Two Step Hybrid Block Method with Two Generalized Off-step Points for Solving Second Ordinary Order Differential Equations Directly

Transcription

1 Global Journal of Pure and Applied Matematics. ISSN Volume 2, Number 2 (206), pp Researc India Publications ttp:// Two Step Hybrid Block Metod wit Two Generalized Off-step Points for Solving Second Ordinary Order Differential Equations Directly Raed, Abdalreem Department of Pysics, College of Art and Sciences, Jordan University of Science and Tecnology. Moammad, Aldalal a Department of matematics, College of Art and Sciences, Univeristi Utara Malaysia. Abstract In tis paper, te new two step ybrid block metod wit two generalized off-step points for solving second order ordinary differential equation directly as been proposed. In te derivation of te metod, power series of order six is used as basis function to obtain te continuous sceme troug collocation and interpolation tecnique. As required by all numerical metods, te numerical properties of te developed block metod wic include consistent, zero stability, convergent and stability region are also establised. te new metod was found to compare favourably wit te existing metods in term of error. AMS subject classification: 65L05, 65L06, 65L20. Keywords: Block metod, Hybrid metod, Second order differential equation, Collocation and Interpolation, Two off step points.

2 520 Raed, Abdalreem and Moammad, Aldalal a. Introduction Tis article considered te solution to te general second order initial value problem (IVPs) of te form y f(x,y,y ), y(a) δ 0,y (a) δ.x [a,b]. () Equations () often arises in several areas of science and engineering an suc as biology, pysics and cemical. Generally, tese equations ave not exact solution. Tus, numerical metods become very crucial. In literature, numerous numerical metods for solving Equation () ave been proposed, for example, Euler metod, linear multistep metod, Runge-Kutta, predictor - corrector metod, ybrid metod and block metod (see [3], [5] and [2]). However, tese metods ave teir setbacks wic affects on teir accuracy and efficiency. Recently, ybrid block metods for solving equations () directly ave been proposed. In te latter, te researcers ave tried to combine advantages of direct, block and ybrid metods (see [], [4], [5], [6] and [7]) to overcome te zero stability problem in linear multistep as well as to avoid setbacks in reduction metods and generating numerical results concurrently. 2. Metodology In tis part, two step ybrid block metod wit two generalized off-step points i.e x n+s and x n+r for solving () is derived. Let te approximate solution of () to be te power series polomial of te form: were, y(x) q+d i0 i x [x n,x n+ ] for n 0,, 2,...,N, ( ) x i xn a i. (2) ii q denotes of te number of interpolation points wic is equal to te order of differential equation, iii d represents te number of collocation points, iv x n x n is constant step size of partition of interval [a,b] wic is given by a x 0 <x < <x N <x N b. Differentiating (2) twice gives y (x) f(x,y,y ) q+d i2 i(i ) 2 a i ( ) x i 2 xn. (3)

3 Two Step Hybrid Block Metod wit Two Generalized Off-step Points 52 Interpolating (2) at x n+s, x n+r and collocating (3) at all points in te selected interval produces seven equations wic can be written in matrix of te form: s s 2 s 3 s 4 s 5 s 6 r r 2 r 3 r 4 r 5 r s 2s 2 20s 3 30s r 2r 2 20r 3 30r a 0 a a 2 a 3 a 4 a 5 a 6 y n+s y n+r f n f n+s f n+r f n+ f n+2 (4) Employing Gaussian elimination metod to (4) gives te unknown values of a i s, i 0()6 wic are ten substituted back into equation (2) to produce a continuous implicit sceme of te form y(x) α i (x)y n+i + is,r Te first derivative of equation (5) gives were y (x) α s (x n x + r) (r s) α r (x x n s) (r s) is,r x α i(x)y n+i + i0,s,r, i0,s,r, β i (x)f n+i (5) x β i(x)f n+i (6) β 0 (x n x + s)(x n x + r) (20rs 4 ) ( 4 r 4 4 r 3 s 6 4 r 3 4 r 2 s r 2 s +0 4 r 2 4 rs rs rs + 4 s s s r 3 x 3 r 3 x n 3 r 2 sx + 3 r 2 sx n 6 3 r 2 x r 2 x n 3 rs 2 x + 3 rs 2 x n rsx rx

