generalized two-hybrid one-step implicit third derivative block method ...

International Journal of Pure and Applied Mathematics Volume 112 No. 3 2017, 497-517 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v112i3.5

AP ijpam.eu

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT THIRD DERIVATIVE BLOCK METHOD FOR THE DIRECT SOLUTION OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS Mohammad Alkasassbeh1 , Zurni Omar2 1,2 Department

of Mathematics College of Art and Sciences Universiti Utara Sintok Kedah, 0060, MALAYSIA

Abstract:

In this article, a general two-hybrid one-step implicit third derivative block

method is developed for the direct solution of the second order initial value problems through interpolation and collocation approach. To derive this method, the approximate basis function is interpolated at the values {xn , xn+r } while its second and third derivatives are collocated at all points {xn , xn+r , xn+s , xn+1 } in the given interval. The new developed method produces better accuracy if compared to the existing methods when solving the same problems. AMS Subject Classification: 35J25, 35Q60, 65L06, 65L05 Key Words:

hybrid block method, second order ordinary differential equation, direct

solution, third derivative, interpolation and collocation

1. Introduction In this article, we are interested in solving the following second order ordinary differential equation ′′

y = f (x, y, y ′ ), y(a) = y0 , y ′ (a) = y1 , a ≤ x ≤ b.

(1)

There are many methods available for approximating (1). One of the methods Received:

September 30, 2016

Revised:

November 21, 2016

Published:

February 7, 2017

c 2017 Academic Publications, Ltd.

url: www.acadpubl.eu

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M. Alkasassbeh, Z. Omar

is called a block method which initially developed by [2] to provide starting values for predictor-corrector schemes. The usage of this method was further extended by [11] and [15] to get better approximation for solving (1). However, block methods are sometimes confined by zero-stability barrier (see [16]). Hence, several mathematicians introduced hybrid block methods to overcome this drawback of block methods. In hybrid block methods step and off-step points are combined to form a single block for solving ODEs, see [5],[9],[10]. In order to enhance the accuracy of the solution, [13] suggested second derivative methods which can solve stiff ODEs. For the same reason, [12] also proposed a Simpson’s type second derivative method to approximate the solution of first order stiff ODEs system. Based on the previous work mentioned above, we attempt to develop a new generalized two-hybrid one-step third derivative implicit method for solving second order ODEs directly using interpolation and collocation approach which can improve the accuracy.

2. Development of the Method The following power series is used as a basis function for an approximate solution to (1) y(x) =

2v+u−1 X j=0

aj (

x − xn j ) , h

(2)

where u and v are the number of interpolation and collocation points respectively. Then, the second and third derivatives are

′′

y (x) =

2v+u−1 X

h2 (j

2v+u−1 X

h3 (j

j=2

′′′

y (x) =

j=3

aj j! x − xn j−2 ( ) = f (x, y, y ′ ), − 2)! h

(3)

aj j! x − xn j−3 ( ) = g(x, y, y ′ ). − 3)! h

(4)

Interpolating (2) at xn+ˆu , for u ˆ = {0, r} and collocating (3) and (4) at all points xn+ˆv , for vˆ = {0, r, s, 1} where {0 < r < s < 1}, and then combining the

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GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

resulted equations gives a system of equations in matrix form 1 1  0  0  0   0  0  0  0 0 

0 0 r r2 0 h22 0 h22 0 h22 0 h22 0 0 0 0 0 0 0 0

0 r3 0

0 r4 0

6r h2 6s h2 6 h2 6 h3 6 h3 6 h3 6 h3

12r 2 h2 12s2 h2 12 h2

0

24r h3 24s h3 24 h3

··· ··· ··· ··· ··· ··· ··· ··· ··· ···

0 r9 0

    a0 yn  a  y   1   n+r   a   f    2 7  n  9!r        7!h72  a3  fn+r  9!s      7!h2  a4  fn+s  . 9!    =  a5  fn+1   7!h2     gn  0  a6      g  9!r 6  a  7 n+r     3 6!h6  g  9!s  a8  n+s 6!h3 9! a9 gn+1 3

(5)

6!h

Solving the unknown coefficients a′j s in (5) using matrix manipulation and substituting them back into Equation (2) yields y(x) = α0 yn + αr yn+r + h2 [βr fn+r + βs fn+s +

1 X

1 X

βi fn+i ] + h3 [γr gn+r + γs gn+s +

γi gn+i ], (6)

i=0

i=0

where n = 0, 1, 2, ..., N − 1, h = xn − xn−1 is the constant step size for the partition πN of the interval [a, b] which is given by πN = [a = x0 < x1 < ... < xN −1 < xN = b], α0 , αr , β0 , βr , βs , β1 , γ0 , γr , γs and γ1 are undetermined constants listed in Appendix I. Differentiating (6) with respect to x produces 1

y ′ (x) =

X ∂ ∂ ∂ ∂ ∂ α0 yn + αr yn+r + h2 [ βr fn+r + βs fn+s + βi fn+i ] ∂x ∂x ∂x ∂x ∂x i=0

1 X ∂ ∂ 3 ∂ + h [ γr gn+r + γs gn+s + γi gn+i ]. (7) ∂x ∂x ∂x i=0

Evaluating (6) at the non interpolating points {xn+s , xn+1 } and (7) at all points xn+i , i = {0, r, s, 1} produces the following general equations in block form (0)

A

(1)

Ym+1 = A

Ym +

1 X i=0

(i)

B Fm+i +

1 X i=0

D (i) Gm+i ,

(8)

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M. Alkasassbeh, Z. Omar

where

A(0)

1 0  0 = 0  0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

B (0)

 0  0  0 = 0   0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

(0)  B16 (0)  B26  (0)  B36   (0) , B46   (0)  B56  (0) B66



0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

(0)  D16 (0)  D26  (0)  D36   (0) , D46   (0)  D56 



D (0)

0  0  0 = 0   0 0

 0 0  0 , 0  0 1

(0)

D66

A(1)

 0 0  0 = 0  0 0

B (1)

D (1)

0 0 0 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

 0 rh 0 sh  0 h , 0 1  0 1 0 1



B11  (1) B21  (1) B 31 = B (1)  41  (1) B51 (1) B61

(1)

B12 (1) B22 (1) B32 (1) B42 (1) B52 (1) B62

(1)

(1)  B13 (1)  B23  (1)  B33   (1) , B43   (1)  B53  (1) B63



(1)

D12 (1) D22 (1) D32 (1) D42 (1) D52 (1) D62

(1)

(1)  D13 (1)  D32  (1)  D33   (1) . D43   (1)  D53 

D11  (1) D21  (1) D 31 = D (1)  41  (1) D51 (1) D61

(1)

D63

T  ′ ′ ′ , Ym+1 = yn+r , yn+s , yn+1 , yn+r , yn+s , yn+1   T ′ ′ Ym = yn−s , yn−r , yn , yn−s , yn−r , yn′ ,  T Fm = fn−5 , fn−4 , fn−3 , fn−2 , fn−1 , fn , T  Fm+1 = fn+r , fn+s , fn+1 , T  Gm = gn−5 , gn−4 , gn−2 , gn−1 , gn ,  T Gm+1 = gn+r , gn+s , gn+1 . Here, the non-zero entries of B (0) , B (1) , D (0) and D (1) are listed in Appendix II.

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GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

3. Analysis of the Method 3.1. Zero Stability Definition 1. The hybrid block method formula (8) is said to be zero stable if no root (Rz ) of the first characteristic equation ρ(R) has modulus greater than one i.e | Rz |≤ 1 and if Rz = 1 then the multiplicity of Rz must not exceed two . To show that the roots of the first characteristic equation satisfies the prior definition we assume that {r, s} ∈ (0, 1) and hence ρ(R) =det([RA(0) − A(1) ]) = 0    R 0 0 0 0 0  0 R 0 0 0 0  0     0 0 R 0 0 0  0  − ρ(R) =     0 0 0 R 0 0  0  0 0 0 0 R 0  0 0 0 0 0 0 R 0

0 0 0 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

 0 rh 0 sh   0 h  = 0, 0 1  0 1  0 1

which implies R4 (R − 1)2 = 0, whose solutions are R1 = R2 = R3 = R4 = 0,

R5 = R6 = 1.

