A four-step implicit block method with three generalized off-step points for solving fourth order initial value problems directly

Ra’ft Abdelrahim; Zurni Omar

doi:10.1016/j.jksus.2017.06.003

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Review

29 (

); 401-412

doi:

10.1016/j.jksus.2017.06.003

A four-step implicit block method with three generalized off-step points for solving fourth order initial value problems directly

Ra’ft Abdelrahim^{^a,⁎}, Zurni Omar^{^b}

a Department of Mathematics, College of Art and Sciences at Tabarjal, Al Jouf University, Saudi Arabia

b Department of mathematics, College of Art and Sciences, Univeristi Utara Malaysia, Malaysia

⁎Corresponding author. rafatshaab@yahoo.com (Ra’ft Abdelrahim),

Received: 2017-1-24, Accepted: 2017-6-14, Published: 2017-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

The existing hybrid methods for solving ordinary differential equations were only derived using specific off-step points. Thus, this paper proposes a new four-step block method with three generalized off-step points for solving initial value problems of fourth order ordinary differential equations directly. The strategy employed to develop this method is interpolating the basis function at $y_{n + j}, j = 0 (1) 3$ and collocating the fourth derivative of the basis function at all points within the selected interval. The implementation of this method in a block-by-block fashion can overcome the setbacks of applying starting values and predictors which are created in predictor-corrector approach. The convergence analysis of the developed method is performed and the accuracy of the method is tested on several problems. The numerical results indicate that the new method outperforms the existing ones in terms of errors. In addition, the new method does not require much computation when compared with previous methods.

Keywords

Block method

Hybrid block method

Fourth order ordinary differential equations

Collocation and interpolation

Three off-step points

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1 Introduction

In this paper, we consider the numerical solution of the fourth order IVPs of the form

(1)

y^{⁗} = f (x, y, y^{'}, y^{″}, y^{‴}), x \in [a, b]

with initial conditions

y (a) = ω_{0}, y^{'} (a) = ω_{1}, y^{″} (a) = ω_{2}, y^{‴} (a) = ω_{3}

Eq. (1) arises in wide fields of science and engineering. This equation can be solved by converting it into system of four equations of first-order IVPs and then suitable numerical methods for first order is employed (see Henrici, 1962; Lambert, 1973). However, this approach increases the number of equations and therefore some researchers have opted hybrid block methods for solving Eq. (1) directly, for instance, among of these researchers are (Yap and Ismail, 2015; Abdelrahim and Omar, 2015; Hussain et al., 2015; Omar and Abdelrahim, 2016; Jator, 2008; Awoyemi, 2005 ). In hybrid block methods, not only the numerical approximation can be computed at more than one point in the same time, but computational burden and zero stability barrier can also be avoided, see Adesanya et al. (2012) and Kayode (2008). However, the existing hybrid block methods only focus on the specific off step points. Hence, the aim of this work is to generalize three off step points of four step block method. As a results, the new method will be more robust and flexible.

2 Derivation of the method

This section shows the derivation of a four-step hybrid block method with three generalized off-step points $x_{n + s_{1}}, x_{n + s_{2}}$ and $x_{n + s_{3}}$ for solving (1).

Consider the polynomial of the form:

(2)

y (x) = \sum_{i = 0}^{q + d - 1} a_{i} {(\frac{x - x_{n}}{h})}^{i} .

as an approximate solution of (1), where

x \in [x_{n}, x_{n + 4}]

for

n = 0, 4, 8, \dots, N - 4, q

is the number of interpolation points which is equal to the order of differential equation, dis the number of collocation points and

h = x_{n} - x_{n - 1}

is constant step size of partition of interval

[a, b]

which is given by

a = x_{0} < x_{1} < \dots < x_{N - 1} < x_{N} = b

The next step in the derivation of the hybrid block method involves differentiating (2) four times to give

(3)

y^{⁗} (x) = f (x, y, y^{'}, y^{″}, y^{‴}) = \sum_{i = 4}^{q + d - 1} \frac{i (i - 1) (i - 2) (i - 3)}{h^{4}} a_{i} {(\frac{x - x_{n}}{h})}^{i - 4} .