4 522 Raed, Abdalreem and Moammad, Aldalal a 9 3 rsx n 0 3 rx n + 3 s 3 x 3 s 3 x n 6 3 s 2 x s 2 x n sx 0 3 sx n 6 2 rx rxx n 6 2 rx 2 n + 2 s 2 x s 2 xx n + 2 s 2 x 2 n 62 sx sxx n xx n 0 2 x 2 n + rx3 3rx 2 x n + 3rxx 2 n rx3 n +sx 3 3sx 2 x n + 3sxx 2 n sx3 n 62 sx 2 n 02 x r 2 x r 2 xx n + 2 r 2 x 2 n 2 rsx rsxx n 2 rsx 2 n + 9x3 27x 2 x n +27xx 2 n 9x3 n 2x4 + 8x 3 x n 2x 2 x 2 n + 8xx3 n 2x4 n ) β s (x n x + s)(x n x + r) (60 4 s(s )(s 2)(r s)) (4 r r 3 s 6 4 r r 2 s r 2 s +0 4 r rs rs rs 2 4 s s s r 3 x 3 r 3 x n + 3 r 2 sx 3 r 2 sx n 6 3 r 2 x r 2 x n + 3 rs 2 x 3 rs 2 x n 6 3 rsx rsx n rx 0 3 rx n 2 3 s 3 x s 3 x n s 2 x 9 3 s 2 x n 0 3 sx sx n + 2 r 2 x r 2 x 2 n + 2 rsx rsxx n + 2 rsx 2 n 62 rx rxx n 6 2 rx 2 n 22 s 2 x sx sxx n +9 2 sx 2 n 02 x xx n 0 2 x 2 n + rx3 3rx 2 x n +3rxx 2 n rx3 n 2sx3 + 6sx 2 x n 6sxx 2 n + 2sx3 n + 9x3 27x 2 x n +8xx 3 n 22 r 2 xx n s 2 xx n 2 2 s 2 x 2 n + 27xx2 n 9x3 n 2x4 +8x 3 x n 2x 2 x 2 n 2x4 n ) β r (x n x + s)(x n x + r) (60 4 r(r )(r 2)(r s)) (24 r 4 4 r 3 s 9 4 r 3 4 r 2 s r 2 s r 2 4 rs rs rs 4 s s s rsx sx n 9 2 rx 2 n + 23 r 3 x 2 3 r 3 x n 3 r 2 sx + 3 r 2 sx n 9 3 r 2 x r 2 x n 3 rs 2 x + 3 rs 2 x n 6 3 rsx n rx 0 3 rx n 3 s 3 x + 3 s 3 x n s 2 x 6 3 s 2 x n 0 3 sx r 2 x r 2 xx n r 2 x 2 n 2 rsx rsxx n 2 rsx 2 n 92 rx 2 8xx 3 n 2 s 2 x s 2 xx n 2 s 2 x 2 n +6 2 sx sxx n sx 2 n + 02 x xx n x 2 n + 2rx3 6rx 2 x n + 6rxx 2 n 2rx3 n sx3 + 3sx 2 x n 3sxx 2 n + sx3 n +8 2 rxx n 9x x 2 x n 27xx 2 n + 9x3 n + 2x4 8x 3 x n +2x 2 x 2 n + 2x4 n ) β (x n x + s)(x n x + r) (60 4 (s )(r )) ( 4 r 4 4 r 3 s 4 4 r 3 4 r 2 s r 2 s 4 rs rs s s r 3 x 3 r 3 x n 3 r 2 sx + 3 r 2 sx n 4 3 r 2 x r 2 x n