Therefore, the developed method is zero stable. 3.2. Order of the Method ˆ associated with the hybrid block methods formula (8) The linear operator L according to [3] and [4] is said to be of order p if ˆ L{y(x); h} = A(0) Ym − A(1) Ym+1 −

1 X

B (i) Fm+i −

i=0

1 X

D (i) Gm+i ,

i=0

expanding in Taylor series and combining like terms ˆ L{y(x); h} =

∞ X i=0

Ci hi y (i) = 0,

(9)

502

M. Alkasassbeh, Z. Omar

where C0 = C1 = ... = Cp+1 = 0 and Cp+2 6= 0. The term Cp+2 is called the error constant and the local truncation error is given by : tn+k = Cp+2 y p+2 hp+2 (xn ) + O(hp+3 ). Equation (9) can be expressed as shown below    E1 0 E2  0     E3  0   =  , E4  0     E5  0 

E6

0

where the values E1 , E2 , E3 , E4 , E5 and E6 are listed in Appendix III. Comparing the coefficients of y i and hi produces C0 = C1 = ... = C9 = 0 with vector of error constants C10 =  6 2

r s (3r4 − 10r3 s2 − 10r3 + 9r2 s3 + 36r2 s2 + 9r2 − 36rs3 − 36rs2 + 42s2 )  s7 (9r2 s4 − 36r2 s3 + 42r2 s − 10rs5 + 36rs4 − 36rs3 + 3s5 − 10s4 + 9s3 )  (42r2 s4 − 36r2 s3 + 9r2 s − 36rs4 + 36rs3 − 10rs + 9s3 − 10s2 + 3)  K 5 2 4 2r s (5r − 15r3 s2 − 15r3 + 12r2 s3 + 48r2 s2 + 12r2 − 42rs3 − 42rs2 + 42s2 )  2s6 (12r2 s4 − 42r2 s3 + 42r2 s − 15rs5 + 48rs4 − 42rs3 + 5s5 − 15s4 + 12s3 ) (84r2 s4 − 84r2 s3 + 24r2 s − 84rs4 + 96rs3 − 30rs + 24s3 − 30s2 + 10),

where K =

1 101606400 ,



   ,  

which implies that the order p of this method is 8. 3.3. Consistency

Definition 2. greater than one.

A block method is said to be consistent if its order p is

Since the order p = 8 of the hybrid block method from the above analysis that is greater than one hence the consistency property is satisfied. 3.4. Convergence Theorem 3 (Henrici, 1962). Consistency and zero stability are sufficient conditions for a linear multi-step method to be convergent.

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GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

The hybrid block formula (8) is convergent since it fulfils both the consistency and zero stability conditions.

4. Numerical Examples In this section the accuracy of the general two hybrid one-step implicit hybrid block formula (8) with order 8 is examined on three test problems, with step 1 . The computed results are then compared with the latest methods. size h = 100 Problem 1. f (x, y, y ′ ) = 100y, y(0) = 1, y ′ (0) = −10. 1 . Exact Solution. y = e−10x , with h = 100 Source. [6]. X value 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Exact solution 0.904837418035959520 0.818730753077981820 0.740818220681717880 0.670320046035639330 0.606530659712633420 0.548811636094026390 0.496585303791409470 0.449328964117221560 0.406569659740599170 0.367879441171442330 0.332871083698079610 0.301194211912202190

Computed solution 0.904837418035959520 0.818730753078006130 0.740818220681788930 0.670320046035777770 0.606530659712859130 0.548811636094358010 0.496585303791865050 0.449328964117819030 0.406569659741356670 0.367879441172378530 0.332871083699213700 0.301194211913554670

Error in new method 0.000000(+00) 2.431388(-14) 7.105427(-14) 1.384448(-13) 2.257083(-13 ) 3.316236(-13) 4.555800(-13) 5.974665(-13) 7.575052(-13) 9.361956(-13) 1.134093(-12) 1.352474(-12)

Error for [6] 7.360845(-11) 2.854670(-10) 6.246097(-10) 1.082590(-9) 1.551924(-9) 2.332691(-9) 3.118856(-9) 4.011705(-9) 5.012991(-9) 6.126237(-9) 7.356706(-9) 8.711392(-9)

Table 1: Comparison of the proposed method with [6]. Problem 2. f (x, y, z) = x(y ′ )2 , y(0) = 1, y ′ (0) = 12 . 1 2+x ), with h = 100 . Exact Solution. y = 1 + ln( 2−x Source. [7].

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M. Alkasassbeh, Z. Omar

X value 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Exact solution 1.05004172927849140 1.10033534773107560 1.15114043593646700 1.20273255405408230 1.25541281188299550 1.30951960420311190 1.36544375427139640 1.42364893019360220 1.48470027859405200 1.54930614433405540

Computed solution 1.05004172927849070 1.10033534773107470 1.15114043593646590 1.20273255405408190 1.25541281188299390 1.30951960420310760 1.36544375427138460 1.42364893019357660 1.48470027859399840 1.54930614433394000

Error in new method 6.661338(-16) 8.881784(-16) 1.110223(-15) 4.440892(-16) 1.554312(-15) 4.218847(-15) 1.176836(-14) 2.553513(-14) 5.351275(-14) 1.154632(-13)

Error for [7] 1.1102(-15) 1.9318(-14) 1.1169(-13) 4.1034(-13) 1.1482(-12) 2.7629(-12) 6.1566(-12) 1.2952(-11) 2.6431(-11) 5.4045(-11)

Table 2: Comparison of the proposed method with [7]. Problem 3. f (x, y, y ′ ) = y ′ , y(0) = 0, y ′ (1) = −1. 1 Exact Solution. y = 1 − ex , with h = 100 . Source. [8]. X value 0.10 0.21 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Exact solution -0.105170918075647710 -0.233678059956743400 -0.349858807576003180 -0.491824697641270570 -0.648721270700128640 -0.822118800390509550 -1.013752707470477500 -1.225540928492468800 -1.459603111156951200 -1.718281828459047300

Computed solution -0.105170918075647630 -0.233678059956743120 -0.349858807576002630 -0.491824697641269630 -0.648721270700126530 -0.822118800390506330 -1.013752707470473100 -1.225540928492462800 -1.459603111156943400 -1.718281828459036700

Error in new method 8.326673(-17) 2.775558(-16) 5.551115(-16) 9.436896(-16) 2.109424(-15 ) 3.219647(-15) 4.440892(-15) 5.995204(-15) 7.771561(-15) 1.065814(-14 )

Error for [8] 2.2360(-13) 1.6425(-12) 3.4625(-12) 8.6628(-12) 1.1338(-11) 2.0317(-11) 3.2476(-11) 4.5463(-11) 6.1781(-11) 8.2113(-11)

Table 3: Comparison of the proposed method with [8].

5. Conclusion A general two hybrid one-step block method of uniform order 8 has been developed for the direct solution of general second order ODEs. The developed method is tested on three problems. Numerical analysis shows that the developed method is consistent and zero stable which implies its convergence. Besides having excellent properties of the numerical method, the numerical results reveals that the new method has perform better than the existing

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

505

methods.

