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Gm2511821187
Gm2511821187
Gm2511821187
Gm2511821187
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Gm2511821187
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Gm2511821187

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  • 1. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187 4-Point Block Method for Direct Integration of First-Order Ordinary Differential Equations M.R. Odekunle*, A.O. Adesanya*, J. Sunday** *(Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria) **(Department of Mathematical Sciences, Adamawa State University, Mubi, Nigeria)ABSTRACT This research paper examines the development and application of block methods.derivation and implementation of a new 4-point More recently, authors like [9], [10], [11], [12],block method for direct integration of first-order [13], [14], [15], [16], have all proposed LMMs toordinary differential equations using generate numerical solution to (1). These authorsinterpolation and collocation techniques. The proposed methods in which the approximateapproximate solution is a combination of power solution ranges from power series, Chebychev’s,series and exponential function. The paper Lagrange’s and Laguerre’s polynomials. Thefurther investigates the properties of the new advantages of LMMs over single step methods haveintegrator and found it to be zero-stable, been extensively discussed by [17].consistent and convergent. The new integrator In this paper, we propose a newwas tested on some numerical examples and Continuous Linear Multistep Method (CLMM), infound to perform better than some existing ones. which the approximate solution is the combination of power series and exponential function. This workKeywords: Approximate Solution, Block Method, is an improvement on [4].Exponential Function, Order, Power SeriesAMS Subject Classification: 65L05, 65L06, 65D30 2. METHODOLOGY: CONSTRUCTION OF THE NEW BLOCK METHOD1. INTRODUCTION Interpolation and collocation procedures Nowadays, the integration of Ordinary are used by choosing interpolation point s at a gridDifferential Equations (ODEs) is carried out using point and collocation points r at all points givingsome kinds of block methods. Therefore, in this rise to   s  r  1 system of equations whosepaper, we propose a new 4-point block method for coefficients are determined by using appropriatethe solution of first-order ODEs of the form: procedures. The approximate solution to (1) is taken y  f ( x, y), y(a)    a  x  b (1) to be a combination of power series and exponentialwhere f is continuous within the interval of function given by: 4 5  jxj y( x)   a j x j  a5 integration [a, b] . We assume that f satisfies (3)Lipchitz condition which guarantees the existence j 0 j 0 j!and uniqueness of solution of (1). The problem (1) with the first derivative given by:occurs mainly in the study of dynamical systems 4 5  j x j 1and electrical networks. According to [1] and [2], y ( x)   ja j x j 1  a5  (4)equation (1) is used in simulating the growth of j 0 ( j  1)! j 1populations, trajectory of a particle, simple where a j ,   for j  0(1)5 and y ( x) is jharmonic motion, deflection of a beam etc.Development of Linear Multistep Methods (LMMs) continuously differentiable. Let the solution of (1)for solving ODEs can be generated using methodssuch as Taylor’s series, numerical integration and be sought on the partition  N : a  x0 < x1 < x 2 < .collocation method, which are restricted by an . . < x n < xn 1 < . . .< x N = b , of the integrationassumed order of convergence, [3]. In this work, wewill follow suite from the previous paper of [4] by   interval a, b , with a constant step-size h , givenderiving a new 4-point block method in a multistep by, h  xn 1  xn , n  0,1,..., N .collocation technique introduced by [5]. Block methods for solving ODEs have Then, substituting (4) in (1) gives:initially been proposed by [6], who used them as 4 5  j x j 1starting values for predictor-corrector algorithm, [7] f ( x, y)   ja j x j 1  a5  (5)developed Milne’s method in form of implicit j 0 j 1 ( j  1)!methods, and [8] also contributed greatly to the 1182 | P a g e