5 Two Step Hybrid Block Metod wit Two Generalized Off-step Points rs 2 x + 3 rs 2 x n rsx 6 3 rsx n + 3 s 3 x 3 s 3 x n 4 3 s 2 x +4 3 s 2 x n 2 2 r 2 xx n + 2 r 2 x 2 n 2 rsx rsxx n 2 rsx 2 n 4 2 rx rxx n 4 2 rx 2 n 22 s 2 xx n + 2 s 2 x 2 n 4 2 sx sxx n 4 2 sx 2 n + rx3 3rx 2 x n + 3rxx 2 n +sx 3 3sx 2 x n + 3sxxn 2 sx3 n + 6x3 8x 2 x n + 2 s 2 x 2 rxn 3 +8xxn r 2 x 2 6xn 3 2x4 + 8x 3 x n 2x 2 xn 2 + 8xx3 n 2x4 n ) β 2 (x n x + s)(x n x + r) (20 4 ( 4 r 4 4 r 3 s 2 4 r 3 4 r 2 s 2 (s 2)(r 2)) +3 4 r 2 s 4 rs rs s s r 3 x 3 r 3 x n 3 r 2 sx + 3 r 2 sx n 2 3 r 2 x r 2 x n 3 rs 2 x + 3 rs 2 x n rsx 3 3 rsx n + 3 s 3 x 3 s 3 x n 2 3 s 2 x s 2 x n + 2 r 2 x r 2 xx n + 2 r 2 x 2 n 2 rsx rsxx n 2 rsx 2 n 22 rx rxx n 2 2 rx 2 n 22 s 2 xx n + 2 s 2 x 2 n 22 sx sxx n 2 2 sx 2 n + rx3 3rx 2 x n + 3rxx 2 n +sx 3 3sx 2 x n + 3sxx 2 n sx3 n + 3x3 9x 2 x n + 9xx 2 n 3x3 n 2x 4 + 8x 3 x n + 2 s 2 x 2 rx 3 n 2x2 x 2 n + 8xx3 n 2x4 n ) Equation (5) is evaluated at te non-interpolatig point x n+ and x n+2 wile Equation (6) is evaluated at all points to give te discrete scemes and its derivative. Te discrete sceme and its derivatives are combined i a matrix form as below [ ] A [2] 2 Y M B [2] (7) A [2] 2 r (r s) (r ) (r s) (r 2) (r s) ((r s)) ((r s)) ((r s)) ((r s)) ((r s)) s (r s) (s ) (r s) (s 2) (r s) ((r s)) ((r s)) ((r s)) ((r s)) ((r s)) ,Y M y n+s y n+r y n+ y n+2 y n+s y n+r y n+ y n+2,

6 524 Raed, Abdalreem and Moammad, Aldalal a ( y n B [2] 2 ), R [] , 2 ( ) f n, f n+s f n+r f n+ f n+2

