1. Cosider the boudary value problem; = A 32+2C, 1 3, 1=17,3= EC Solutio: Let!,, = A 32+2C! = A,! F= A Solve a system of two secod- order iitial value problems: = A 32+2C, 1 3,1=17, 1= # 9 =! 9+! F9 = A 9+9,1 3,91= C!,, = +l,! = 1,! F = 2 Solve a system of two secod-order iitial value problems: = +l,1 2,1=0, 1= # 9 =! 9+! F9 = 9 2 9,1 2,91= Let =, =, C =9, E =9. = = O 1=0 +l J ; N 1= # C = E N C 1=0Q H E = C 2 E M E 1=1P Example 3. Cosider the boudary value problem = C,0 1,0=4,1=1 Solutio: Let!,, = C! =3,! F =0 Solve a system of two secod-order iitial value problem = C,0 1,0=4, 0= # 9 =! 9+! F9 =39,0 1,90=0,9 0=1 Let =, =, C =9, E =9. Solve a system of four first-order iitial value problem = = C O 0=4 ; N 0= # J C = E N C 0=0Q H E =3 C M E 0=1P Let =, =, C =9, E =9. = = A 32+2C J ; C = E H E = A C + E O 1=17 N 1= # N C 1=0Q M E 1=1P Example 2.Cosider the boudary value problem = +l,1 2,1=0,2= l2 Solutio: Let Figure 1. Plot of exact ad approximate solutio.

4 S 99 Md. Mizaur ahma et al.: Numerical Study o the Boudary Value Problem by Usig a Shootig Method 0.7 y"=-y1 2 -y+l(x,[1,2],y(1=0,y(2=l(2(secat 9 =! 9+! F9 =3 9 9,1 2,91= 0. y(x,s 2 Let =, =, C =9, E = exact solutio y=log(x y(x,s 1 y(x,s 0 = = C u u J ; C = E H E =3 C C E O 1= N N 1= # Q N C 1=0 M E 1=1P QQ Figure 2. Plot of exact ad approximate solutio. Figure 4. Plot of exact ad approximate solutio. Example 5.Cosider the boudary value problem = +2 l C,2 3, W 2= +l2,3=+l3 C Figure 3. Plot of exact ad approximate solutio. Figure 1, Figure 2 ad Figure 3 shows that the approximate solutio of Example 1, Example 2 ad Example 3 is at the third iteratio (shot, because the solutio is almost coicidet with the exact solutio amog the three shots. Example 4.Cosider the boudary value problem = C,1 2,1=,2= C Solutio: Let!,, = C! =3,! F = Solve a system of two secod-order iitial value problems: Solutio: Let!,, = +2 l C W! = l,! F =1 Solve a system of two secod-order iitial value problems: = +2 l C W,2 3,2= + l2, 2= # 9 =! 9+! F9 = l 9+9,2 3,92= 0,9 2=1 Let =, =, C =9, E =9. = C,1 2,1=, 1= #

5 S Pure ad Applied Mathematics Joural 2015; 4(3: = = +2 l C O 2= +l 2 N W; J N 2= # Q C = E N C 2=0 H E = C l + E M E 2=1 P QQ [3] Ma amwog ad James S. W. Wog, The shootig method ad o-homogeeous multipoit BVPs of secod-order ODE,. Hidawi Publishig Corporatio, Article D 4012, [4] W. M. Abd-Elhameed, E. H. Doha ad Y. H. Youssri, New wavelates collectio method for solvig secod-order multipoit boudary value problems usig Chebyshev polyomials of third ad fourth kids, Hidawi Publishig Corporatio, Article D , [5] Phag Pei See, Zaariah Abdul Majid ad Mohamed Suleima, Solvig oliear two poit boudary value problem usig two step direct method, Joural of Quality Measuremet ad Aalysis, vol. 7, No. 1, pp , [] Douglas B. Meade, Bala S. Hara ad alph E. White, The shootig techique for the solutio two-poit boudary value problems, 199. [7] M. M. ahma, M.A. Hosse, M. Nurul slamad Md. Shajib Ali, Numerical Solutios of Secod Order Boudary Value Problems by alerki Method with Hermite Polyomials, Aals of Pure ad Applied Mathematics Vol. 1, No. 2, pp , [8] Afet olayoglu Fatullayev, Emie Ca ad Caaoroglu, vestigated umerical solutio of a boudary value problem for a secod order Fuzzy differetial equatio,. TWMS J. Pure Appl. Math, vol. 4, No. 2, pp , Figure 5. Plot of exact ad approximate solutio. Figure 4 ad Figure 5 shows that the approximate solutio of Example 4 ad Example 5 is at the first iteratio, because this solutio is so close to the exact solutio amog the three shoots. 5. Coclusio We have developed a Shootig method to solve o-liear two poit boudary value problem aalytically. The give problems were tested usig three iteratios of shootig method. each figure, we represet the compariso betwee the exact solutio ad each iteratio, which are made i order to solve these problems. The umerical results obtaied by the proposed method are i good agreemet with the exact solutios. efereces [1] D. J. Joes. Use of a shootig method to compute Eige-values of fourth-order two-poit boudary value problems Joural of Computatioal ad Applied Mathematics, vol.47, pp ,1993. [2] Huila Wag, Zige Ouyag ad Liguag Wa, Applicatio of the shootig method to secod order multi- poit itegral boudary value problems, A Spiger ope Joural,Article D 205, [9] A. raas,. B. uether ad J. W. Lee, The shootig method for the umerical solutio of a class of oliear boudary value problems, SAM J. Numer. Aal., vol. 1, No. 5, pp , 200. [10] A. T. Cole ad.. Adeboye, Studied a alterative approach to solutios of oliear two poit boudary value problems, teratioal Joural of formatio ad Commuicatio Techology esearch, vol. 3, No. 4, [11] Tiatia Qiao ad Weiguo Li, Two kids of importat umerical methods forcalculatig periodic solutios Joural of formatio ad Commuicatio sciece, vol. 1, pp , 200. [12] Nguye Trug Hieu, emarks o the shootig method for oliear two-poit boudary value problem, VNU. JOUNAL OF SCENCE, Mathematics-Physics, [13]. D. ussell ad L. F. Shampie, Numerical methods for sigular boudary value problem, SAM. J. Numer. Aal., vol. 12, No. 1, pp. 13-3, 200. [14] Dikar Sharma, am Jiuari ad Sheo umar, Numerical solutio of two-poit boudary value problems usig alerki-fiite elemet method. teratioal Joural of Noliear Sciece, vol. 13, No. 2, pp ,2012. [15] L. F. Shampie,. ladwell ad S. Thompso. Solvig ODEs with MATLAB, [1] L. F. Shampie, J. ierzeka ad M. W. eichelt, Solvig boudary value problem for ordiary differetial equatios i MATLAB with bvp4c, [17] Stephe J. Chapma, MATLAB Programmig for Egieers, Thomso Learig, 2004.