Interpolating the approximate solution (2) at $x_{n + j}, j = 0 (1) 3$ and collocating Eq. (3) at all points in the selected interval produces twelve equations which can be written as a system in matrix form $AX = B$ , where $A = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 & 2048 \\ 1 & 3 & 9 & 27 & 81 & 243 & 729 & 2187 & 6561 & 19683 & 59049 & 177147 \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{(120 s_{1})}{h^{4}} & \frac{(360 s_{1}^{2})}{h^{4}} & \frac{(840 s_{1}^{3})}{h^{4}} & \frac{(1680 s_{1}^{4})}{h^{4}} & \frac{(3024 s_{1}^{5})}{h^{4}} & \frac{(5040 s_{1}^{6})}{h^{4}} & \frac{(7920 s_{1}^{7})}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{120}{h^{4}} & \frac{360}{h^{4}} & \frac{840}{h^{4}} & \frac{1680}{h^{4}} & \frac{3024}{h^{4}} & \frac{5040}{h^{4}} & \frac{7920}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{(120 s_{2})}{h^{4}} & \frac{(360 s_{2}^{2})}{h^{4}} & \frac{(840 s_{2}^{3})}{h^{4}} & \frac{(1680 s_{2}^{4})}{h^{4}} & \frac{(3024 s_{2}^{5})}{h^{4}} & \frac{(5040 s_{2}^{6})}{h^{4}} & \frac{(7920 s_{2}^{7})}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{240}{h^{4}} & \frac{1440}{h^{4}} & \frac{6720}{h^{4}} & \frac{26880}{h^{4}} & \frac{96768}{h^{4}} & \frac{322560}{h^{4}} & \frac{1013760}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{(120 s_{3})}{h^{4}} & \frac{(360 s_{3}^{2})}{h^{4}} & \frac{(840 s_{3}^{3})}{h^{4}} & \frac{(1680 s_{3}^{4})}{h^{4}} & \frac{(3024 s_{3}^{5})}{h^{4}} & \frac{(5040 s_{3}^{6})}{h^{4}} & \frac{(7920 s_{3}^{7})}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{360}{h^{4}} & \frac{3240}{h^{4}} & \frac{22680}{h^{4}} & \frac{136080}{h^{4}} & \frac{734832}{h^{4}} & \frac{3674160}{h^{4}} & \frac{17321040}{h^{4}} \\ 0 & 0 & 0 & 0 & \frac{24}{h^{4}} & \frac{480}{h^{4}} & \frac{5760}{h^{4}} & \frac{53760}{h^{4}} & \frac{430080}{h^{4}} & \frac{3096576}{h^{4}} & \frac{20643840}{h^{4}} & \frac{129761280}{h^{4}} \end{matrix}),$ $\begin{matrix} X = {[a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, a_{11}]}^{T} and \\ B = {[y_{n}, y_{n + 1}, y_{n + 2}, y_{n + 3}, f_{n}, f_{n + s_{1}}, f_{n + 1}, f_{n + s_{2}}, f_{n + 2}, f_{n + s_{3}}, f_{n + 3}, f_{n + 4}]}^{T} . \end{matrix}$

The unknown values of $a_{i}^{'} s, i = 0 (1) 11$ can be obtained by matrix inverse approach, where $X = A^{- 1} B$ . The values obtained are then substituted back into Eq. (2) to produce a continuous implicit scheme of the form

(4)

y (x) = \sum_{i = 0}^{3} α_{i} (x) y_{n + i} + \sum_{i = 0}^{4} β_{i} (x) f_{n + i} + \sum_{i = 1}^{3} β_{s_{i}} (x) f_{n + s_{i}}

Differentiating (4) once, twice and thrice gives

(5)

y^{(j)} (x) = \sum_{i = 0}^{3} α_{i}^{(j)} (x) y_{n + i} + \sum_{i = 0}^{4} β_{i}^{(j)} (x) f_{n + i} + \sum_{i = 1}^{3} β_{s_{i}}^{(j)} (x) f_{n + s_{i}}; j = 1 (1) 3 .

