Development of implicit rational runge kutta schemes for second order ordinary differential equations

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Development of implicit rational runge kutta schemes for second order ordinary differential equations

  1. 1. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org Development of Implicit Rational Runge-Kutta Schemes for Second Order Ordinary Differential Equations Usman Abdullahi S.1*, Mukhtar Hassan S.2 1.Department of Statistics, Federal Polytechnic Bali, Taraba State – Nigeria 2.GNS Department, Federal Polytechnic Bali, Taraba State – Nigeria *E-Mail of corresponding author: usasmut@gmail.com Abstract In this paper, the development of One – Stage Implicit Rational Runge – Kutta methods are considered using Taylor and Binomial series expansion for the direct solution of general second order initial value problems of ordinary differential equations with constant step length. The basic properties of the developed method were investigated and found to be consistent and convergent. Keywords: Implicit Rational Runge Kutta scheme, Second Order Equations, Convergence and Consistent 1. Introduction Consider the numerical approximation first order initial value problems of the form, y ′ = f (x, y ), y ( x0 ) = y 0 a ≤ x ≤ b , (1.1) A Runge-Kutta method is the most important family of implicit and explicit iterative method of approximation of initial value problems of ordinary differential equations. So far many work and schemes have been developed for solving problem (1). The numerical solution of (1.1) is. yn+1 = yn + hφ (xn , yn , h ) (1.2) where s φ (x, y, h ) = ∑ ci ki i =1 r   k r = f  x + hai , y + h ∑ bij k j ,   k1 = f (x, y ), j =1   and with constraints r = 1(1)s (1.3) i ai = ∑ bij , i = 1(1)s j =1 The derivative of suitable parameters , and of higher order term involves a large amount of tedious algebraic manipulations and functions evaluations which is both time consuming and error prone, Julyan and Oreste (1992). The derivation of the Runge – Kutta methods is extensively discussed by Lambert (1973), Butcher (1987), Fatunla (1987), According to Julyan and Oreste (1992) the minimum number of stages necessary for an explicit method to attain order p is still an open problem. Therefore so many new schemes and approximation formula have been derived this includes the work of Ababneh et al. (2009a), Ababneh et al. (2009b) Faranak and Ismail (2010). Since the stability function of the implicit Runge-Kutta scheme is a rational function, Butcher (2003); Hong (1982) first proposes rational form of Runge-Kutta method (1.2), then Okunbor (1987) investigate rational form and derived the explicit rational Runge-Kutta scheme: r yn+1 = yn + h∑ wi K i i =1 r 1 + hyn ∑ vi H i i =1 where = and where + ℎ, = in which , , (1.4) + , , =− ℎ, ! +ℎ∑ +ℎ∑ , , , and =1 1 (1.5) =1 1 = (1.6) "# are arbitrary constants to be determined. 131 (1.7)