7 Two Step Hybrid Block Metod wit Two Generalized Off-step Points (r4 r 3 s 6r 3 r 2 s 2 + 9r 2 s + 0r 2 rs 3 + 9rs 2 30rs + s 4 6s 3 + 0s 2 ) 2 2 (r )(s ) (r 4 r 3 s 5r 3 r 2 s 2 + 8r 2 s + 5r 2 rs 3 + 8rs 2 20rs) 22rs + 5r + s 4 5s 3 + 5s 2 + 5s 3) 3 2 (r 2)(s 2) (r 4 r 3 s 4r 3 r 2 s 2 + 7r 2 s + 2r 2 rs 3 + 7rs 2 20rs 4 5 6rs + 4r + s 4 4s 3 + 2s 2 + 4s) 20rs (r5 r 4 s 6r 4 r 3 s 2 + 9r 3 s + 0r 3 r 2 s 3 + 9r 2 s 2 30r 2 s rs 4 + 9rs 3 30rs 2 + s 5 6s 4 + 0s 3 ) 20rs (r5 r 4 s 6r 4 r 3 s 2 + 9r 3 s + 0r 3 r 2 s 3 + 9r 2 s 2 30r 2 s 8rs 5 + 4rs 4 2rs rs 2 8s 6 68s 5 + 9s 4 0s 3 ) 6 20rs (8r6 + 8r 5 s + 68r 5 4r 4 s 9r 4 + r 3 s 2 + 2r 3 s + 0r 3 + r 2 s 3 9r 2 s 2 30r 2 s + rs 4 9rs rs 2 s 5 + 6s 4 0s 3 ) rs (r5 r 4 s 6r 4 r 3 s 2 + 9r 3 s + 0r 3 r 2 s 3 + 9r 2 s 2 30r 2 s rs 4 + 9rs 3 30rs rs 33r + s 5 6s 4 + 0s 3 33s 59) 20rs (r5 r 4 s 6r 4 r 3 s 2 + 9r 3 s + 0r 3 r 2 s 3 + 9r 2 s 2 30r 2 s rs 4 + 9rs 3 30rs rs 576r + s 5 6s 4 + 0s 3 576s 228) ( 2 r (60(r s)(s )(s 2)) (r4 + r 3 s 6r 3 + r 2 s 2 6r 2 s + 0r 2 + rs 3 6rs 2 + 0rs 2s 4 + 9s 3 0s 2 )) ( 2 s (60(r s)(r )(r 2)) (2r4 r 3 s 9r 3 r 2 s 2 + 6r 2 s + 0r 2 rs 3 + 6rs 2 0rs s 4 + 6s 3 0s 2 )) ( 2 rs (60(r )(s )) (r4 r 3 s 4r 3 r 2 s 2 + 6r 2 s rs 3 +6rs 2 + s 4 4s 3 )) ( 2 rs (20(s 2)(r 2)) (r4 r 3 s 2r 3 r 2 s 2 + 3r 2 s rs 3 +3rs 2 + s 4 2s 3 ))

8 526 Raed, Abdalreem and Moammad, Aldalal a 2 22 ( 2 (r ) (60s(r s)(s 2)) (r4 + r 3 s 5r 3 + r 2 s 2 5r 2 s + 5r 2 + rs 3 5rs 2 + 5rs + 5r 2s 4 + 7s 3 3s 2 3s 3)) ( 2 (s ) (60r(r s)(r 2)) (2r4 r 3 s 7r 3 r 2 s 2 + 5r 2 s + 3r 2 rs 3 +5rs 2 5rs + 3r s 4 + 5s 3 5s 2 5s + 3)) 23 (2 60 (r4 r 3 s 3r 3 r 2 s 2 + 5r 2 s 3r 2 rs 3 + 5rs 2 + 5rs 3r +s 4 3s 3 3s 2 3s + 4)) 24 (2 (s )(r ) (20(r 2)(s 2)) (r4 r 3 s r 3 r 2 s 2 + 2r 2 s r 2 rs 3 +2rs 2 + 2rs r + s 4 s 3 s 2 s + )) D [2] ( 2 (r 2) 2 3 (60s(r s)(s )) (r4 + r 3 s 4r 3 + r 2 s 2 4r 2 s + 2r 2 + rs 3 4rs 2 + 2rs + 4r 2s 4 + 5s 3 )) D [2] ( 2 (s 2) 2 32 (60r(r s)(r )) (2r4 r 3 s 5r 3 r 2 s 2 + 4r 2 s rs 3 + 4rs 2 2rs s 4 + 4s 3 2s 2 4s)) 33 (2 (r 2)(s 2) (60(r )(s )) (r4 r 3 s 2r 3 r 2 s 2 + 4r 2 s 4r 2 rs 3 34 ( rs 2 + 8rs 8r + s 4 2s 3 4s 2 8s + 6)) 20 (r4 r 3 s r 2 s 2 + r 2 s rs 3 + rs 2 + 2rs + s 4 8)) (60s(r s)(s )(s 2)) (r5 + r 4 s 6r 4 + r 3 s 2 6r 3 s + 0r 3 +r 2 s 3 6r 2 s 2 + 0r 2 s + rs 4 6rs 3 + 0rs 2 2s 5 + 9s 4 0s 3 )) (60r(r s)(r )(r 2)) (2r5 r 4 s 9r 4 r 3 s 2 + 6r 3 s + 0r 3 r 2 s 3 +6r 2 s 2 0r 2 s rs 4 + 6rs 3 0rs 2 s 5 + 6s 4 0s 3 ) (60(r )(s )) r5 r 4 s 4r 4 r 3 s 2 + 6r 3 s r 2 s 3 + 6r 2 s 2 rs 4 + 6rs 3 + s 5 4s 4 ) (20(s 2)(r 2)) (r5 r 4 s 2r 4 r 3 s 2 + 3r 3 s r 2 s 3 + 3r 2 s 2 rs 4 +3rs 3 + s 5 2s 4 ))

