DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS. Ra ft Abdelrahim 1, Z. Omar 2

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1 International Journal of Pure and Applied Matematics Volume 6 No. 6, -9 ISSN: - (printed version); ISSN: -95 (on-line version) url: ttp:// doi:.7/ijpam.v6i. PAijpam.eu DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS Ra ft Abdelraim, Z. Omar, Department of Matematics Scool of Quantitative Sciences College of Art and Sciences Univeristi Utara Malaysia MALAYSIA Abstract: In tis article, boundary value problems of second order linear ordinary differential equations are solved directly using ybrid block one-step metod. Power series of order six is adopted as basis function to derive tis metod troug collocation and interpolation approac. Sooting metod is employed to transform boundary value problem into initial value problem. Ten ybrid block one step is used to approximate te solution. Te performance of te new metod is sown by solving some boundary value problems examined by previous metods. AMS Subject Classification: 5J5, 5Q6, 65L6, 65L5 Key Words: ybrid one step, block metod, second order boundary value problem, tree off step points, sooting metod. Introduction Matematical models are expressed by boundary value problem (BVPs) and initial value problem (IVPs) of ordinary differential equations in order to elp Received: December, 5 Publised: February 7, 6 Correspondence autor c 6 Academic Publications, Ltd. url:

2 R. Abdelraim, Z. Omar in understanding te pysical penomena. BVPs ave long been a wide area of study because of teir vast application in sciences suc as pysics, engineering, biology and cemistry. In general, it is not easy to solve tese types of problems analytically []. Several numerical metods ave been derived to approximate te solutions for BVPs. According to [7], te using of spline metods for solving BVPs was initially investigated by Bickley in 96. In te same work, Adomian Decomposition Metod (ADM) as also been widely usedfor solving BVPs ( see [6] and [9]). In, [] proposed different finite numerical metod for solving two-point BVPs. Recently, [] presented tree-points block one-step metod for solving second order linear Diriclet and Neumann BVPs directly. However, tese metods ave dealt wit some difficulties for solving BVPs wic leads to inefficiently in term of error. Our focus in tis paper is to derive new one step ybrid block metod wit tree off step points for solving second order linear BVPs. It wort to igligt tat, one step ybrid block metod combines te advantages of block and ybrid metods( see [], [5]) wic overcomes te zero stability barrier in linear multistep metod [].. Derivation of Metod Te following second order two points linear (BVPs) of te form: y p(x)y +q(x)y +r(x), q(x), x [a,b]. y(a) α, y(b) λ, () is considered. In our strategy, te first step is employing sooting metod to transform equation () into two IVPs as below: y p(x)y +q(x)y +r(x), y (a) α, y (a), y p(x)y +q(x)y, y (a), y (a). () Ten, te developed one step ybrid block metod is used to solve () directly. Finally, te approximate solution for() is obtained as following: y(x) y (x)+ λ y (b) y (x). y (b)

3 DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY... In order to determine y (x) and y (x), a power series of te form y(x) v+m i ( ) i x xn a i. x [x n,x n+] () is considered for approximation solution of (), were n,,,...,n, v represents te number of interpolation points, m 5 denotes te number of collocation points, x n x n and [a,b] is divided as following a x < x <... < x N < x N b. Initially, equation () is interpolated at points x n+ derivative is collocated at all points i.e x n, x n+ will produce te following system of equations ( ) ( ) (6 ) ( ) (7 ) 6 6 ( ) (7 ) ( ) (7 ) 6 (7 ) 6, x n+ a a a a a a 5 a 6, x n+, x n+ and its second and x n+. Tis y n+ y n+ f n Gaussian elimination metod is used in () to find te values of a i s, i ()6 and ten substituted into equation ()to obtain a continuous linear multistep metod of te form: y(x) i, α i (x)y n+si + β i (x)i + i Te first derivative of equation (5) are given by were y (x) i, α α d dx α i(x)y n+si + i ( ((x x n)) ) ( ((x x n)) ) i,, d dx β i(x)i + i,, () β i i. (5) d dx (x)β ii, (6)

4 R. Abdelraim, Z. Omar β (x x n) β β β + ((x x n) 6 ) (5 ) (56(x x n) ) (5) (6(x x n)) 5 (7(x x n) ) ( ) ((x x n)) ( (9(x x n) ) () ((x x n)) 6 β (7(x x n) ) (7 ) + ((x x n)) 7 (9(x x n) ) () (9(x x n)) (7(x x n) ) (5 ) + ((x x n) ) ( ) ((x x n) 5 ) ( ) + ( ) 6 + ((x x n) 5 ) (5 ) + ( ) 675 ((x x n) 6 ) (5 ) (9(x x n) ) () (6(x x n) 5 ) ( ) + (9 ) 7 + (7(x x n) 6 ) (5 ) (9(x x n) ) (5 ) + (5(x x n) 5 ) ( ) + (7 ) 7 (7(x x n) 6 ) (5 ) (x x n) () 5 + (x x n) 6 (5 ) ((x x n) 5 ) ( ) Equation(5)is evaluated at te non-interpolating point i.e x n+, x n+ and equation (6) are evaluated at all points to produce te discrete scemes and its derivatives. Te discrete sceme and its derivatives are combined in matrix of te form block as below A [] Y [] m B [] R [] + D [] R [] + E [] R [], (7) were A [] 5, Y [] m y n+ y n+ y n+ y n+,