  2. 2. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org =∑ , (1.8) is imposed to ensure consistency of the method. In view of these inadequacies of the explicit schemes and the superior region of absolute stability associated with implicit schemes, Ademuluyi and Babatola (2000) generate implicit rational Runge-Kutta and generates also the parameters so that the resulting numerical approximation method shall be A-stable and will have low bound for local truncation error. Since then many new rational Runge – Kutta schemes have been developed for the solution of first order initial value problems and found out to give better estimates. Among these authors are: Ademuluyi and Babatola (2001), Odekunle (2001), Odekunle et al. (2004), Bolarinwa (2005), Babatola et al. (2007), Bolarinwa et al. (2012) and Abhulimen and Uluko (2012) The schemes are absolutely stable, consistent and convergent and was used to approximate a variety of first order differential equations. However, the methods are presently receiving more attention as efficient schemes for the solutions of various types of first order initial value problems are considered. 2. Derivation of the Scheme Consider the second order initial value problems y′′ = f (x, y, y′), y (x0 ) = y0 , y ′(x0 ) = y0 a ≤ x ≤ b , (2.1) The general $ − stage Runge-Kutta scheme for general second order initial value problems of ordinary differential equations of the form (1) as defined by Jain (1984) is s yn+1 = yn + hy′ + ∑ cr k r n r =1 (2.2) 1 s ∑ cr′ kr h r =1 (2.3) and ′ yn+1 = y′ + n where r  h2  1 r ′ Kr = f  xn + ci h, y n + hci yn + ∑ aij k j , y ′ + ∑ bij k j , i = 1(1)s n   2  h j =1 j =1  (2.4) i 1 i ci = ∑ aij = ∑ bij , i(1)r 2 j =1 j =1 with The rational form of (2.2) and (2.3) can be defined as s ′ yn + hyn + ∑ wr K r yn+1 = r =1 s ′ 1 + y n ∑ vr H r r =1 y′ +1 = n yn + 1+ (2.5) 1 s ∑ wr′ K r h r =1 1 ' s y n ∑ v′ H r r h r =1 (2.6) where Kr = Hr = s  h2  1 s ′ ′ f  xn + ci h, y n + hci y n + ∑ aij K j , y n + ∑ bij K j , i = 1(1) s  2  h j =1 j =1   h2 2 s   1 s g  xn + d i h, zn + hd i z ′ + ∑ α ij H j , z ′ + ∑ β ij H j , i = 1(1) s n n   h j =1 j =1   with constraints i ci = ∑ aij = j =1 1 i ∑ bij , i = 1(1)r 2 j 132 (2.7) (2.8)
  3. 3. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 i d i = ∑ α ij = j =1 in which , , ′ =− www.iiste.org 1 i ∑ β ij , i = 1(1)r 2 j , ! , ′ (A) & = (B) "# The constraint equations are to ensure consistency of the method, ℎ is the step size and the parameters , , , , ' , ( are constants called the parameters of the method. Using Bobatola etal (2007), the following procedures are adapted. K H ( x , y , y′ ) i. Obtain the Taylor series expansion of r and r about the point n n n and binomial series expansion of right side of (2.1) and (2.2). iv. Insert the Taylor series expansion into (2.1) and (2.2) respectively. K H ( x , y , y′ ) v. Compare the final expansion of r and r about the point n n n to the Taylor series expansion of yn+1 y′ ( x , y , y′ ) and n+1 about n n n in the powers of h . Normally the numbers of parameters exceed the number of equations, these parameters are