9 Two Step Hybrid Block Metod wit Two Generalized Off-step Points (60s(r s)(s )(s 2)) (r5 + r 4 s 6r 4 + r 3 s 2 6r 3 s + 0r 3 + r 2 s 3 6r 2 s 2 + 0r 2 s 8rs 5 4rs rs 3 50rs 2 56s 5 36s s 3 ) (60r(r s)(r )(r 2)) (2r5 r 4 s 9r 4 r 3 s 2 + 6r 3 s + 0r 3 r 2 s 3 +6r 2 s 2 0r 2 s rs 4 + 6rs 3 0rs 2 + 8s s 5 9s 4 + 0s 3 ) (60(r )(s )) (r5 r 4 s 4r 4 r 3 s 2 + 6r 3 s r 2 s 3 + 6r 2 s 2 8rs 5 +4rs 4 4rs 3 8s 6 50s 5 + 6s 4 ) ) (20(s 2)(r 2) (r5 r 4 s 2r 4 r 3 s 2 + 3r 3 s r 2 s 3 + 3r 2 s 2 8rs 5 +4rs 4 7rs 3 8s 6 32s 5 + 3s 4 ) (60s(r s)(s )(s 2)) (8r6 + 68r 5 r 4 s 9r 4 r 3 s 2 + 6r 3 s + 0r 3 r 2 s 3 + 6r 2 s 2 0r 2 s rs 4 + 6rs 3 0rs 2 + 2s 5 9s 4 + 0s 3 ) ) (60r(r s)(r )(r 2)) (8r5 s + 56r 5 + 4r 4 s + 36r 4 r 3 s 2 54r 3 s 30r 3 r 2 s 3 + 6r 2 s r 2 s rs 4 + 6rs 3 0rs 2 s 5 +6s 4 0s 3 ) (60(r )(s )) (8r6 + 8r 5 s + 50r 5 4r 4 s 6r 4 + r 3 s 2 + 4r 3 s +r 2 s 3 6r 2 s 2 + rs 4 6rs 3 s 5 + 4s 4 ) (20(s 2)(r 2)) (8r6 + 8r 5 s + 32r 5 4r 4 s 3r 4 + r 3 s 2 + 7r 3 s +r 2 s 3 3r 2 s 2 + rs 4 3rs 3 s 5 + 2s 4 ) (60s(r s)(s )(s 2)) (r5 + r 4 s 6r 4 + r 3 s 2 6r 3 s + 0r 3 + r 2 s 3 6r 2 s 2 + 0r 2 s + rs 4 6rs 3 + 0rs 2 33r 2s 5 + 9s 4 0s 3 59) (60r(r s)(r )(r 2)) (2r5 r 4 s 9r 4 r 3 s 2 + 6r 3 s + 0r 3 r 2 s 3 +6r 2 s 2 0r 2 s rs 4 + 6rs 3 0rs 2 s 5 + 6s 4 0s s + 59) (60(r )(s )) (r5 r 4 s 4r 4 r 3 s 2 + 6r 3 s r 2 s 3 + 6r 2 s 2 rs 4 +6rs 3 40rs + 7r + s 5 4s 4 + 7s 66)