References [1] P. Henrici, Discrete Variable Methods in ODE, John Wiley and Sons, New York (1962). [2] W.E. Milne, Numerical Solution of Differential Equations, John Wiley and Sons, New York (953). [3] S.O. Fatunla, Numerical Methods for Initial Value Problems in Ordinary Differential Equation, Academic Press, New York (1988). [4] J.D. Lambert, Computational Methods in ODEs, John Wiley and Sons, New York (1973). [5] W. Gragg, H.J. Stetter, Generalized multistep predictor-corrector methods, J. Assoc. Comput. Mach., 11 (1964), 188-209. [6] F.M. Kolawole, Continuous hybrid block stomer cowell methods for solutions of second order ordinary differential equations, Journal of Mathematical and Computational Science, 4 (2014), 118-127. [7] A. O. Adesanya, M. K. Fasasi, and T. A. Anake, Three steps hybrid block method for the solution of general second order ordinary differential equations, J.P. Journal of Applied Mathematics, 8 (2013), 1-11. [8] O. Olanegan, D. O. Awoyemi, B. G. Ogunware, and F. O. Obarhua, Continuous doublehybrid point method for the solution of second order ordinary differential equations, International Journal of Advanced Scientific and Technical Research, 5 (2015), 549-562. [9] W.H. Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 11 (1974), 321-331. [10] S. N. Jator, Solving second order initial value problems by a hybrid multistep method without predictors, Applied Mathematics and Computation, 217 (2010), 4036-4046. [11] C. W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM Journal of Numerical Analysis, 2 (1964), 69-86. [12] R.K. Sahi, S.N. Jator, N. A. Khan, a simpson’s-type second derivative method for stiff systems, International journal of pure and applied mathematics, 81 (2012), 619-633. [13] F. Ngwane, S. Jator, Block hybrid-second derivative method for stiff systems, Int. J. Pure Appl. Math., 80 (2012), 543-559. [14] L. F. Shampine, H. A. Watts, Block implicit one-step methods, Mathematics of Computation, 23 (1969), 731-740. [15] J.D. Rosser, A Runge-kutta for all seasons, SIAM, Rev., 9 (1967), 417-452. [16] G.G. Dahlquist, Numerical integration of ordinary differential equations, Math. Scand., 4 (1956), 69-86. [17] Gupta, G.K. Implementing second-derivative multistep methods using Nordsieck polynomial representation, Math. Comp., 32 (1978), 13-18.

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Appendix I

α0 =

(xn − x + hr) , (hr)

αr =

(x − xn ) , (hr)

β0 = (((x − xn )2 /2 − ((x − xn )4 (3r2 s2 + 4r2 s + 3r2 + 4rs2 + 4rs + 3s2 ))/ (12h2 r2 s2 ) + ((x − xn )9 (r + s + rs))/(36h7 r3 s3 ) + ((x − xn )5 (r3 s3 + 4r3 s2 + 4r3 s + r3 + 4r2 s3 + 8r2 s2 + 4r2 s + 4rs3 + 4rs2 + s3 ))/(10h3 r3 s3 ) +(hr(x − xn )(−10r5 s − 10r5 + 36r4 s2 + 53r4 s + 36r4 − 36r3 s3 − 108r3 s2 −108r3 s − 36r3 + 90r2 s3 + 24r2 s2 + 90r2 s + 168rs3 + 168rs2 − 882s3 ))/ (2520s3 ) − ((x − xn )8 (4r2 s + 4r2 + 4rs2 + 11rs + 4r + 4s2 + 4s))/(56h6r3 s3 ) − ((x − xn )6 (4r3 s2 + 7r3 s + 4r3 + 4r2 s3 + 20r2 s2 + 20r2 s + 4r2 + 7rs3 +20rs2 + 7rs + 4s3 + 4s2 ))/(30h4 r3 s3 ) + ((x − xn )7 (r3 s + r3 + 4r2 s2 + 8r2
s + 4r2 + rs3 + 8rs2 + 8rs + r + s3 + 4s2 + s))/(21h5 r3 s3 ) ,

βr = (((x − xn )8 (7r3 + 7r2 s + 7r2 − 8rs2 − 13rs − 8r + 4s2 + 4s))/(56h6 r3 (r

−s)3 (r − 1)3 ) − (hr(x − xn )(105r6 − 385r5 s − 385r5 + 468r4 s2 + 1457r4 s +468r4 − 180r3 s3 − 1836r3 s2 − 1836r3 s − 180r3 + 720r2 s3 + 2418r2 s2 + 720r2 s − 966rs3 − 966rs2 + 378s3 ))/(2520(r − s)3 (r − 1)3 ) − ((x − xn )6 (−7r3 s2 − 28r3 s − 7r3 + 5r2 s3 + 13r2 s2 + 13r2 s + 5r2 + 5rs3 + 4rs2 + 5rs −4s3 − 4s2 ))/(30h4 r3 (r − s)3 (r − 1)3 ) + ((x − xn )9 (2r − s + 2rs − 3r2 ))/ (36h7 r3 (r − s)3 (r − 1)3 ) + ((x − xn )7 (−7r3 s − 7r3 + 2r2 s2 − 2r2 s + 2r2 + 2rs3 + 7rs2 + 7rs + 2r − s3 − 4s2 − s))/(21h5 r3 (r − s)3 (r − 1)3 ) − (s(x− xn )5 (7r3 s + 7r3 − 5r2 s2 − 7r2 s − 5r2 + rs2 + rs + s2 ))/(10h3 r3 (r − s)3 (r
−1)3 ) − (s2 (x − xn )4 (5r − 3s + 5rs − 7r2 ))/(12h2 r2 (r − s)3 (r − 1)3 )) , βs = ((((x − xn )7 (−2r3 s + r3 − 2r2 s2 − 7r2 s + 4r2 + 7rs3 + 2rs2 − 7rs + r+ 7s3 − 2s2 − 2s))/(21h5 s3 (r − s)3 (s − 1)3 ) + ((x − xn )6 (5r3 s2 + 5r3 s − 4r3 −7r2 s3 + 13r2 s2 + 4r2 s − 4r2 − 28rs3 + 13rs2 + 5rs − 7s3 + 5s2 ))/(30h4 s3 (r − s)3 (s − 1)3 ) + ((x − xn )9 (r − 2s − 2rs + 3s2 ))/(36h7 s3 (r − s)3 (s − 1)3 ) −((x − xn )8 (−8r2 s + 4r2 + 7rs2 − 13rs + 4r + 7s3 + 7s2 − 8s))/(56h6s3 (r −s)3 (s − 1)3 ) − (hr5 (x − xn )(−20r4 s + 10r4 + 75r3 s2 + 25r3 s − 36r3 − 63r2 s3 − 243r2 s2 + 108r2 s + 36r2 + 252rs3 + 138rs2 − 198rs − 294s3 + 210s2 )) /(2520s3(r − s)3 (s − 1)3 ) + (r(x − xn )5 (−5r2 s2 + r2 s + r2 + 7rs3 − 7rs2 + rs + 7s3 − 5s2 ))/(10h3 s3 (r − s)3 (s − 1)3 ) − (r2 .(x − xn )4 (3r − 5s − 5rs+