5 DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY... 5 B [], R [] ( yn y n ), D [] ( ) 6 (9 ) ( 9) ( 579) 76 6 () ( 7) 7, R [] ( fn ), E [] (99 ) 7 (7 ) 96 ( ) ( ) ( ) ( ) (6 ) 5 () (6) 9 () (9) 675 () () 675 () 5 (9) 5 (59 ) (9 ) 5 ( ) 675 (7 ) () 5 () () 75 (99) (97 ) 66 ( ) 5 (67) 96 (7) 97 () 5 (7) 6 and R [] Multiplying Equation (7) by (A [] ) produces te following ybrid block metod I [] Y [] m B [] R [] ++ D[] R [] + Ē [] R [], () were I [], B[] (), D[] ( ) 9 ( ) ( ) ( ) () 7 (5) (7) 5 (7),

6 6 R. Abdelraim, Z. Omar Ē [] ) Numerical Results In order to confirm te performance of te new one step ybrid block metod, te following tree second linear BVPS were solved and compared wit existing metods in [6], [7] and [] as demonstrated in Tables -. Te notation below are used in te tables: MAXE maximum error of te computed solution EAD extended adomian decomposition metod in [6] BVP direct tree-point block one-step metod in [] ECBIM extended cubic b-spline metod minimizing using Newton s metod in [7] OSHBT implementation of te one step ybrid block metod wit tree off-step points. Problem. Exact solution: Problem. y (x) y(x)+cos(x), y(), y(), x [,]. y(x) cos()+sin()+cos()+ e x sin() + cos()+sin() cos() e x sin() + cos(x). y (x) y (x)+e x, y(), y(). Exact solution: y(x) x( e x ).

7 DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY... 7 x MAXE BVP[ 5] MAXE EAD [] MAXE OSHBT.e 7.7e e.e 7.7e 7.9e.e 7.5e 6.e.e 7.e 6.5e 5.9e 7.5e 6.5e 6.65e 7.7e 7.6e 7.e 7.7e 7.e Table : Comparison of te new metod wit [6] and [], solving Problem wit.5 Problem. y (x) ( x )y (x)+y(x)+( x )e x, y(), y(), x [,]. Exact solution y(x) x( e x ).. Conclusion Hig accurate one step ybrid block metod for solving second order linear Boundary value problem directly as been successfully developed. Te new metod outperformed te existing metods wen solving te same BVPs of second order ODEs directly. References [] T. A. Anake, Continuous implicit ybrid one-step metods for te solution of initial value problems of general second-order ordinary differential equations (Unpublised doctoral dissertation). Covenant University,

8 R. Abdelraim, Z. Omar x MAXE BVP[ 5] MAXE ECBIM [] MAXE OSHBT..e.e..5e.5e.e..9e.e..9e.6e.5.e 5.95e.6.96e 6.6e.7.76e 6.7e..6e 6.9e.9.e 7 5.e Table : Comparison of te new metod wit [] and [7], solving Problem wit. [] A. Sagir, An accurate computation of block ybrid metod for solving stiff ordinary differential equations, Journal of Matematics,,,pp. -. [] J. D. Lambert, Computational Metods in Ordinary Differential Equation, Jon Wiley and Sons Inc, London,97. [] D. O. Awoyemi, A P-stable Linear Multistep Metod for Solving Tird Order Ordinary Differential Equation, Inter. J. Computer Mat,,,pp [5] Z.Omar and Ra ft, Abdelraim, Developing a Single Step Hybrid Block Metod wit Generalized Tree Off -step Points for te Direct Solution of Second Order Ordinary Differential Equations, International Journal of Matematical Analysis, 9,6, 5, pp [6] J. Bongsoo,Two-point boundary value problems by te extended Adomian decomposition metod, Journal of Computational and Applied Matematics, 9,,, pp [7] N. A. Hamid, A. A. Majid and A. I. Ismail, Extended cubic B-spline Metod for Linear Two-Point Boundary Value Problems, Sains Malaysiana,,,,pp. 5-9 [] Q. Fang,T. Tsuciya, and T.Yamamoto,Finite difference, finite element and finite volume metods applied to two-point boundary value problems,journal of Computational and Applied Matematics,9,,,pp. 9-9

9 DIRECTLY SOLVING SECOND ORDER LINEAR BOUNDARY... 9 x EXACT SOLUTION COMPUTED SOLUTION ERROR e e e e e e e e e.. -..e 7 Table : Numerical results of te new metod, solving Problem wit. [9] H. A. Emad,E. Abdelalim and R. Randolp, Advances in te Adomian decomposition metod for solving two-point nonlinear boundary value problems wit Neumann boundary conditions, Computer and Matematics wit Applications, 6,,pp [] Z. A. Majid, M.M. Hasni & N. Senu, Solving second order linear Diriclet and Neumann boundary value problems by block metod. Int J Appl Mat,,(),(), 7-76.

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