  • 2. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187Now, interpolating (3) at point xn  s , s  0 and A  [a0 , a1 , a2 , a3 , a4 , a5 ]Tcollocating (5) at points xn  r , r  0(1)4 , leads to U  [ yn , f n , f n 1 , f n  2 , f n 3 , f n  4 ]Tthe following system of equations: and AX  U (6)where    2 xn  3 xn  4 xn  5 xn  2 3 4 5  1 xn 2 xn 3 xn 4 xn  1   xn         2 6 24 120      3 xn  4 xn  5 xn  2 3 4  0 1 2 xn 2 3xn 3 4 xn     xn  2       2 6 24     0   3 xn 1  4 xn 1  5 xn 1   2 3 4     xn 1    2 3 2  1 2 xn 1 3 xn 1 4 xn 1   2 6 24  X      3 xn  2  4 xn  2  5 xn  2   2 3 4 0     xn  2     2 3 2 1 2 xn  2 3 xn  2 4 xn  2   2 6 24    0   3 xn 3  4 xn 3  5 xn 3   2 3 4    2 xn 3     2 3 1 2 xn 3 3 xn 3 4 xn 3    2 6 24        3 xn  4  4 xn  4  5 xn  4   2 3 4     xn  4    2 3 2 0 1 2 xn  4 3 xn  4 4 xn  4    2 6 24   Solving (6), for a j s, j  0(1)5 and substituting Ym   yn1 , yn 2 , yn3 , yn 4  Tback into (3) gives a continuous linear multistep y n   yn 3 , yn 2 , yn 1 , yn  Tmethod of the form: F(Ym )   f n 1 , f n  2 , f n 3 , f n  4  4y ( x )   0 ( x ) yn  h   j ( x ) f n  j T (7) j 0 f (y n )   f n3 , f n2 , f n1 , f n  Twhere  0  1 and the coefficients of f n  j gives: 1 0 0 0 0 0 0 1 0  1 0 0 0 0 0 1 1  A(0)  , E ,0  (6t 5  75t 4  350t 3  750t 2  720t )  0 0 1 0 0 0 0 1 720      0 0 0 1 0 0 0 1 (12t  135t  520t  720t )  11   5 4 3 2 360   251   0 0 0  1   720  2  (3t 5  30t 4  95t 3  90t 2 )  60  0 0 0 29  1   90 3   (12t 5  105t 4  280t 3  240t 2 )  d   360 0 27   0 0 1   80 4  (6t  45t  110t  90t ) 5 4 3 2   720   14 0 0 0  (8)  45 where t  ( x  xn ) h . Evaluating (7) att  1(1)4 gives a block scheme of the form:A(0) Ym  Ey n  hdf (y n )  hbF (Ym ) (9)where 1183 | P a g e
  • 3. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187  323 11 53 19  Expanding (10) using Taylor series and comparing  360 30 360 720  the coefficients of h gives:   L  y ( x); h  c0 y ( x)  c1hy ( x)  c2 h 2 y ( x)  62 4 2 1   45   ...  c p h p y p ( x)  c p 1h p 1 y p 1 ( x)  ... 15 45 90b   51 9 21 3  (11)  40 10 40 80  3.1.1 Definition  64 8 64 14  The linear operator L and the associated   continuous linear multistep method (7) are said to  45 15 45 45  be of order p if3. ANALYSIS OF BASIC c0  c1  c2  ...  c p and c p 1  0. c p 1 isPROPERTIES OF THE NEW BLOCKMETHOD called the error constant and the local truncation3.1 Order of the New Block Method error is given by: Let the linear operator L  y( x); h tnk  cp1h( p1) y( p1) ( xn )  O(h p2 ) (12)associated with the block (9) be defined as: For our method:L y( x); h  A(0)Ym  Eyn  hdf ( yn )  hbF (Ym ) (10)  251 323 11 53 19   720 360 30 360 720   f n  1 0   yn 1  1   0 0  29 62 4 2 1   f  0   1 0 0   yn  2  1  90 90    n 1 L  y ( x); h        yn   h  45 15 45  f n2   0 0 0 1 0   yn 3  1  27 51 9 21 3         80 80   f n 3  0 0 0 1  yn  4  1 40 10 40    14 64 8 64 14   f n  4     45 45 15 45 45  (13)Expanding (13) in Taylor series gives:  ( h) j j 251h  h j 1 j 1  323 j 11 53 19  j ! yn  yn  yn   yn  (1)  (2) j  (3) j  (4) j  j 0 720 j 0 j !  360 30 360 720  (2h) j  j 29h  h j 1  62 4 2 1   0 yn  yn  yn   ynj 1  (1) j  (2) j  (3) j  (4) j     j 0 j ! 90 j 0 j !  45 15 45 90   0  (3h) j 27 h  h j 1 j 1  51 j 9   0 21 j 3 j yn  yn  j yn   yn  (1)  (2)  (3)  (4)  j    j 0 j ! 80 j 0 j!  40 10 40 80   0  j 1  (4h) y j  y  14h y  h y j 1  64 (1) j  8 (2) j  64 (3) j  14 (4) j  j  j ! n j 0 n 45 n  j ! n  45 j 0  15 45 45     (14)  c1  c2  c3  c4  c5  0, c6  1.88(02), 1.11(02), 1.88(02),  8.47(03)  . THence, c0Therefore, the new block method is of order five.3.2 Zero Stability3.2.1 Definition every root satisfying zs  1 have multiplicity not The block method (9) is said to be zero- exceeding the order of the differential equation.stable, if the roots zs , s  1, 2,..., k of the first Moreover, as h  0,  ( z )  z ( z  1) where r  characteristic polynomial  ( z) defined by  is the order of the differential equation, r is the ( z )  det( zA (0)  E) satisfies zs  1 and 1184 | P a g e