chosen to ensure that (one or more of the following conditions are satisfied. iv. Minimum bound of local truncation error exists. v. The method has maximized interval of absolute stability. vi. Minimized computer storage facilities are utilized. To derive a One – stage scheme, we set s = 1 in equations (2.5), (2.6), (2.7) and (2.8) to have ′ y + hyn + w1K1 yn+1 = n 1 + y′ v1H1 n (2.9) and y′ +1 = n yn + 1+ 1 ′ w1 K1 h 1 ' ′ yn v1 H1 h (2.10) where ℎ! 1 = * + ℎ, + ℎ ′ + +, ′ + + ,, = 1 1 $ 2 ℎ and ℎ! 1 = * + ℎ, + ℎ ′ +' , ′ + ( ,, = 1 1 $ 2 ℎ with constraints = = & =' = ( ! ! where , , , , ' , ( , . , .′ , / and /′ are all constants to be determined. Equation (2.9) can be written as −1 ' yn+1 = yn + hyn + w1k1 (1 + yn v1H1 ) ( ) Expanding the bracket and neglecting 2nd and higher orders gives ' yn+1 = yn + hyn + w1k1 (1 − yn v1 H1 ) ( ) Expanding (2.14) and re-arranging, gives + ℎ ′ − !/ + ℎ 0 = Equation (2.10) can be written as ′ 0 = ′ + .′ + 31 + 2 2 / 1 ′ /′ + . − 4  ′ 1 ′  1 ′ ′  ′ yn+1 =  yn + w1k1 1 − yn v1H1  h   h  Expanding the binomial and re-arranging also gives ′ 0 = ′ + .′ + − 3 ′! /′ + 5 ′ .′ /′ + 4 2 2 2 Now, the Taylor’s series expansion of yn +1 about xn is given as 133 2.11 2.11 (2.12) (2.13) (2.14) / . + (2.15) (2.16) (2.17)
  4. 4. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 0 and ′ = 0 where 11 = 111 = 1: = 1: +ℎ ′ + ℎ! ′′ ℎ7 ′′′ ℎ9 : + + + … 2! 3! 4! = ′ + ℎ ′′ + , , ′ = + ′ " + "1 = ∆ ! ! ?? + ′ "" + "1"1 + 2 (2.20) = ∆! + "1 ∆ + " ∆= B B? 2.18 ℎ! ′′′ ℎ7 : + + … 2! 3! ? Since www.iiste.org + ′ B B" 1 "" A 2.19 +2 ?"1 + "1 ∆ B + (2.21) B"1 Using the Taylor’s series of the function of three variables we have from 2 1 + = +* ℎ ?+ ℎ ′ + + + "1 , ! ℎ ℎ 1 1 + C ℎ ! ?? + 2 ℎ ℎ ′ + + ?" + 2 ℎ * +, 2! ℎ ! 1 1 +2 ℎ ′ + + * + , ""1 + * + , "1"1 D + … ℎ ℎ Simplifying further and arranging the equation in powers of ℎ gives, 2G 2 + = E ! + "1 + +! ""1 F + 25 ! E + + 2H " + + ?"1 + ?"1 ! + ℎ +! "" ′ + + 1 + + ! "" ""1 F + 1 1 E ?+ + ?" + + "" F + E ! ?? + ! 1 ?" + ! ′! "" F + 0 ℎJ (2.22) " + 9 Equation (2.22) is implicit; one cannot proceed by successive substitution. Following Lambert (1973), we can assume that the solution for + may be express in the form + = ℎK + ℎ! L + ℎ7 M + ℎ9 N + 0 ℎJ (2.23) Substituting equation (2.23) into (2.22) gives ℎ + = E ℎK + ℎ! L + ℎ7 M "1 + ℎK + ℎ! L ! ""1 F 2 ℎ! + E + ℎK + ℎ! L " + ℎK + ℎ! L ?"1 + ! ℎK ! "" 2 ℎ7 1 1 1 + ℎK + ℎ! L ""1 F + E ? + ℎK ?" + ℎK "" F " + 2 9 ℎ + E ! ?? + ! 1 ?" + ! ′! "" F + 0 ℎJ 2.24 4 On equating powers of ℎ from equation (2.22) and (2.23), gives 1 K = 0, L = , M = ∆ , since = ? + " + 1/2 "1 = ! ! ! ! ! ! N = ∆ + ∆ "1 + " 9 Substituting K , L , M and N into (2.23) gives. 25 2G 2H ! (2.25) ! ! + = + ∆ + ∆ + ∆ "1 + " ! ! ! (2.26) Similarly, expanding in Taylor’s series about , , 1 , from (2.11b), we have ! ℎ ℎ ! ! ! 1 P +' = P( ( ( Q1 + ' ( QQ1 R + Q + ?Q1 + ' QQ + QQ1 R 2 2 7 9 ℎ ℎ ! 1 ! 1 ! ! P + P ' ' 1 ′ QQ R ? + Q + ?Q + QQ R + ?? + ?Q + 2 4 J +0 ℎ 2.27 Equation (2.27) is also implicit which cannot be proceed by successive substitution. Assuming a solution of the equation is of the form = ℎT + ℎ! U + ℎ7 V + ℎ9 W + 0 ℎJ 2.28 Substituting the values of in (2.28) into equation (2.27) and equating powers of ℎ of the equation, we can get the following after substitutions: 134