Eq. (4) is evaluated at the non-interpolating point $x_{n + s_{i}}, x_{n + 4}, i = 1 (1) 3$ while Eqs. (5) is evaluated at $x_{n}$ to give the discrete schemes and its derivatives at $x_{n}$ . The discrete schemes and its derivatives at $x_{n}$ are combined in a matrix form as below

(6)

{AY}_{M}^{{[3]}_{4}} = B^{{[3]}_{4}} R_{1}^{{[3]}_{4}} + h^{4} [D^{{[3]}_{4}} R_{2}^{{[3]}_{4}} + E^{{[3]}_{4}} R_{3}^{{[3]}_{4}}]

where,

A = (\begin{matrix} 1 & \frac{- (s 1 (s 1 - 2) (s 1 - 3))}{2} & 0 & \frac{(s 1 (s 1 - 1) (s 1 - 3))}{2} & 0 & \frac{- (s 1 (s 1 - 1) (s 1 - 2))}{6} & 0 \\ 0 & \frac{- (s 2 (s 2 - 2) (s 2 - 3))}{2} & 1 & \frac{(s 2 (s 2 - 1) (s 2 - 3))}{2} & 0 & \frac{- (s 2 (s 2 - 1) (s 2 - 2))}{6} & 0 \\ 0 & \frac{- (s 3 (s 3 - 2) (s 3 - 3))}{2} & 0 & \frac{(s 3 (s 3 - 1) (s 3 - 3))}{2} & 1 & \frac{- (s 3 (s 3 - 1) (s 3 - 2))}{6} & 0 \\ 0 & - 4 & 0 & 6 & 0 & - 4 & 1 \\ 0 & \frac{- 3}{h} & 0 & \frac{3}{(2 h)} & 0 & \frac{- 1}{(3 h)} & 0 \\ 0 & \frac{5}{h^{2}} & 0 & \frac{- 4}{h^{2}} & 0 & \frac{1}{h^{2}} & 0 \\ 0 & \frac{- 3}{h^{3}} & 0 & \frac{3}{h^{3}} & 0 & \frac{- 1}{h^{3}} & 0 \end{matrix}), Y_{M}^{{[3]}_{4}} = (\begin{matrix} y_{n + s_{1}} \\ y_{n + 1} \\ y_{n + s_{2}} \\ y_{n + 2} \\ y_{n + s_{3}} \\ y_{n + 3} \\ y_{n + 4} \end{matrix}),

B^{{[3]}_{4}} = (\begin{matrix} \frac{- ((s 1 - 1) (s 1 - 2) (s 1 - 3))}{6} & 0 & 0 & 0 \\ \frac{- ((s 2 - 1) (s 2 - 2) (s 2 - 3))}{6} & 0 & 0 & 0 \\ \frac{- ((s 3 - 1) (s 3 - 2) (s 3 - 3))}{6} & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 \\ \frac{- 11}{(6 h)} & - 1 & 0 & 0 \\ \frac{2}{h^{2}} & 0 & - 1 & 0 \\ \frac{- 1}{h^{3}} & 0 & 0 & - 1 \end{matrix}), R_{1}^{{[3]}_{4}} = (\begin{matrix} y_{n} \\ y_{n}^{'} \\ y_{n}^{″} \\ y_{n}^{‴} \end{matrix}), R_{2}^{{[3]}_{4}} = [\begin{matrix} f_{n} \end{matrix}],