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

507


7s2 ))/(12h2 s2 (r − s)3 (s − 1)3 ))) ,

β1 = (((x − xn )8 (4r2 s − 8r2 + 4rs2 − 13rs + 7r − 8s2 + 7s + 7))/(56h6 (r − 1)3 (s − 1)3 ) − ((x − xn )7 (r3 s − 2r3 + 4r2 s2 − 7r2 s − 2r2 + rs3 −7rs2 + 2rs + 7r − 2s3 − 2s2 + 7s))/(21h5 (r − 1)3 (s − 1)3 ) − ((x− xn )6 (−4r3 s2 + 5r3 s + 5r3 − 4r2 s3 + 4r2 s2 + 13r2 s − 7r2 + 5rs3 + 13 rs2 − 28rs + 5s3 − 7s2 ))/(30h4 (r − 1)3 (s − 1)3 ) + ((x − xn )9 (2r + 2s −rs − 3))/(36h7 (r − 1)3 (s − 1)3 ) + (hr5 (x − xn )(10r4 s − 20r4 − 36r3 s2 + 25r3 s + 75r3 + 36r2 s3 + 108r2 s2 − 243r2 s − 63r2 − 198rs3 + 138 rs2 + 252rs + 210s3 − 294s2))/(2520(r − 1)3 (s − 1)3 ) − (rs(x − xn )5 (r2 s2 + r2 s − 5r2 + rs2 − 7rs + 7r − 5s2 + 7s))/(10h3 (r − 1)3 (s − 1)3 )
−(r2 s2 (x − xn )4 (5r + 5s − 3rs − 7))/(12h2 (r − 1)3 (s − 1)3 )) , γ0 = (((x − xn )(xn − x + hr)(18h7 r6 s − 5h7 r7 + 18h7 r6 − 18h7 r5 s2 − 72h7 r5 s − 18h7 r5 + 84h7 r4 s2 + 84h7 r4 s − 126h7r3 s2 − 5h6 r6 x + 5h6 r6 xn + 18h6 r5 sx − 18h6 r5 sxn + 18h6 r5 x − 18h6 r5 xn − 18h6 r4 s2 x + 18h6 r4 s2 xn − 72h6 r4 sx + 72h6 r4 sxn − 18h6 r4 x + 18h6 r4 xn + 84h6 r3 s2 x − 84h6 r3 s2 xn + 84h6 r3 sx − 84h6 r3 sxn − 126h6r2 s2 x + 126h6r2 s2 xn − 5h5 r5 x2 +10h5 r5 xxn − 5h5 r5 x2n + 18h5 r4 sx2 − 36h5 r4 sxxn + 18h5 r4 sx2n + 18h5 r4 x2 − 36h5 r4 xxn + 18h5 r4 x2n − 18h5 r3 s2 x2 + 36h5 r3 s2 xxn − 18h5 r3 s2 x2n − 72h5 r3 sx2 + 144h5r3 sxxn − 72h5 r3 sx2n − 18h5 r3 x2 + 36h5 r3 xxn − 18h5 r3 x2n + 84h5 r2 s2 x2 − 168h5 r2 s2 xxn + 84h5 r2 s2 x2n + 84h5 r2 sx2 − 168 h5 r2 sxxn + 84h5 r2 sx2n + 294h5 rs2 x2 − 588h5rs2 xxn + 294h5 rs2 x2n − 5h4 r4 x3 + 15h4 r4 x2 xn − 15h4 r4 xx2n + 5h4 r4 x3n + 18h4 r3 sx3 − 54h4 r3 sx2 xn +54h4 r3 sxx2n − 18h4 r3 sx3n + 18h4 r3 x3 − 54h4 r3 x2 xn + 54h4 r3 xx2n − 18h4 r3 x3n − 18h4 r2 s2 x3 + 54h4 r2 s2 x2 xn − 54h4 r2 s2 xx2n + 18h4 r2 s2 x3n − 72h4 r2 sx3 + 216h4r2 sx2 xn − 216h4 r2 sxx2n + 72h4 r2 sx3n − 18h4 r2 x3 + 54h4 r2 x2 xn − 54h4 r2 xx2n + 18h4 r2 x3n − 336h4 rs2 x3 + 1008h4rs2 x2 xn − 1008h4r s2 xx2n + 336h4 rs2 x3n − 336h4rsx3 + 1008h4rsx2 xn − 1008h4rsxx2n + 336h4 rsx3n − 126h4 s2 x3 + 378h4 s2 x2 xn − 378h4s2 xx2n + 126h4 s2 x3n − 5h3 r3 x4 + 20h3 r3 x3 xn − 30h3 r3 x2 x2n + 20h3 r3 xx3n − 5h3 r3 x4n + 18h3 r2 sx4 − 72h3 r2 sx3 xn + 108h3r2 sx2 x2n − 72h3 r2 sxx3n + 18h3 r2 sx4n + 18h3 r2 x4 − 72h3 r2 x3 xn + 108h3 r2 x2 x2n − 72h3 r2 xx3n + 18h3 r2 x4n + 108h3rs2 x4 − 432h3 rs2 x3 xn + 648 h3 rs2 x2 x2n − 432h3 rs2 xx3n + 108h3 rs2 x4n + 432h3 rsx4 − 1728h3 rsx3 xn +

508

M. Alkasassbeh, Z. Omar 2592h3 rsx2 x2n − 1728h3rsxx3n + 432h3 rsx4n + 108h3rx4 − 432h3 rx3 xn +648h3 rx2 x2n − 432h3 rxx3n + 108h3 rx4n + 168h3s2 x4 − 672h3s2 x3 xn + 1008 h3 s2 x2 x2n − 672h3s2 xx3n + 168h3s2 x4n + 168h3sx4 − 672h3 sx3 xn + 1008h3sx2 x2n − 672h3sxx3n + 168h3sx4n − 5h2 r2 x5 + 25h2 r2 x4 xn − 50h2 r2 x3 x2n + 50h2 r2 x2 x3n − 25h2 r2 xx4n + 5h2 r2 x5n − 150h2rsx5 + 750h2 rsx4 xn − 1500h2rsx3 x2n + 1500h2rsx2 x3n − 750h2 rsxx4n + 150h2 rsx5n − 150h2 rx5 + 750h2 rx4 xn − 1500h2 rx3 x2n + 1500h2rx2 x3n − 750h2rxx4n + 150h2rx5n − 60h2 s2 x5 + 300h2s2 x4 xn − 600h2 s2 x3 x2n + 600h2s2 x2 x3n − 300h2 s2 xx4n + 60h2 s2 x5n − 240h2 sx5 + 1200h2 sx4 xn − 2400h2sx3 x2n + 2400h2sx2 x3n − 1200h2sxx4n + 240h2sx5n − 60 h2 x5 + 300h2x4 xn − 600h2 x3 x2n + 600h2x2 x3n − 300h2 xx4n + 60h2 x5n + 55hrx6 −330hrx5 xn + 825hrx4 x2n − 1100hrx3 x3n + 825hrx2 x4n − 330hrxx5n + 55hrx6n +90hsx6 − 540hsx5 xn + 1350hsx4 x2n − 1800hsx3x3n + 1350hsx2 x4n − 540hsx x5n + 90hsx6n + 90hx6 − 540hx5 xn + 1350hx4x2n − 1800hx3 x3n + 1350hx2 x4n − 540hxx5n + 90hx6n − 35x7 + 245x6 xn − 735x5 x2n + 1225x4 x3n − 1225x3 x4n + 735
x2 x5n − 245xx6n + 35x7n ))/(2520h6 r2 s2 ) ,

γr = ((x − xn )9 /(72h6 r2 (r − s)2 (r − 1)2 ) − ((x − xn )6 (r2 s2 + 4r2 s + r2 − rs3 − 2rs2 + rs − 2s3 − 2s2 ))/(30h3 r2 (r − s)3 (r − 1)2 ) − (s2 (x − xn )4 )/(12hr(r −s)2 (r − 1)2 ) + (h2 r2 (x − xn )(5r4 − 15r3 s − 15r3 + 12r2 s2 + 48r2 s + 12r2 −42rs2 − 42rs + 42s2 ))/(1260(r − s)2 (r − 1)2 ) − ((x − xn )8 (r2 + rs + 2r −2s2 − 2s))/(56h5 r2 (r − s)3 (r − 1)2 ) + ((x − xn )7 (2r2 s + 2r2 − rs2 + 2rs +r − s3 − 4s2 − s))/(42h4 r2 (r − s)3 (r − 1)2 ) − (s(x − xn )5 (−2r2 s − 2r2
+2rs2 + rs + s2 ))/(20h2 r2 (r − s)3 (r − 1)2 )) , γs = ((r2 s(x − xn )4 (r − 1))/(12h(s − 1)(rs − s2 )2 (r + s − rs − 1)) + ((x − xn )6 (hr9 s3 + hr9 s2 − 2hr9 s − hr8 s4 − 3hr8 s2 + 4hr8 s − hr7 s4 − 3hr7 s3 + 4hr7 s2 +8hr6 s4 − 4hr6 s2 − 4hr6 s − 8hr5 s4 + 3hr5 s3 + 3hr5 s2 + 2hr5 s + hr4 s4 − h r4 s2 + hr3 s4 − hr3 s3 ))/(30h4 r3 s2 (r − s)2 (r − 1)2 (s − 1)2 (rs − s2 )(r + s − rs −1)) − ((x − xn )8 (2hr8 s2 − 2hr8 s − hr7 s3 − 3hr7 s2 + 4hr7 s − hr6 s4 + 2hr6 s3 − hr6 s2 + 3hr5 s4 + hr5 s2 − 4hr5 s − 3hr4 s4 − 2hr4 s3 + 3hr4 s2 + 2hr4 s+ hr3 s4 + hr3 s3 − 2hr3 s2 ))/(56h6 r3 s(r − s)(r − 1)2 (s − 1)(r + s − rs − 1)(r2 s2 −r2 s3 + 2rs4 − 2rs3 − s5 + s4 )) + ((x − xn )(5h3 r13 s2 − 5h3 r13 s − 14h3 r12 s3 −19h3 r12 s2 + 33h3 r12 s + 9h3 r11 s4 + 87h3 r11 s3 − 9h3 r11 s2 − 87h3 r11 s − 63h3