  • 4. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187  ( z , h)   ( z )  h ( z )  0 satisfies (0)order of the matrices A and E (see [11] fordetails). zs  1, s  1, 2,..., n whereFor our new integrator, f 1 0 0 0 0 0 0 1 h   h and   . 0 1 0 0 0 0 0 1 y ( z)  z     0 (15) We shall adopt the boundary locus method for the 0 0 1 0 0 0 0 1 region of absolute stability of the block method.     Substituting the test equation y   y into the 0 0 0 1 0 0 0 1 block formula gives, ( z )  z 3 ( z  1)  0,  z1  z2  z3  0, z4  1 A (0) Ym (r )  Ey n (r )  h Dyn (r )  h BYm (r ). Hence, the new block method is zero-stable. (16)3.3 Convergence Thus: The new block method is convergent byconsequence of Dahlquist theorem below.  A(0)Ym (r )  Eyn (r )  h( r , h)     (17)3.3.1 Theorem [18]  Dyn (r )  BYm (r )  The necessary and sufficient conditions Writing (17) in trigonometric ratios gives:that a continuous LMM be convergent are that it be  A(0)Ym ( )  Eyn ( ) consistent and zero-stable. h( , h)     (18)  Dyn ( )  BYm ( ) 3.4 Region of Absolute Stability where r  ei . Equation (18) is our3.4.1 Definition characteristic/stability polynomial. Applying (18) to The method (9) is said to be absolutely our method, we have:stable if for a given h , all the roots z s of thecharacteristic polynomial (cos 2 )(cos 3 )(cos  )  (cos 2 )(cos 3 )(cos 4 )(cos )h( , h)  (19) 1 1 (cos 2 )(cos 3 )(cos  )  (cos 2 )(cos 3 )(cos 4 )(cos ) 5 5which gives the stability region shown in fig. 1 below. 5 y 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 -1 x -2 -3 -4 -5 Fig. 1: Region of Absolute Stability of the 4-Point Block Method4. NUMERICAL IMPLEMENTATIONS EOAS- Error in [4]We shall use the following notations in the tables EBM- Error in [13]below; EMY- Error in [12]ERR- |Exact Solution-Computed Result| 1185 | P a g e
  • 5. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187Problem 1 This problem was solved by [12] and [4].Consider a linear first order IVP: They adopted block methods of order four to solvey   y, y(0)  1, 0  x  2, h  0.1 (20) the problem (20). We compare the result of ourwith the exact solution: new block method (9) (which is of order five) with their results as shown in table 1 below.y ( x)  e x (21)Table 1: Performance of the New Block Method on y   y, y(0)  1, 0  x  1, h  0.1_____________________________________________________________________________________ x Exact solution Computed Solution ERR EOAS EMY---------------------------------------------------------------------------------------------------------------------------------------0.1000 0.9048374180359595 0.9048374025462963 1.5490(-08) 2.3231(-07) 2.5292(-06)0.2000 0.8187307530779818 0.8187307437037037 9.3743(-09) 1.0067(-07) 2.0937(-06)0.3000 0.7408182206817178 0.7408182037500000 1.6932(-08) 3.2505(-07) 2.0079(-06)0.4000 0.6703200460356393 0.6703200296296297 1.6406(-08) 4.6622(-07) 1.6198(-06)0.5000 0.6065306597126334 0.6065306344848305 2.5228(-08) 3.4071(-07) 3.1608(-06)0.6000 0.5488116360940264 0.5488116163781553 1.9716(-08) 4.8161(-07) 2.7294(-06)0.7000 0.4965853037914095 0.4965852802878691 2.3504(-08) 5.6328(-07) 2.5457(-06)0.8000 0.4493289641172216 0.4493289421226676 2.1995(-08) 4.4956(-07) 2.1713(-06)0.9000 0.4065696597405991 0.4065696328791496 2.6862(-08) 5.3518(-07) 3.1008(-06)1.0000 0.3678794411714423 0.3678794189516900 2.2220(-08) 5.7870(-07) 2.7182(-06)---------------------------------------------------------------------------------------------------------------------------------------Problem 2 This problem was solved by [13] and [4].Consider a linear first order IVP: They adopted a self-starting block method of order y  xy, y(0)  1, 0  x  2, h  0.1 (22) six and four respectively to solve the problem (22).with the exact solution: We compare the result of our new block method (9) (which is of order five) with their results as shown x2 in table 2. y ( x)  e 2 (23)Table 2: Performance of the New Block Method on y  xy, y(0)  1, 0  x  1, h  0.1_____________________________________________________________________________________ x Exact solution Computed Solution ERR EOAS EBM---------------------------------------------------------------------------------------------------------------------------------------0.1000 1.0050125208594010 1.0050122069965277 3.1386(-07) 5.2398(-07) 5.29(-07)0.2000 1.0202013400267558 1.0202011563888889 1.3364(-07) 1.6913(-07) 1.77(-07)0.3000 1.0460278599087169 1.0460275417187499 3.1819(-07) 8.7243(-07) 8.99(-07)0.4000 1.0832870676749586 1.0832870577777778 9.8972(-09) 3.0098(-06) 3.09(-06)0.5000 1.1331484530668263 1.1331477578536935 