{R_{3}}^{{[3]}_{4}} = (\begin{matrix} f_{n + s_{1}} \\ f_{n + 1} \\ f_{n + s_{2}} \\ f_{n + 2} \\ f_{n + s_{3}} \\ f_{n + 3} \\ f_{n + 4} \end{matrix}), D^{{[3]}_{4}} = (\begin{matrix} D_{11} \\ D_{21} \\ D_{31} \\ D_{41} \\ D_{51} \\ D_{61} \\ D_{71} \end{matrix}), and E^{{[3]}_{4}} = (\begin{matrix} E_{11} & E_{12} & E_{13} & E_{14} & E_{15} & E_{16} & E_{17} \\ E_{21} & E_{22} & E_{23} & E_{24} & E_{25} & E_{26} & E_{27} \\ E_{31} & E_{32} & E_{33} & E_{34} & E_{35} & E_{36} & E_{37} \\ E_{41} & E_{42} & E_{43} & E_{44} & E_{45} & E_{46} & E_{47} \\ E_{51} & E_{52} & E_{53} & E_{54} & E_{55} & E_{56} & E_{57} \\ E_{61} & E_{62} & E_{63} & E_{64} & E_{65} & E_{66} & E_{67} \\ E_{71} & E_{72} & E_{73} & E_{74} & E_{75} & E_{76} & E_{77} \end{matrix}) .

The elements of $D^{{[3]}_{4}}$ and $E^{{[3]}_{4}}$ are given in A.

Multiplying Eq. (6) by the inverse of A gives a four-step hybrid block method (main block) with three generalized off-step points of the form

(7)

Y_{M}^{{[3]}_{4}} = A^{- 1} B^{{[3]}_{4}} R_{1}^{{[3]}_{4}} + h^{4} [A^{- 1} D^{{[3]}_{4}} R_{2}^{{[3]}_{4}} + A^{- 1} E^{{[3]}_{4}} R_{3}^{{[3]}_{4}}]

which can be seen in Appendix B.

In order to get the derivatives (first, second and third) of the block, the value of the block at $y_{n + 1}, y_{n + 2}$ and $y_{n + 3}$ are substituted into the first, second and third derivatives of the discrete schemes and this produces first, second and third derivatives of the block.

3 Analysis of the Method

3.1

3.1 Order of the Method

In finding order of the new block method (7), we define the linear difference operator L associated with (7)

(8)

L [y (x); h] = Y_{m}^{{[3]}_{4}} - A^{- 1} B^{{[3]}_{4}} R_{1}^{{[3]}_{4}} - h^{4} [A^{- 1} D^{{[3]}_{4}} R_{2}^{{[3]}_{4}} + A^{- 1} E^{{[3]}_{4}} R_{3}^{{[3]}_{4}}]

Expanding $Y_{m}^{{[3]}_{4}}$ and $R_{3}^{{[3]}_{4}}$ components in Taylors series respectively and collecting their terms in powers of h yields

(9)

L [y (x), h] = \bar{C_{0}} y (x) + \bar{C_{1}} {hy}^{'} (x) + {\overline{C}}_{2} h^{2} y^{″} (x) + \dots

Definition 3.1

The hybrid block method (7) and its linear operator (8) are said to have order p, if $\bar{C_{0}} = \bar{C_{1}} = \bar{C_{2}} = \dots = {\overline{C}}_{p + 3} = 0$ and ${\overline{C}}_{p + 4} \neq 0$ with error constants vector ${\overline{C}}_{p + 4}$ .

Expanding (8) about $x = x_{n}$ using Taylor series gives the orders of all integrators in the main block method which are ${[8, 8, 8, 8, 8, 8, 8]}^{T}$ . Therefore, the new hybrid block method is consistent since each integrator has order greater than one.

3.2

3.2 Zero stability

Following Fatunla(1991), the zero stability of the method will obtain by finding the roots of the first characteristic function $Π (x)$ as given below, where I is $7 \times 7$ identity matrix and ${\overline{B}}^{{[3]}_{4}}$ is coefficients matrix of $y_{n}$ , $\begin{matrix} Π (x) = & | x I - {\overline{B}}^{{[3]}_{4}} | \\ = & |x (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) - (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})| \\ = & x^{6} (x - 1) \end{matrix}$

Solving $Π (x) = 0$ yields $x = 0, 0, 0, 0, 0, 0, 1$ . Hence, our method is zero stable since $Π (x)$ having roots such that $| x_{z} | ⩽ 1$ , and when $| x_{z} | = 1$ , the multiplicity of $x_{z}$ doesn’t exceed four.