  5. 5. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 T = 0, 1 2 U = , V = www.iiste.org 1 2 ∆ & W = 1 4 ∆ ! ! +( ∆ Q1 +' Q (2.29 ) Substituting equation (2.29) into equation (2.28) gives ℎ! ℎ7 ℎ9 ! ! = + ∆ + ∆ +( ∆ Q1 + ' Q 2 2 2 Using equations (2.23) and (2.28) into equations (2.15) and (2.17) respectively gives 1 + ℎ 1 − !/ + ℎ / ℎ! U + ℎ7 V + ℎ9 W 0 = + P. − / . ℎ! U + ℎ7 V + ℎ9 W R ℎ! L + ℎ7 M + ℎ9 N Expanding the brackets and re-arranging in powers of ℎ gives 1 + ℎ 1 + ℎ! . L − ! / U + ℎ7 . M − ! / V − / U + 0 ℎ9 0 = 1 Also for 0 2.30 (2.31) gives 1 + . 1 ℎ! L + ℎ7 M + ℎ9 N ℎ 1 1 − X ′! / 1 + ! 1 . 1 / 1 ℎ ! L + ℎ 7 M + ℎ 9 N Y ℎ ! U + ℎ 7 V + ℎ 9 W ℎ ℎ Expanding the brackets and re-arrange in powers of ℎ gives 1 1 + ℎ . 1 L − ′! / 1 U + ℎ! . 1 M − ′! / 1 V − 1 . 1 / 1 L U 0 = + ℎ7 . 1 N − ′! / 1 W − 1 . 1 / 1 L V − 1 . 1 / 1 M U + 0 ℎ9 2.32 Comparing the corresponding powers in ℎ of equations (2.31) and (2.32) with equations (2.18) and (2.19) we obtain . − 1/ = 1 ! ! ! .. ∆ ! .1 ∆ ! .1 − = 0 − ! ! − 1 1! ! / / 1! 1 ! / ∆ − = 1! 1 ∆ − ! 1 ! 1 / .1 /1 = ∆ Z = ∆ ! ! (By using the equations in (2.25) and (2.29)) Since from equation (1.7) =− ∆ [# 5 "# , ? =− [ 5 "# , Q = −2 [# "# (2.33) + " , Q1 = −2 [# "# + "1 , 1 =− A "# 5 "# and = + 1 Q+ Q1 Using those equations into equation (2.33), we get the following simultaneous equations . +/ = 1 = . +/ 7 1 1 . +/ = 2 .1 + /1 = 1 Equation (2.35) has (4) equations with (6) unknowns; there will not be a unique solution for There will be a family of one-stage scheme of order four. i. Choosing the parameters 1 2 1 ′ ′ w1 = , v1 = , c1 = a11 = b11 = 0, w1 = 0, v1 = 2, d1 = α11 = , β11 = 1 3 3 2 arbitrarily the following scheme is obtain. 1 hy′ + K1 n 3 yn+1 = 2 1 + yn H1 3 and y′ n ′ yn +1 = 2 1 + H1 h 135 (2.34) (2.35) (2.35). (2.36) (2.37)
  6. 6. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org where h2 f ( xn , y n , y ′ ) n 2 h2  1 1 1 1  ′ H1 = f  x n + h, z n + hz ′ + H 1 , z n + H 1  n 2  2 2 2 h  , since =' = ( ! Choosing the parameters 1 1 1 1 ′ ′ w1 = v1 = , c1 = a11 = , d1 = α11 = , w1 = 2, v1 = 0, b11 = 1, β11 = 2 2 6 3 From (2.35) setting Then, 1 y n + hy ′ + K 1 n 2 y n +1 = 1 1 + yn H 1 2 K1 = ii. (2.38) and ′ ′ yn+1 = yn + 2 K1 h (2.39) where K1 = h2  1 1 1 1  1 ′ ′ f  xn + h, yn + hyn + K1 , yn + K1  c =a = b 2  2 2 2 h  , since 1 11 2 11 H1 = h2  1 1 1 1  1 ′ ′ g  xn + h, zn + hzn + H1 , zn + H1  2  6 6 6 3h  , since d 1= α11 = 2 β11 and 3. CONVERGENCE A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size tends to zero. Convergent = lim2→a | − 0 | 0 In other words, if the discretiation error at 0 tends to zero as ℎ → ∞, i.e if − 0 |→0 d 0 =| as & → ∞ (3.1) 0 From equation (2.5), 1 s ′ yn + ∑ wr K r h r =1 ′ yn+1 = 1 ' s ′ 1 + y n ∑ vr H r h r =1 (3.2) 1 seems to satisfy the equation of the form 0 1 s ′ y ( xn ) + ∑ wr K r h r =1 y ′(xn+1 ) = + Tn+1 s 1 ′ 1 + y′( xn )∑ vr H r h r =1 while the exact solution Where e 0 is a local truncation error. Subtracting equation (3.3) from (3.2) gives 1 s 1 s yn + ∑ w′ K r y (xn ) + ∑ w′ K r r r h r =1 h r =1 y′ +1 − y′( xn+1 ) = − + Tn+1 n s 1 ' s 1 ′ ′ 1 + y n ∑ vr H r 1 + y′(xn )∑ vr H r h r =1 h r =1 Adopting equation (3.4) gives 136 (3.3) (3.4)
  7. 7. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org s 1 s 1 s  1    1 ' s   ′ 1 + y ′(xn )∑ v′ H r  yn + ∑ w′ K r  − 1 + yn ∑ v′ H r  y (xn ) + ∑ wr K r  r r r h h r =1 h r =1 r =1    h r =1   +T en+1 =  n +1 s s  1 '  1  1 + yn ∑ v′ H r 1 + y ′(xn )∑ v′ H r  r r r =1  h r =1  h  Expanding the brackets and re-arranging gives s  s 1  en + 2 ( y′ − y′(xn )) ∑ w′ K r ∑ v′ H r  n r r h r =1   r =1 en+1 = + Tn+1 s s  1 '  1  ′ ′ 1 + yn ∑ vr H r 1 + y′(xn )∑ vr H r  r =1  h r =1  h  This implies that s 1  s  ′ ′ en + en 2  ∑ wr K r ∑ vr H r  h  r =1 r =1  en+1 = + Tn+1 s s  1 '  1  ′ 1 + yn ∑ vr H r 1 + y′(xn )∑ v′ H r   r r =1  h r =1  h  s  1  s  ′ ′ en 1 + 2  ∑ wr K r ∑ vr H r  r =1   h  r =1 en+1 = + Tn+1 s s  1 '  1  ′ H r 1 + y′(xn )∑ vr H r  ′ 1 + yn ∑ vr r =1  h r =1  h  From equations (3.7), setting s  1  s  ′ ′ An = 1 + 2  ∑ wr K r ∑ vr H r   r =1   h  r =1 and e Then =e 0 d 0 = f# g# h# (3.6) (3.7) s  1 ' s   1  ′ Bn = 1 + yn ∑ v′ H r  Cn = 1 + y′(xn )∑ vr H r  r r =1  h r =1 ,  h  , d +e Let L = max L > 0, M = max M > 0 and K = max K < 0 then (3.8) becomes, f d 0 ≤ d +e f (3.5) (3.8) gh = + < 1, then gh d 0 ≤ +d + e (3.9) If & = 0, then from (3.9), d = +da + e d! = +d + e = + ! da + +e + e by substituting the value of d d7 = +d! + e = + 7 da + + ! e + e Continuing in this manner, we get the following d 0 = + 0 da + ∑n 0a + n e (3.10) f Since = + < 1, then one can see that as & → ∞, d 0 → 0. This proves that the scheme converges. gh 7. CONSISTENCY A scheme is said to be consistent, if the difference equation of the computation formula exactly approximates the differential equation it intends to solve as the step size ends to zero. To prove if equation (2.5) is consistent, subtract from both side of (2.5), then Set s y n+1 − yn = ′ y n + hyn + ∑ wr K r r =1 s 1 + y n ∑ vr H r − yn r =1 (4.1) s yn+1 − yn = y n + hy′ + ∑ wr K r − yn − y n r =1 2 n s ∑ vr H r r =1 s 1 + y n ∑ vr H r r =1 137
  8. 8. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 s y n+1 − y n = www.iiste.org s 2 ′ hy n + ∑ wr K r − y n ∑ vr H r r =1 r =1 s 1 + y n ∑ vr H r r =1 (4.2) but Kr = s  h2  1 s ′ ′ f  x n + c i h, y n + hc i y n + ∑ a ij K j , y n + ∑ bij K j    2  h j =1 j =1  and s   1 s ′ ′ g  x n + d i h, z n + hd i z n + ∑ α ij H j , z n + ∑ β ij H j    h j =1 j =1   Then (4.2) becomes Hr = h2 2 s hy ′ + ∑ wr n y n +1 − y n = r =1 s s s   1 s 1 s h2  h2  2 ′ ′ ′ f  xn + ci h, y n + hci y n + ∑ aij K j , y n + ∑ bij K j  − y n ∑ v r g  xn + d i h, z n + hd i z n + ∑ α ij H j , z ′ + ∑ β ij H j  n     