6.9521(-07) 1.7466(-06) 1.91(-06)0.6000 1.1972173631218102 1.1972169551828156 4.0794(-07) 4.1710(-06) 4.48(-06)0.7000 1.2776213132048868 1.2776205400192113 7.7319(-07) 9.6465(-06) 1.02(-05)0.8000 1.3771277643359572 1.3771270561253053 7.0821(-07) 6.7989(-06) 7.74(-05)0.9000 1.4993025000567668 1.4992996318515335 2.8682(-06) 1.2913(-05) 1.44(-05)1.0000 1.6487212707001282 1.6487192043043850 2.0664(-06) 2.6575(-05) 2.93(-05)---------------------------------------------------------------------------------------------------------------------------------------5. CONCLUSION natural laws of growth and decay. Pacific In this paper, we have proposed a new 4- Journal of Science and Technology, 12(1),point block numerical method for the solution of 237-243, 2011.first-order ordinary differential equations. The block [2] P. Kandasamy, K. Thilagavathy and K.integrator proposed was found to be zero-stable, Gunavathy, Numerical methods (S. Chandconsistent and convergent. The new method was also and Company, New Delhi-110 055, 2005).found to perform better than some existing methods. [3] J. O. Ehigie, S. A. Okunugo, and A. B. Sofoluwe, A generalized 2-step continuous implicit linear multistep method of hybrid type. Journal of Institute of MathematicsReferences and Computer Science, 2(6), 362-372, 2010. [1] J. Sunday, On Adomian decomposition [4] M. R. Odekunle, A. O. Adesanya and J. method for numerical solution of ordinary Sunday, A new block integrator for the differential equations arising from the solution of initial value problems of first- 1186 | P a g e
  • 6. M.R. Odekunle, A.O. Adesanya, J. Sunday / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September-October 2012, pp.1182-1187 order ordinary differential equations, differential equation, Intern. J. Comp. Math, International Journal of Pure and Applied 77, 117-124, 2001. Science and Technology, 11(2), 92-100, [18] G. G. Dahlquist, Numerical integration of 2012. ordinary differential equations, Math. [5] P. Onumanyi, D. O. Awoyemi, S. N. Jator Scand., 4(1), 33-50, 1956. and U. W. Sirisena, New linear multistep methods with continuous coefficients for first Order IVPS, Journal of the Nigerian Mathematical Society, 13, 37-51, 1994. [6] W. E. Milne, Numerical solution of differential equations (New York, Willey, 1953). [7] J. B. Rosser, A Runge-Kutta method for all Seasons. SIAM Review, 9, 417-452, 1967. . [8] L. F. Shampine and H. A. Watts, Block implicit one-step Methods, Mathematics of Computation, 23(108), 731-740, 1969. [9] J. C. Butcher, Numerical Methods for Ordinary Differential Equation (West Sussex: John Wiley & Sons, 2003). [10] Zarina, B. I., Mohammed, S., Kharil, I. and Zanariah, M, Block Method for Generalized Multistep Adams Method and Backward Differentiation Formula in solving first- order ODEs, Mathematika, 25-33, 2005. [11] D. O. Awoyemi, R. A. Ademiluyi and E. Amuseghan, Off-grids exploitation in the development of more accurate method for the solution of ODEs, Journal of Mathematical Physics, 12, 379-386, 2007. [12] U. Mohammed, and Y. A. Yahaya, Fully implicit four point block backward difference formula for solving first-order initial value problems, Leonardo Journal of Sciences, 16(1) ,21-30, 2010. [13] A. M. Badmus and D. W. Mishelia, Some uniform order block method for the solution of first-order ordinary differential equations, Journal of Nigerian Association of Mathematical Physics, 19 ,149-154, 2011. [14] E. A. Areo, R. A. Ademiluyi, and P.O. Babatola, Three steps hybrid linear multistep method for the solution of first-order initial value problems in ordinary differential equations, Journal of Mathematical Physics, 19, 261-266, 2011. [15] E. A. Ibijola, Y. Skwame and G. Kumleng, Formulation of hybrid method of higher step-sizes through the continuous multistep collocation, American Journal of Scientific and Industrial Research, 2, 161-173, 2011. [16] J. P. Chollom, I. O. Olatunbosun, and S. A. Omagu, A class of A-stable block explicit methods for the solution of ordinary differential equations. Research Journal of Mathematics and Statistics, 4(2), 52-56, 2012. [17] D. O. Awoyemi, A new sixth-order algorithm for general second-order ordinary 1187 | P a g e

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