10 528 Raed, Abdalreem and Moammad, Aldalal a (20(s 2)(r 2)) (r5 r 4 s 2r 4 r 3 s 2 + 3r 3 s r 2 s 3 + 3r 2 s 2 rs 4 +3rs 3 0rs 3r + s 5 2s 4 3s 33) (60s(r s)(s )(s 2)) (r5 + r 4 s 6r 4 + r 3 s 2 6r 3 s + 0r 3 + r 2 s 3 6r 2 s 2 + 0r 2 s + rs 4 6rs 3 + 0rs 2 576r 2s 5 + 9s 4 0s 3 228) (60r(r s)(r )(r 2)) (2r5 r 4 s 9r 4 r 3 s 2 + 6r 3 s + 0r 3 r 2 s 3 +6r 2 s 2 0r 2 s rs 4 + 6rs 3 0rs 2 s 5 + 6s 4 0s s + 228) (60(r )(s )) (r5 r 4 s 4r 4 r 3 s 2 + 6r 3 s r 2 s 3 + 6r 2 s 2 rs 4 +6rs 3 80rs 496r + s 5 4s 4 496s 632) (20(s 2)(r 2)) (r5 r 4 s 2r 4 r 3 s 2 + 3r 3 s r 2 s 3 + 3r 2 s 2 rs 4 +3rs rs 656r + s 5 2s 4 656s 86) Multiplying Equation (7) by te inverse of A [] 2 gives IY m B [ ] [2] (8) were I is 8 8 identity matrix and B [] 2 s r , , Ē []

11 Two Step Hybrid Block Metod wit Two Generalized Off-step Points 529 wit (2 s 2 (40r 0s 5rs + 2rs 2 + 6s 2 s 3 )) (20r) 2 (2 r 2 (0r 40s + 5rs 2r 2 s 6r 2 + r 3 )) (20s) 3 (2 (35rs 8s 8r + 3)) (20rs) 4 (22 (5rs s r)) (5rs) 5 (s(20s 60r + 30rs 5rs2 + 8rs 3 5s s 3 + 8s 4 )) (20r) 6 (r(20r 60s + 30rs 5r2 s + 8r 3 s 5r r 3 + 8r 4 )) (20s) 7 8 ((50rs 33s 33r 59)) 20rs) ((5rs 72s 72r 266)) (5rs) (2 s 2 (20r 0s 5rs + 3rs 2 + 9s 2 2s 3 )) (60(s )(s 2)(r s)) 2 ( 2 s 4 (s 2 6s + 0)) (60r(r )(r 2)(r s)) 3 (2 s 4 (4s 0r + 2rs s 2 )) (60(r )(s )) 4 (2 s 4 (2s 5r + 2rs s 2 )) (20(r 2)(s 2)) 2 ( 2 r 4 (r 2 6r + 0)) (60s(s )(s 2)(r s)) 22 (2 r 2 (0r 20s + 5rs 3r 2 s 9r 2 + 2r 3 )) (60(r )(r 2)(r s)) 23 (2 r 4 (0s 4r 2rs + r 2 )) (60(s )(r )) 24 (2 r 4 (5s 2r 2rs + r 2 )) (20(s 2)(r 2))

12 530 Raed, Abdalreem and Moammad, Aldalal a 3 32 ( 2 (8r 3)) (60s(s )(s 2)(r s)) ( 2 (8s 3)) (60r(r )(r 2)(r s)) 33 (2 (5rs 7s 7r + 4)) (60(s )(r )) 34 (2 (5rs 2s 2r + )) (20(s 2)(r 2)) 4 42 (4 2 r) (5s(s )(s 2)(r s)) (4 2 s) (5r(r )(r 2)(r s)) 43 (42 (5rs 4s 4r + 4)) (5(s )(r )) 44 (22 (r + s 2)) (5(s 2)(r 2)) 5 (s(60r 40s 60rs + 5rs2 + 8rs s s 3 )) (60(s )(s 2)(r s)) 52 (s3 (8s s 2 5s + 20)) (60r(r )(r 2)(r s)) 53 (s3 (20r 0s 5rs + 8rs 2 + 5s 2 + 8s 3 )) (60(s )(r )) 54 (s3 (0r 5s 5rs + 8rs s 2 + 8s 3 )) (20(s 2)(r 2)) 6 (r3 (8r r 2 5r + 20)) (60s(s )(s 2)(r s)) 62 (r(60s 40r 60rs + 5r2 s + 8r 3 s + 45r r 3 )) (60(r )(r 2)(r s)) 63 (r3 (20s 0r 5rs + 8r 2 s + 5r 2 + 8r 3 )) (60(s )(r ))