2  2  h j =1 h j =1 j =1 r =1 j =1   2 s s  1 s h   xn + d i h, z n + hd i z ′ + ∑ α ij H j , z ′ + ∑ β ij H j  1 + y n ∑ vr g n n  2  h j =1 r =1 j =1   Dividing the above equation throughout by h and taking the limit as h tends to zero on both sides gives y − yn lim n+1 = y′ n h→0 h y′ Again recall that from (2.6), subtracting n on both sides gives 1 s y ′ + ∑ w′ K r n r h r =1 ′ ′ y n +1 − y ′ = − yn n s 1 ' ′ ∑ v′ H r 1 + yn r h r =1 1 s 1 ' s y ′ + ∑ w′ K r − y ′ − y ′ 2 n ∑ v′ H r n r n r h r =1 h r =1 y ′ +1 − y ′ = n n s 1 ′ 1 + y n' ∑ v′ H r r h r =1 Simplify further gives 1 s 1 ' s ∑ wr′ K r − h y′ 2 n ∑ vr′ H r r =1 ′ h y ′ +1 − y n = r =1 n 1 ' s ′ ′ 1 + y n ∑ vr H r h r =1 (4.3) (4.4) Substituting the values of Kr and Hr (4.4) becomes s s  1 ' s  1 s h2  1 s h2  1 s ′ ∑ w′r K r 2 f  xn + ci h, yn + hci yn + ∑ aij K j , y′n + h ∑ bij K j  − h y′ 2 n ∑ v′r H r 2 g  xn + di h, zn + hdi z′n + ∑α ij H j , z′n + h ∑ β ij H j      h r =1 j =1 j =1 r =1 j =1 j =1     y ′ +1 − y ′ = n n 2 s s s  1 ' 1 h  1 + y′ ∑ v′ g  xn + d i h, z n + hd i z ′ + ∑α ij H j , z ′ + ∑ β ij H j  n r n n  h r =1 2  h j =1 j =1   Dividing all through by ℎ and taking the limit as ℎ tends to zero on both sides gives s s  1 ' s  1 s h2  1 s h2  1 s ′ ∑ w′r K r 2 f  xn + ci h, yn + hci y′n + ∑ aij K j , yn + h ∑ bij K j  − h y′ 2 n ∑ v′r H r 2 g xn + di h, zn + hdi z′n + ∑α ij H j , z′n + h ∑ β ij H j      h r =1 j =1 j =1 r =1 j =1 j =1     y ′ +1 − y ′ = n n s  1 ' s 1 s h2  1 + y ′ ∑ v′ g  xn + d i h, z n + hdi z ′ + ∑α ij H j , z ′ + ∑ β ij H j  n r n n  h r =1 2  h j =1 j =1   lim h→0 yn+1 − yn = h 1 2 s s    ' 1  1 s 1 s ′ f  xn + ci h, yn + hci yn + ∑aij K j , y′ + ∑bij K j  − y′2 n g xn + di h, zn + hdi z′ + ∑αij H j , z′ + ∑ βij H j  n n n     h j =1 2  h j=1 j =1 j =1    s  1  1 s 1 + y′' g xn + di h, zn + hdi z′ + ∑αij H j , z′ + ∑βij H j  n n n  2  h j =1 j =1   but by definition 138
  9. 9. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.11, 2013 www.iiste.org ′ f n = y ′ 2 g ( xn , z n , z n ) n hence the above equation becomes y − yn lim n+1 = fn h→0 h Hence, the numerical method is consistent. Conclusion The new numerical schemes derived follows the techniques of rational form of Runge – Kutta methods proposed by Hong (1982) which was adopted by Okunbor (1987) and Ademiluyi and Babatola (2000) by using Taylor and Binomial expansion in stages evaluation. The order condition obtained in this research is up to five (5) and the stage is up to three (3). This is an improvement on the work of earlier authors. Due it convergence and consistency of the new schemes, the scheme will be of high accuracy for direct numerical solution of general second order ordinary differential equations. The steps to the derivation of the new schemes are presented in the methodology while the analysis of the schemes proved to be consistent, convergent. The implementation of the schemes will be highlighted in the forthcoming paper. References Ababneh O. Y., Ahmad R, and Ismail E. S. (2009a): Design