3.3

3.3 Convergence

Theorem 3.2

Theorem 3.2 Henrici, 1962

Consistency and zero stability are sufficient conditions for a linear multistep method to be convergent.

Following the theorem in the above the developed method (7) is convergent since it is zero stable and consistent.

3.4

3.4 Implementation of the method

The developed method is implemented by combining all the hybrid integrators for IVPs simultaneously without additional method to predict the starting values. We continue by explicitly obtaining initial conditions at x, $n = 0, 4, \dots, N - 4$ using the computed values $y (x_{n + 4}) = y_{n + 4}$ over subintervals $[x_{0}, x_{4}], \dots, [x_{N - 4}, x_{N}]$ . For example, using (7), $n = 0,_{M} = 0, (y_{s_{1}}, y_{1}, y_{s_{2}}, y_{2}, y_{s_{3}}, y_{3}, y_{4})$ where its first, second and third derivatives are simultaneously obtained over the subinterval $[x_{0}, x_{4}]$ as $y_{0}^{(j)} (x_{0}), j = 0 (1) 3$ are known from the IVP (1), for $n = 4,_{M} = 1, (y_{s_{1} + 4}, y_{5}, y_{s_{2} + 4}, y_{6}, y_{7}, y_{s_{3} + 4}, y_{8})$ with its derivatives are simultaneously obtained over the subinterval $[x_{4}, x_{8}]$ as $y_{4}^{(j)} (x_{4}), j = 0 (1) 3$ are known from the previous block and so on. Hence, the method is applied over non overlapping subintervals and the solutions derived are more accurate than those obtained in the existing method. We remind that, the system of these equations has a unique solution since the determinant of its matrix $\neq 0$ when $s_{i} \in (x_{n + (i - 1)}, x_{n + i}), i = 1 (1) 3$ . We also note that, for linear problems, we solve (7) directly from the start with Gaussian elimination using partial pivoting and for nonlinear problems, we use a modified Newton–Raphson method.

4 Numerical results

In this section, the following IVPs available in previous literatures which are solved to special case of the new method when $s_{1} = \frac{1}{8}, s_{2} = \frac{5}{3}$ and $s_{3} = \frac{7}{2}$ (chosen arbitrarily) in order to compare the performance of new method with the existing ones in [3,4–9]. It is worth noting that the block method proposed is in a generalized form and hence can take varying off-step point values which gives room for flexibility. The absolute errors at different value of end point x with several h were carried out using flexible Matlab code as depicted in Tables 1,2,3,4.

Table 1 Comparison of the new method with some existing methods for solving problem 1.

h	Method	$E (x_{N})$ at x = 2.	Steps
	New method	$8.07 e^{- 10}$	5
	Yap and Ismail (2015)	$1.74 e^{- 8}$	5
0.1	Adams	$2.11 e^{- 3}$
	Jator	$1.26 e^{- 4}$	20

	New method	$3.22 e^{- 12}$	10
	Yap and Ismail (2015)	$8.45 e^{- 11}$	10
0.05	Adams	$5.37 e^{- 4}$
	Jator	$1.91 e^{- 6}$	40

	New method	$7.30 e^{- 16}$	20
	Yap and Ismail (2015)	$3.69 e^{- 13}$	20
0.025	Adams	$5.09 e^{- 5}$
	Jator	$2.96 e^{- 8}$	80

Table 2 Comparison of the new method with some existing methods for solving problem 2.

h	Method	$E (x_{N})$ at x = 1.01325	Steps.
	New method	$1.65 e^{- 18}$	80
0.003125	Kayode (2008)	$1.58 e^{- 7}$	320

	New method	$2.95 e^{- 17}$	3
0.103125	Awoyemi et al. (2015)	$5.69 e^{- 6}$	2

h		$E (x_{N})$ at x = 1
	New method	$1.45 e^{- 18}$	3
0.1	Adesanya et al (2012)	$8.04 e^{- 16}$	2