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

509

r10 s4 − 201h3 r10 s3 + 151h3r10 s2 + 113h3 r10 s + 177h3 r9 s4 + 179h3 r9 s3 − 284 h3 r9 s2 − 72h3 r9 s − 243h3r8 s4 + 9h3 r8 s3 + 216h3 r8 s2 + 18h3 r8 s + 162h3 r7 s4 −102h3 r7 s3 − 60h3 r7 s2 − 42h3 r6 s4 + 42h3 r6 s3 ))/(2520hrs2 (r − s)2 (r − 1)2 (s − 1)2 (rs − s2 )(r + s − rs − 1)) + ((x − xn )5 (2hr9 s3 − hr9 s2 − hr9 s − 2hr8 s4 − 3hr8 s3 + 2hr8 s2 + 3hr8 s + 4hr7 s4 − hr7 s3 − 3hr7 s + hr6 s3 − 2hr6 s2 + hr6 s − 4hr5 s4 + 3hr5 s3 + hr5 s2 + 2hr4 s4 − 2hr4 s3 ))/(20h3 r3 s2 (r − s)2 (r − 1)(s − 1)(rs − s2 )(r + s − rs − 1)2 ) + ((x − xn )9 (hr7 s2 −hr7 s − hr6 s3 − 2hr6 s2 + 3hr6 s + 3hr5 s3 − 3hr5 s − 3hr4 s3 + 2hr4 s2 + h r4 s + hr3 s3 − hr3 s2 ))/(72h7 r3 s2 (r − s)2 (r − 1)(s − 1)(rs − s2 )(r2 s2 − 2r2 s + r2 − 2rs2 + 4rs − 2r + s2 − 2s + 1)) − ((x − xn )7 (hr9 s2 − hr9 s + hr8 s3 − hr8 s − 2hr7 s4 − 3hr7 s3 − 3hr7 s2 + 8hr7 s + 4hr6 s4 + 4hr6 s3 − 8hr6 s −4hr5 s3 + 3hr5 s2 + hr5 s − 4hr4 s4 + 3hr4 s3 + hr4 s + 2hr3 s4 − hr3 s3 − h r3 s2 ))/(42h5 rs2 (r − s)(r − 1)(s − 1)2 (rs − s2 )(r + s − rs − 1)(r2 s − r3 s−
r3 + r4 ))) , γ1 = (((x − xn )(5h3 r13 s4 − 5h3 r13 s3 − 33h3 r12 s5 + 19h3 r12 s4 + 14h3 r12 s3 + 87h3 r11 s6 + 9h3 r11 s5 − 87h3 r11 s4 − 9h3 r11 s3 − 113h3r10 s7 − 151h3 r10 s6 +201h3 r10 s5 + 63h3 r10 s4 + 72h3 r9 s8 + 284h3 r9 s7 − 179h3 r9 s6 − 177h3 r9 s5 − 18h3 r8 s9 − 216h3r8 s8 − 9h3 r8 s7 + 243h3 r8 s6 + 60h3 r7 s9 + 102h3 r7 s8 − 162h3 r7 s7 − 42h3 r6 s9 + 42h3 r6 s8 ))/(2520hrs2 (r − s)2 (r − 1)2 (s− 1)2 (rs − s2 )(r + s − rs − 1)) − ((x − xn )4 (hr8 s4 − hr8 s5 + 3hr7 s6 − 2h r7 s5 − hr7 s4 − 3hr6 s7 + 3hr6 s5 + hr5 s8 + 2hr5 s7 − 3hr5 s6 − hr4 s8 + hr4 s7 ))/(12h2 r2 (r − s)(r − 1)2 (s − 1)2 (rs − s2 )2 (r + s − rs − 1)) − ((x − xn )6 (hr9 s4 − 2hr9 s5 + hr9 s3 + 4hr8 s6 − 3hr8 s5 − hr8 s3 + 4hr7 s6 − 3hr7 s5 − h r7 s4 − 4hr6 s8 − 4hr6 s7 + 8hr6 s5 + 2hr5 s9 + 3hr5 s8 + 3hr5 s7 − 8hr5 s6 − h r4 s9 + hr4 s7 − hr3 s9 + hr3 s8 ))/(30h4 r3 s2 (r − s)2 (r − 1)2 (s − 1)2 (rs − s2 ) (r + s − rs − 1)) − ((x − xn )5 (2hr9 s4 − hr9 s5 − hr9 s6 + 3hr8 s7 + 2hr8 s6 − 3hr8 s5 − 2hr8 s4 − 3hr7 s8 − hr7 s6 + 4hr7 s5 + hr6 s9 − 2hr6 s8 + hr6 s7 + h r5 s9 + 3hr5 s8 − 4hr5 s7 − 2hr4 s9 + 2hr4 s8 ))/20h3 r3 s2 (r − s)2 (r − 1)(s − 1) (rs − s2 )(r + s − rs − 1)2 ) − ((x − xn )9 (hr7 s3 − hr7 s4 + 3hr6 s5 − 2hr6 s4 − hr6 s3 − 3hr5 s6 + 3hr5 s4 + hr4 s7 + 2hr4 s6 − 3hr4 s5 − hr3 s7 + hr3 s6 ))/(72 h7 r3 s2 (r − s)2 (r − 1)(s − 1)(rs − s2 )(r2 s2 − 2r2 s + r2 − 2rs2 + 4rs − 2r+ s2 − 2s + 1)) − (s2 (r − s)2 (x − xn )8 (2r + 2s + 1))/(56h5 (r − 1)(r + s − rs

510

M. Alkasassbeh, Z. Omar −1)(r2 s2 − r2 s3 + 2rs4 − 2rs3 − s5 + s4 )) + ((x − xn )7 (hr9 s3 − hr9 s4 − hr8 s5 + hr8 s3 + 8hr7 s6 − 3hr7 s5 − 3hr7 s4 − 2hr7 s3 − 8hr6 s7 + 4hr6 s5 + 4hr6 s4 +hr5 s8 + 3hr5 s7 − 4hr5 s6 + hr4 s9 + 3hr4 s7 − 4hr4 s6 − hr3 s9 − hr3 s8 + 2hr3 s7 ))/(42h5 rs2 (r − s)(r − 1)(s − 1)2 (rs − s2 )(r + s − rs − 1)(r2 s − r3 s − r3
+r4 )) .

Appendex II

h2 r 2 −10r 5 s − 10r 5 + 36r 4 s2 + 53r 4 s + 36r 4 − 36r 3 s3 − 108r 3 s2 − 2520s3
108r 3 s − 36r 3 + 90r 2 s3 + 24r 2 s2 + 90r 2 s + 168rs3 + 168rs2 − 882s3 ,

(0)

B16 = −

(0)

h2 s2 −36r 3 s3 + 90r 3 s2 + 168r 3 s − 882r 3 + 36r 2 s4 − 108r 2 s3 + 24 2520r 3
r 2 s2 + 168r 2 s − 10rs5 + 53rs4 − 108rs3 + 90rs2 − 10s5 + 36s4 − 36s3 ,

B26 = −

(0)

h2 −882r 3 s3 + 168r 3 s2 + 90r 3 s − 36r 3 + 168r 2 s3 + 24r 2 s2 − 2520r 3 s3
108r 2 s + 36r 2 + 90rs3 − 108rs2 + 53rs − 10r − 36s3 + 36s2 − 10s , hr =− −5r 5 s − 5r 5 + 16r 4 s2 + 23r 4 s + 16r 4 − 14r 3 s3 − 40r 3 s2 − 40 420s3
r 3 s − 14r 3 + 28r 2 s3 + 28r 2 s + 56rs3 + 56rs2 − 210s3 , hs 14r 3 s3 − 28r 3 s2 − 56r 3 s + 210r 3 − 16r 2 s4 + 40r 2 s3 − 56r 2 s+ = 420r 3
5rs5 − 23rs4 + 40rs3 − 28rs2 + 5s5 − 16s4 + 14s3 , h = 210r 3 s3 − 56r 3 s2 − 28r 3 s + 14r 3 − 56r 2 s3 + 40r 2 s − 16r 2 − 420r 3 s3
28rs3 + 40rs2 − 23rs + 5r + 14s3 − 16s2 + 5s) .