of New Diagonally Implicit Runge–Kutta Methods for Stiff Problems. Journal of Applied Mathematical Sciences, 3(45): 2241 – 2253. Ababneh O. Y., Ahmad R, and Ismail E. S. (2009b): New Multi-step Runge-Kutta Method. Journal of Applied Mathematical Sciences, 3(4):2255 – 2262. Abhulimen C. E and Uloko J. A (2012). A Class of an Implicit Stage-two Rational Runge-Kutta Method for Solution of Ordinary Differential Equations. Journal of Applied Mathematics and Bioinformatics, 2(3): 17-31. Ademiluyi, R. A, Babatola, P. O. (2000): Implicit Rational Runge-Kutta scheme for integration of stiffs ODE’s. Nigerian Mathematical Society, (NMS) Journal. Babatola P. O., Ademiluyi R. A and Areo E. A., (2007): One-Stage Implicit Rational Runge-Kutta Schemes for treatment of Discontinuous Initial value problems, Journal of Engineering and Applied Sciences 2(1): 96 – 104. Bolarinwa, B. (2005): A Class of Semi – Implicit Rational Runge – Kutta scheme for solving ordinary differential equations with derivative discontinuities. M. Tech thesis, Federal University of Technology, Akure; Nigeria. Unpublished. Bolarinwa B, Ademiluyi R. A, Oluwagunwa A. P. and Awomuse B. O. (2012): A Class of Two-Stage SemiImplicit Rational Runge - Kutta Scheme for Solving Ordinary Differential Equations. Canadian Journal of Science and Engineering Mathematics, 3(3):99 – 111. Butcher, J. C. (1987) Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods (Wiley). Butcher, J. C. (2003): Numerical methods for ordinary differential equations. John wiley and sons. Faranak R. and Ismail F., (2010): Fifth Order Improved Runge-Kutta Method for Solving Ordinary Differential Equations. Journal of applied informatics and remote sensing, 129 – 133. Hong, Y. F, (1982): A Class of A – Stable Explicit Scheme, Computational and Asymptotic Method for Boundary and Interior Layer. Proceeding of ball II conference, trinity college Dublin, 236 – 241. Jain, M. K. (1984): Numerical Solutions of Differential Equations, (Second Edition), Wiley Eastern Limited. Julyan H. E. and Oreste P, (1992): The Dynamics of Runge–Kutta Methods. Internal Journal of Bifurcation and Chaos, 2, 427–449. Lambert, J. D. (1973): Computational methods in ordinary differential equations. New York: John wiley and sons. Lotkin M. (1951): On the accuracy of Runge – Kutta Methods. MTAC 5 128 – 132 Nystrom, E. J. (1925): Verber, Die Numericche Integration von Differential-gleichagen, Acta sos., Sc. Fenn, 5013,1-55. Odekunle M. R (2001): Some Semi-Implicit Rational Runge-Kutta Schemes. Bagale Journal of Pure and Applied Sciences, 1(1):11-14. Odekunle, M. R, Oye, N. D, Adee, S. O. (2004): A class of inverse Runge-Kutta schemes for the numerical integration of singular problems. Journal of Applied Mathematics and Computing. Elsevier, 158:149-158. Okunbor, D. I. (1987): Explicit rational Runge-Kutta schemes for stiff system of ordinary differential equations. M.Sc Thesis, University of Benin, Benin city. (Unpublished). 139
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