13 Two Step Hybrid Block Metod wit Two Generalized Off-step Points (r3 (0s 5r 5rs + 8r 2 s + 33r 2 + 8r 3 )) (20(s 2)(r 2)) ((33r + 59)) (60s(s )(s 2)(r s)) ((33s + 59)) (60r(r )(r 2)(r s)) ((40rs 7s 7r + 66)) (60(s )(r )) ((3r + 3s + 0rs + 33)) (20(s 2)(r 2)) (4(36r + 33)) (5s(s )(s 2)(r s)) (4(36s + 33)) (5r(r )(r 2)(r s)) (4(3r + 3s + 5rs + 02)) (5(s )(r )) ((5rs 82s 82r 02)) (5(s 2)(r 2)) 3. Analysis of te Metod 3.. Order of te Metod Te linear difference operator L associated wit (8) is defined as L[y(x); ] IY M B [ ] [2] (9) were y(x) is an arbitrary test function continuously differentiable on [a,b]. Y M and 3 components are expanded in Taylor s series respectively and its terms are collected in powers of to give L[y(x),] C 0 y(x) + C y (x) + C 2 y (x) + (0) Definition 3.. Hybrid block metod (8) and associated linear operator (9) are said to be of order p, ifc 0 C 2 C p+2 0 and C p+2 0 wit error vector constants C p+2.

14 532 Raed, Abdalreem and Moammad, Aldalal a Expanding (8) in Taylor series about x n gives (s) j j j y n (s)y n (s2 (40r 0s 5rs + 2rs 2 + 6s 2 s 3 )) (20r) j0 (s2 (20r 0s 5rs + 3rs 2 + 9s 2 2s 3 )) (s) j (60(s )(s 2)(r s)) j0 (s 4 (s 2 6s + 0)) (60r(r )(r 2)(r s)) j0 + (s4 (4s 0r + 2rs s 2 )) (r) j (60(r )(s )) j0 j0 j (s4 (2s 5r + 2rs s 2 )) (20(r 2)(s 2)) j0 y n yj n y n ()y n (r2 (0r 40s + 5rs 2r 2 s 6r 2 + r 3 )) (20s) (r 4 (r 2 6r + 0)) (s) j (60s(s )(s 2)(r s)) j0 (r2 (0r 20s + 5rs 3r 2 s 9r 2 + 2r 3 )) (60(r )(r 2)(r s)) j0 (r4 (0s 4r 2rs + r 2 )) (r) j (60(s )(r )) (r4 (5s 2r 2rs + r 2 )) (20(s 2)(r 2)) j0 (r) j j j y n (r)y ((35rs 8s 8r + 3)) n (20rs) j0 (r 4 (r 2 6r + 0)) (s) j (60s(s )(s 2)(r s)) j0 ( 2 (8s 3)) (60r(r )(r 2)(r s)) j0 (2 (5rs 7s 7r + 4)) (r)j (60(s )(r )) j0 (2 (5rs 2s 2r + )) (20(s 2)(r 2)) j0 (2) j j j y n (2)y n (22 (5rs s r)) y n (5rs) j0 (4 2 r) (s) j (5s(s )(s 2)(r s)) j0 (4 2 s) (5r(r )(r 2)(r s)) j0 (42 (5rs 4s 4r + 4)) (r)j (22 (r + s 2)) (5(s )(r )) (5(s 2)(r 2)) j0 j0 y n j0 y n y n y n