B36 = − (0)

B46

(0)

B56

(0)

B66

(1)

h2 r 2 105r 6 − 385r 5 s − 385r 5 + 468r 4 s2 + 1457r 4 s+ 2520(r − s)3 (r − 1)3 468r 4 − 180r 3 s3 − 1836r 3 s2 − 1836r 3 s − 180r 3 + 720r 2 s3 + 2418r 2 s2 +
720r 2 s − 966rs3 − 966rs2 + 378s3 ,

B11 = −

(1)

B12 =

h2 r 6 −20r 4 s + 10r 4 + 75r 3 s2 + 25r 3 s − 36r 3 − 63r 2 s3 2520s3 (r − s)3 (s − 1)3

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

511


−243r 2 s2 + 108r 2 s + 36r 2 + 252rs3 + 138rs2 − 198rs − 294s3 + 210s2 ,

h2 r 6 10r 4 s − 20r 4 − 36r 3 s2 + 25r 3 s + 75r 3 + 36r 2 s3 2520(r − 1)3 (s − 1)3
+108r 2 s2 − 243r 2 s − 63r 2 − 198rs3 + 138rs2 + 252rs + 210s3 − 294s2 ,

(1)

B13 = −

h2 s6 −63r 3 s2 + 252r 3 s − 294r 3 + 75r 2 s3 − 243r 2 s2 2520r 3 (r − s)3 (r − 1)3 )
+138r 2 s + 210r 2 − 20rs4 + 25rs3 + 108rs2 − 198rs + 10s4 − 36s3 + 36s2 ,

(1)

B21 = −

(1)

B22 =

−h2 s2 −180r 3 s3 + 720r 3 s2 − 966r 3 s + 378r 3 + 468r 2 s4 − 2520(r − s)3 (s − 1)3 1836r 2 s3 + 105s6 + 2418r 2 s2 − 966r 2 s − 385rs5 + 1457rs4 − 1836rs3 +
720rs2 − 385s5 + 468s4 − 180s3 ,

h2 s6 36r 3 s2 − 198r 3 s + 210r 3 − 36r 2 s3 + 108r 2 s2 + 2520(r − 1)3 (s − 1)3
138r 2 s − 294r 2 + 10rs4 + 25rs3 − 243rs2 + 252rs − 20s4 + 75s3 − 63s2 ,

(1)

B23 = −

h2 −294r 3 s2 + 252r 3 s − 63r 3 + 210r 2 s3 + 138r 2 s2 2520r 3 (r − s)3 (r − 1)3
−243r 2 s + 75r 2 − 198rs3 + 108rs2 + 25rs − 20r + 36s3 − 36s2 + 10s ,

(1)

B31 = −

(1)

B32 =

(1)

h2 210r 3 s2 − 198r 3 s + 36r 3 − 294r 2 s3 + 138r 2 s2 2520s3 (r − s)3 (s − 1)3
+108r 2 s − 36r 2 + 252rs3 − 243rs2 + 25rs + 10r − 63s3 + 75s2 − 20s ,

h2 −378r 3 s3 + 966r 3 s2 − 720r 3 s + 180r 3 + 966r 2 s3 2520(r − 1)3 (s − 1)3 −2418r 2 s2 + 1836r 2 s − 468r 2 − 720rs3 + 1836rs2 − 1457rs + 385r + 180
s3 − 468s2 + 385s − 105 , hr 105r 6 − 350r 5 s − 350r 5 + 388r 4 s2 + 1187r 4 s + 388r 4 = 420(r − s)3 (r − 1)3 −140r 3 s3 − 1342r 3 s2 − 1342r 3 s − 140r 3 + 490r 2 s3 + 1554r 2 s2 + 490r 2 s−
574rs3 − 574rs2 + 210s3 ,

B33 = −

(1)

B41

(1)

B42 =

(1)

hr 5 −10r 4 s + 5r 4 + 35r 3 s2 + 10r 3 s − 16r 3 − 28r 2 s3 − 420s3 (r − s)3 (s − 1)3
98r 2 s2 + 46r 2 s + 14r 2 + 98rs3 + 42rs2 − 70rs − 98s3 + 70s2 ,

B43 = −

hr 5 5r 4 s − 10r 4 − 16r 3 s2 + 10r 3 s + 35r 3 + 14r 2 s3 420(r − 1)3 (s − 1)3

512

M. Alkasassbeh, Z. Omar


+46r 2 s2 − 98r 2 s − 28r 2 − 70rs3 + 42rs2 + 98rs + 70s3 − 98s2 ,

(1)

hs5 −28r 3 s2 + 98r 3 s − 98r 3 + 35r 2 s3 − 98r 2 s2 + 420r 3 (r − s)3 (r − 1)3
42r 2 s + 70r 2 − 10rs4 + 10rs3 + 46rs2 − 70rs + 5s4 − 16s3 + 14s2 , −hs = −140r 3 s3 + 490r 3 s2 − 574r 3 s + 210r 3 + 388r 2 s4 420(r − s)3 (s − 1)3 −1342r 2 s3 + 105s6 + 1554r 2 s2 − 574r 2 s − 350rs5 + 1187rs4 − 1342rs3
+490rs2 − 350s5 + 388s4 − 140s3 ,

B51 = −

(1)

B52

(1)

hs5 14r 3 s2 − 70r 3 s + 70r 3 − 16r 2 s3 + 46r 2 s2 + 42r 2 s 420(r − 1)3 (s − 1)3
−98r 2 + 5rs4 + 10rs3 − 98rs2 + 98rs − 10s4 + 35s3 − 28s2 , h =− −98r 3 s2 + 98r 3 s − 28r 3 + 70r 2 s3 + 42r 2 s2 − 3 420r (r − s)3 (r − 1)3
98r 2 s + 35r 2 − 70rs3 + 46rs2 + 10rs − 10r + 14s3 − 16s2 + 5s , h 70r 3 s2 − 70r 3 s + 14r 3 − 98r 2 s3 + 42r 2 s2 + = 420s3 (r − s)3 (s − 1)3
46r 2 s − 16r 2 + 98rs3 − 98rs2 + 10rs + 5r − 28s3 + 35s2 − 10s , h −210r 3 s3 + 574r 3 s2 − 490r 3 s + 140r 3 + 574r 2 =− 420(r − 1)3 (s − 1)3 s3 − 1554r 2 s2 + 1342r 2 s − 388r 2 − 490rs3 + 1342rs2 − 1187rs + 350
r + 140s3 − 388s2 + 350s − 105 .

B53 = −

(1)

B61

(1)

B62

(1)

B63

(0)

D16 =

(0)

D26 =

(0)

D36 =

(0)

D46 =

h3 r 3 5r 4 − 18r 3 s − 18r 3 + 18r 2 s2 + 72r 2 s + 18r 2 − 84rs2 − 84rs 2520s2
+126s2 ,

h3 s3 18r 2 s2 − 84r 2 s + 126r 2 − 18rs3 + 72rs2 − 84rs + 5s4 − 18s3 2520r 2
+18s2 ,

h3 126r 2 s2 − 84r 2 s + 18r 2 − 84rs2 + 72rs − 18r + 18s2 − 18s 2520r 2 s2 +5) , h2 r 2 5r 4 − 16r 3 s − 16r 3 + 14r 2 s2 + 56r 2 s + 14r 2 − 56rs2 − 56rs 840s2
+70s2 ,

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

(0)

D56 =

(0)

513

h2 s2 14r 2 s2 − 56r 2 s + 70r 2 − 16rs3 + 56rs2 − 56rs + 5s4 − 16s3 840r 2
+14s2 ,


h2 2 2 2 2 2 2 70r s − 56r s + 14r − 56rs + 56rs − 16r + 14s − 16s + 5 . 840r 2 s2 h3 r 3 =− 5r 4 − 15r 3 s − 15r 3 + 12r 2 s2 + 48r 2 s + 12r 2 − 1260(r − s)2 (r − 1)2
42rs2 − 42rs + 42s2 ,
h3 r 6 2 2 3 = 18r − 42s + 36rs − 9r s − 18r + 5r , 2520s2 (r − s)2 (s − 1)2
h3 r 6 5r 3 − 18r 2 s − 9r 2 + 18rs2 + 36rs − 42s2 , = 2 2 2520(r − 1) (s − 1)
h3 s6 =− 42r − 18s − 36rs + 9rs2 + 18s2 − 5s3 , 2 2 2 2520r (r − s) (r − 1) h3 s3 =− 12r 2 s2 − 42r 2 s + 42r 2 − 15rs3 + 48rs2 − 1260(r − s)2 (s − 1)2
42rs + 5s4 − 15s3 + 12s2 ,
h3 s6 2 2 2 3 2 18r s − 42r − 18rs + 36rs + 5s − 9s , = 2520(r − 1)2 (s − 1)2
h3 =− 9r + 18s − 36rs + 42rs2 − 18s2 − 5 , 2 2 2 2520r (r − s) (r − 1)
h3 =− 18r + 9s − 36rs + 42r 2 s − 18r 2 − 5 , 2 2 2 2520s (r − s) (s − 1) h3 =− 42r 2 s2 − 42r 2 s + 12r 2 − 42rs2 + 48rs − 15r 1260(r − 1)2 (s − 1)2
+12s2 − 15s + 5 ,

D66 = (1)

D11

(1)

D12

(1)

D13

(1)

D21

(1)

D22

(1)

D23

(1)

D31

(1)

D32

(1)

D33

(1)

h2 r 2 15r 4 − 40r 3 s − 40r 3 + 28r 2 s2 + 112r 2 s + 28r 2 840(r − s)2 (r − 1)2
−84rs2 − 84rs + 70s2 ,
h2 r 5 14r − 28s + 28rs − 8r 2 s − 16r 2 + 5r 3 , = 2 2 2 840s (r − s) (s − 1)
h2 r 5 3 2 2 2 2 = 5r − 16r s − 8r + 14rs + 28rs − 28s , 840(r − 1)2 (s − 1)2

D41 = −

(1)

D42

(1)

D43

514 (1)

M. Alkasassbeh, Z. Omar


h2 s5 28r − 14s − 28rs + 8rs2 + 16s2 − 5s3 2 2 2 840r (r − s) (r − 1) h2 s2 28r 2 s2 − 84r 2 s + 70r 2 − 40rs3 + 112rs2 − =− 840(r − s)2 (s − 1)2
84rs + 15s4 − 40s3 + 28s2 ,
h2 s5 = 14r 2 s − 28r 2 − 16rs2 + 28rs + 5s3 − 8s2 , 2 2 840(r − 1) (s − 1)
h2 2 2 =− 8r + 16s − 28rs + 28rs − 14s − 5 , 840r 2 (r − s)2 (r − 1)2
h2 =− 16r + 8s − 28rs + 28r 2 s − 14r 2 − 5 , 2 2 2 840s (r − s) (s − 1) h2 =− 70r 2 s2 − 84r 2 s + 28r 2 − 84rs2 + 112rs− 840(r − 1)2 (s − 1)2
40r + 28s2 − 40s + 15 .

D51 = − (1)

D52

(1)

D53

(1)

D61

(1)

D62

(1)

D63

Appindex III

E1 =

P

(i) ∞ hi r i i=0 ( i! )yn

− yn − hryn′ −

1 2 (2) 2 3 2520s3 (h yn r (882s

+ 10r5 (1 + s)

−168rs2 (1 + s) + 36r3 (1 + s)3 − 6r2 s(15 + 4s + 15s2 ) − r4 (36+ P∞ hi+2 yn(i+2) 1 i 2 6 53s + 36s2 ))) − i=0 [ 2520(−1+r) 3 (r−s)3 ) (r) (r (105r + 378 i! s3 − 385r5 (1 + s) − 966rs2 (1 + s) + 6r2 s(120 + 403s + 120s2 ) + r4

(468 + 1457s + 468s2 ) − 36r3 (5 + 51s + 51s2 + 5s3 )) −

1 (2520(r−s)3 (−1+s)3 s3 )) 2

(r6 (10r4 (−1 + 2s) + 42s2 (−5 + 7s) + r3 (36 − 25s − 75s ) − 6rs(−33 +23s + 42s2 ) + 9r2 (−4 − 12s + 27s2 + 7s3 ))(s)i −

1 (2520(−1+r)3 (−1+s)3 ) 2

(r6 (10r4 (−2 + s) + 42s2 (−7 + 5s) + r3 (75 + 25s − 36s ) − 6rs(−42 −23s + 33s2 ) + 9r2 (−7 − 27s + 12s2 + 4s3 ))] −

1 3 (3) 3 4 (2520s2 ) (h yn r (5r +

126s2 − 18r3 (1 + s) − 84rs(1 + s) + 18r2 (1 + 4s + s2 ))) +

P∞

i=0

(i+3) hi+3 yn i! 2

1 i 3 4 2 3 [ (1260(−1+r) 2 (r−s)2 )) r ((r (5r + 42s − 15r (1 + s) − 42rs(1 + s) + 12r

(1 + 4s + s2 ))) − (1 + 2s)))) −

1 i 6 3 (2520(r−s)2 (−1+s)2 s2 ) s ((r (5r

1 6 3 (2520(−1+r)2 (−1+s)2 ) ((r (5r

− 42s − 9r2 (2 + s) + 18r

 − 42s2 + 18rs(2 + s) − 9r2 (1 + 2s))) ,

515

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT...

E2 =

hP ∞

(i) hi si i=0 ( i! )yn

− yn − hsyn′ −

1 2 (2) 2 3 (2520r 3 ) (h yn s (2s (18

− 18s + 5s2 ) − 12r2 s

(14 + 2s − 9s2 + 3s3 ) + 6r3 (147 − 28s − 15s2 + 6s3 ) + rs2 (−90 + 108s− P∞ hi+2 yn(i+2) 1 i 6 3 [ (2520(−1+r) 53s2 + 10s3 ))) − i=0 3 r 3 (r−s)3 ) (r) ((s (21r (14 − 12s i!

+3s2 ) − 2s2 (18 − 18s + 5s2 ) + rs(198 − 108s − 25s2 + 20s3 ) − 3r2 (70 + 46s −81s2 + 25s3 )))) +

1 i 2 3 (2520(r−s)3 (−1+s)3 ) (s) ((s (s (180 2 3 2

− 468s + 385s2 − 105s3 )

+6r3 (−63 + 161s − 120s + 30s ) − 6r s(−161 + 403s − 306s2 + 78s3 ) + rs2 (−720 + 1836s − 1457s2 + 385s3 )))) − 2

3

2

2

1 6 2 (2520(−1+r)3 (−1+s)3 )) (((s (s (−63 2 3

+ 75s

−20s ) + 6r (35 − 33s + 6s ) − 6r (49 − 23s − 18s + 6s ) + rs(252 − 243s (3)

1 3 3 2 2 2 +25s2 + 10s3 ))))] − [ (2520r 2 ) (h yn (s (6r (21 − 14s + 3s ) − 6rs(14 − 12s + 3s ) P∞ hi+3 yn(i+3) 1 i 6 +s2 (18 − 18s + 5s2 ))))] − i=0 840i! [ (2520(−1+r) 2 r 2 (r−s)2 ) r ((s (r(−42 + 36s

−9s2 ) + s(18 − 18s + 5s2 )))) −

1 i 3 2 (1260(r−s)2 (−1+s)2 )) s (((s (6r (7

− 7s + 2s2 ) − 3rs i 6 2 (−7+3s)+s2 (−9+5s))) , (14 − 16s + 5s2 ) + s2 (12 − 15s + 5s2 )))) (s (−18r(−2+s)s+6r ] 2 2 (2520(−1+r) (−1+s) ) P (i) ∞ hi 1 ′ 2 (2) 2 E3 = i=0 ( i! )yn − yn − hyn − 2520r 3 s3 (h yn (2s(5 − 18s + 18s ) + r(10 − 53s +108s2 − 90s3 ) − 12r2 (3 − 9s + 2s2 + 14s3 ) + 6r3 (6 − 15s − 28s2 + 147s3 )) (i+2) P hi+2 yn 1 i 3 2 [ (2520(−1+r) − ∞ 3 r 3 (r−s)3 ) r ((21r (3 − 12s + 14s ) − 2s(5 − 18s i=0 i! +18s2 ) − 3r2 (25 − 81s + 46s2 + 70s3 ) + r(20 − 25s − 108s2 + 198s3 )))+ 1 i (2520(r−s)3 (−1+s)3 s3 ) s ((s(−20

+ 75s − 63s2 ) + 6r3 (6 − 33s + 35s2 ) − 6r2 (6 − 18s 1 (2520(−1+r)3 (−1+s)3 ) ((105 2 3 2

−23s2 + 49s3 ) + r(10 + 25s − 243s2 + 252s3))) +

−385s + 468s2 − 180s3 + 6r3 (−30 + 120s − 161s + 63s ) − 6r (−78 + 306s −403s2 + 161s3 ) + r(−385 + 1457s − 1836s2 + 720s3 )))] −

1 3 (3) (2520r 2 s2 ) (h yn (5

−18s + 18s2 − 6r(3 − 12s + 14s2 ) + 6r2 (3 − 14s + 21s2 ))) −

P∞

i=0

(i+3) hi+3 yn i!