15 Two Step Hybrid Block Metod wit Two Generalized Off-step Points 533 By comparing te coefficient of, we obtain te order of te metod to be [5, 5, 5, 5] T wit error constant vector s 4 (70r 28s 42rs + 7rs 2 + 2s 2 4s 3 ) r 4 (70s 28r 42rs + 7r 2 s + 2r r 3 ) C (56rs 2s 2r + 60) (7rs + 396) Zero Stability Te ybrid block metod (8) is said to be zero stable if te first caracteristic polomial π(x) aving roots suc tat x z, and if x z, ten, te multiplicity of x z must not greater tan two. In order to find te zero-stability of main block ˆB [2] 2 [y n+s,y n+r,y n+,y n+2 ] T in (8), we only consider te solution of te first caracteristic polomial were I is 4 4 identity matrix, tat is (r) xi ˆB [2] 2 x x 3 (x ) wic implies x 0, 0,,. Hence, our metod is zero stable for all s, r (0, ) 3.3. Consistency Te one step ybrid block metod (8) is said to be consistent if its order greater tan or equal one i.e. P. Tis proves tat our metod is consistent for all s, r (0, ) Convergence Teorem 3.2. [Henrici, 962] Consistency and zero stability are sufficient conditions for a linear multistep metod to be convergent. Since te metod is consistent and zero stable, it implies te metod is convergent for all s, r (0, ) Numerical Results In finding te accuracy of our metods, te following second order ODEs are examined. Te new block metods solved te same problems te existing metods solved in order to compare results in terms of error.

16 534 Raed, Abdalreem and Moammad, Aldalal a Table : Comparison of te new metod wit Kayode and Adeyeye (203) for solving Problem, 0 x exact solution computed solution in new metod error in our metod, s 4, r 3 errors in [2] e e e e e e e e e e 5 Table 2: Comparison of te new metod wit Awoyemi et al. (20) for solving problem 2, 320 x exact solution computed solution in new metod s error in our metod,s 3 4, r 3 errors in [3] e e e e e e e e e e 8 Problem 2: y y 0, y(0) 0, y (0), 0.. Exact solution: y(x) e x Problem : y x(y ) 2 0, y(0), y (0) 2, 320. Exact solution: y(x) + ( ) 2 + x 2 ln 2 x 4. Conclusion A two step ybrid block metod wit two generalized off-step points was developed. Te metod was tested to be convergent wit order five for all general off-step point belong to selected interval. Te derived metod was applied to solve bot non-linear and linear second ODEs problems witout converting to te equivalents system of first order ODEs. New generated results confirm te accuracy of te new metods in terms of error.

17 Two Step Hybrid Block Metod wit Two Generalized Off-step Points 535 References [] S. N. Jator, Solving second order initial value problems by a ybrid multistep metod witout predictors, Applied Matematics and Computation, (200), 27(8), [2] D. O.Awoyemi, E.A.Adebile, A. O.Adesanya & T.A.Anake, Modified block metod for te direct solution of second order ordinary differential equations, International Journal of Applied Matematics and Computation, (20)., 3(3), [3] Kayode, & O. Adeyey, Two-step two-point ybrid metods for general second order differential equations, African Journal of Matematics and Computer Science Researc, (203), 6(0), [4] T. A. Anake, D. O. Awoyemi and A. A. Adesanya, A one step metod for te solution of general second order ordinary differential equations, International Journal of science and Tecnology, (202b), 2(4), [5] J.D. Lambert, Computational metods in ordinary differential equations, (973). [6] T. A Anake, D. O. Awoyemi and A. A. Adesanya, One-step implicit ybrid block metod for te direct solution of general second order ordinary differential equations, IAENG International Journal of Applied Matematics, (202a), 42(4), [7] A. Sagir, An accurate computation of block ybrid metod for solving stiff ordinary differential equations, Journal of Matematics, (202), 4, 8 2. [8] Henrici, P. (962). Discrete variable metods in ordinary differential equations.

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