1 i 2 2 [ (2520(−1+r) 2 r 2 (r−s)2 ) r ((5 − 18s + 18s + r(−9 + 36s − 42s )))+ 1 i (2520(r−s)2 (−1+s)2 s2 ) s ((5

− 9s + 18r(−1 + 2s) − 6r2 (−3 + 7s)))

1 (1260(−1+r)2 (−1+s)2 ) ((−5

 + 15s − 12s2 − 6r2 (2 − 7s + 7s2 ) + 3r(5 − 16s + 14s2 )))] ,

E4 =

P

(i+1) ∞ hi r i i=0 ( i! )yn

− yn′ −

(2) 1 3 (420s3 ) (hyn r(210s

+ 5r5 (1 + s) − 56rs2 (1 + s)

−r4 (16 + 23s + 16s2 ) − 28r2 (s + s3 ) + 2r3 (7 + 20s + 20s2 + 7s3 )))− P∞ hi+1 yn(i+2) 1 i 6 3 5 2 [ (420(−1+r) 3 (r−s)3 ) r (r(105r + 210s − 350r (1 + s) − 574rs i=0 i!

516

M. Alkasassbeh, Z. Omar

(1 + s) + 14r2 s(35 + 111s + 35s2 ) + r4 (388 + 1187s + 388s2) − 2r3 (70 + 671s 1 i 5 4 (420(r−s)3 (−1+s)3 s3 )) s (−((r (5r (−1 2 2 2

+671s2 + 70s3 ))) +

+ 2s) + 14s2 (−5 + 7s)

+r3 (16 − 10s − 35s ) − 14rs(−5 + 3s + 7s ) + 2r (−7 − 23s + 49s2 + 14s3 )))) + (420(−1+r)13 (−1+s)3 )) (−r5 (5r4 (−2 + s) + 14s2 (−7 + 5s) + r3 (35 + 10s − 16s2 ) +14rs(7 + 3s − 5s2 ) + 2r2 (−14 − 49s + 23s2 + 7s3 )))] −

1 2 (3) 2 4 (840s2 ) (h yn (r (5r

+70s2 − 16r3 (1 + s) − 56rs(1 + s) + 14r2 (1 + 4s + s2 )))) +

P∞

i=0

(i+3) hi+2 yn i! 2

1 i 2 4 2 3 [ (840(−1+r) 2 (r−s)2 )) r (−((r (15r + 70s − 40r (1 + s) − 84rs(1 + s) + 28r 1 i 5 3 (840(r−s)2 (−1+s)2 s2 ) s ((r (5r

− 28s − 8r2 (2 + s) + 14r(1 + 2s))))  + (840(−1+r)12 (−1+s)2 ) ((r5 (5r3 − 28s2 + 14rs(2 + s) − 8r2 (1 + 2s))))] , P (i+1) (2) ∞ hi si 1 3 2 3 E5 = − yn′ − (420r 3 )) (hyn s(s (14 − 16s + 5s ) + 14r (15 − 4s− i=0 ( i! )yn (i+2) P hi+1 yn 2s2 + s3 )−8r2 s(7 − 5s2 + 2s3 ) + rs2 (−28 + 40s − 23s2 + 5s3 ))) − ∞ i=0 i! (1 + 4s + s2 )))) +

[ (420(−1+r)13 r3 (r−s)3 ) ri ((s5 (s2 (−14 + 16s − 5s2 ) + 14r3 (7 − 7s + 2s2 ) − 7r2

(10 + 6s − 14s2 + 5s3 ) + 2rs(35 − 23s − 5s2 + 5s3 )))) +

1 (420(r−s)3 (−1+s)3 )) 2 3

si (s(s3 (140 − 388s + 350s2 − 105s3 ) + 14r3 (−15 + 41s − 35s + 10s )−

2r2 s(−287 + 777s − 671s2 + 194s3) + rs2 (−490 + 1342s − 1187s2 + 350 s3 ))) +

1 5 2 (420(−1+r)3 (−1+s)3 )) (−s (s (−28 2 3

+ 35s − 10s2 ) + 14r3 (5 − 5s + s2 )

+r2 (−98 + 42s + 46s − 16s ) + rs(98 − 98s + 10s2 + 5s3 )))] −

1 (840r 2 )

(3)

(h2 yn (s2 (14r2 (5 − 4s + s2 ) − 8rs(7 − 7s + 2s2 ) + s2 (14 − 16s + 5s2 )))) P∞ hi+2 yn(i+3) + i=0 [ (840(−1+r)12 r2 (r−s)2 ) ri ((s5 (−4r(7 − 7s + 2s2 ) + s(14 − 16s i!

+5s2 )))) +

1 i 2 2 (840(r−s)2 (−1+s)2 )) s (−((s (14r (5

− 6s + 2s2 ) − 4rs(21 − 28s+  5 2 2 (−8+5s))) 10s2 ) + s2 (28 − 40s + 15s2 )))) + (s (14r (−2+s)−4rs(−7+4s)+s ] , (840(−1+r)2 (−1+s)2 ) P

(i+1) ∞ hi i=0 ( i! )yn

(2) 1 (420r 3 s3 )) (hyn (s(5

− 16s + 14s2 ) + r(5 − 23s + 40s2 P∞ hi+1 yn(i+2) −28s3 ) − 8r2 (2 − 5s + 7s3 ) + 14r3 (1 − 2s − 4s2 + 15s3 ))) − i=0 i!

E6 =

− yn′ −

[ (420(−1+r)13 r3 (r−s)3 ) ri ((s(−5 + 16s − 14s2 ) + 14r3 (2 − 7s + 7s2 ) − 7r2 (5−

14s + 6s2 + 10s3 ) + 2r(5 − 5s − 23s2 + 35s3 ))) +

1 i (420(r−s)3 (−1+s)3 s3 ) s ((s(−10 2 3

+35s − 28s2 ) + 14r3 (1 − 5s + 5s2 ) + r2 (−16 + 46s + 42s − 98s ) + r(5 + 10s −98s2 + 98s3 ))) +

1 (420(−1+r)3 (−1+s)3 )) ((105

− 350s + 388s2 − 140s3 + 14r3 (−10

+35s − 41s2 + 15s3 ) − 2r2 (−194 + 671s − 777s2 + 287s3 ) + r(−350 + 1187s

GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT... −1342s2 + 490s3)))] −

− 16s + 14s2 + 14r2 (1 − 4s + 5s2 )

(i+3) hi+2 yn [ (840(−1+r)12 r2 (r−s)2 ) ri ((5 − 16s + 14s2 − 4r i! 1 i 2 (840(r−s)2 (−1+s)2 s2 ) s ((5 + r (14 − 28s) − 8s + 4r(−4 + 7s)))

−8r(2 − 7s + 7s2 ))) − (2 − 7s + 7s2 ))) +

1 i+2 (3) yn (5 (840r 2 s2 ) (h

517

P∞

i=0

+ (840(−1+r)12 (−1+s)2 ) ((−15 + 40s − 28s2 − 14r2 (2 − 6s + 5s2 ) + 4r(10 − 28s+
21s2 ))) .

518