Table 3 Comparison of the new method with Yap and Ismail (2015) for solving problem 1.

x	AE in new method	AE in Yap (2015).
0.2	$3.512997 \times 10^{- 13}$	$2.318519 \times 10^{- 13}$
0.4	$4.183300 \times 10^{- 12}$	$2.260324 \times 10^{- 12}$
0.6	$1.430233 \times 10^{- 11}$	$1.965140 \times 10^{- 11}$
0.8	$3.592435 \times 10^{- 11}$	$9.914494 \times 10^{- 11}$
1.0	$7.276201 \times 10^{- 11}$	$3.311345 \times 10^{- 10}$
1.2	$1.336016 \times 10^{- 10}$	$9.000018 \times 10^{- 10}$
1.4	$2.234540 \times 10^{- 10}$	$2.117600 \times 10^{- 9}$
1.6	$3.579606 \times 10^{- 10}$	$4.550582 \times 10^{- 9}$
1.8	$5.433495 \times 10^{- 10}$	$9.117964 \times 10^{- 9}$
2.0	$8.079574 \times 10^{- 10}$	$1.740907 \times 10^{- 8}$

Table 4 Comparison of the new method with Awoyemi et al. (2015) and Kayode (2008) for solving problem 2.

x	AE in Kayode (2008)	AE in Awoyemi et al. (2015)	AE in new method.
0.103125	$0.48355417 \times 10^{- 16}$	$2.11164 \times 10^{- 13}$	$2.269123 \times 10^{- 19}$
0.206250	$0.13933299 \times 10^{- 15}$	$5.69866 \times 10^{- 12}$	$4.554620 \times 10^{- 19}$
0.306250	$0.66893539 \times 10^{- 15}$	$6.80311 \times 10^{- 10}$	$3.891211 \times 10^{- 19}$
0.406250	$0.20129384 \times 10^{- 14}$	$2.20723 \times 10^{- 9}$	$6.038588 \times 10^{- 19}$
0.506250	$0.46736053 \times 10^{- 14}$	$1.27407 \times 10^{- 8}$	$9.727741 \times 10^{- 19}$
0.603125	$0.91874598 \times 10^{- 14}$	$3.45612 \times 10^{- 6}$	$8.900363 \times 10^{- 19}$
0.703125	$0.16069038 \times 10^{- 13}$	$6.55238 \times 10^{- 6}$	$8.843511 \times 10^{- 19}$
0.803125	$0.25407974 \times 10^{- 13}$	$9.58653 \times 10^{- 6}$	$6.690965 \times 10^{- 19}$
0.903125	$0.38108926 \times 10^{- 13}$	$1.04933 \times 10^{- 6}$	$2.258860 \times 10^{- 18}$
1.003125	$0.54051538 \times 10^{- 13}$	$5.69624 \times 10^{- 6}$	$1.653371 \times 10^{- 18}$

The following notations are used in the tables.

h:step size.
Step: total number of steps taken to obtain solution.
$E (x_{N})$ : magnitude of the maximum error of the computed solution.
AE: absolute error.

\begin{matrix} Problem 1 : y^{iv} = y^{‴} + y^{″} + y^{'} + 2 y, y (0) = 1, \\ y^{'} (0) = 0, y^{″} (0) = 0, y^{‴} (0) = 30,. \\ Exact solution : y (x) = 2 e^{2 x} - 5 e^{- x} + 3 \cos x - 9 \sin x \\ Problem 2 : y^{iv} = - y^{″}, y (0) = 0, y^{'} (0) = \frac{- 1.1}{72 - 50 π}, y^{″} (0) = \frac{1}{144 - 100 π}, \\ y^{‴} (0) = \frac{1.2}{144 - 100 π} \\ Exact solution : y (x) = 1 - x - \cos x - \frac{1.2 \sin x}{144 - 100 π} \end{matrix}

In general, the results from Tables 1, show that the performance of four-step hybrid block methods for solving fourth order ODEs directly using three specific hybrids points is better than existing methods in terms of error. These are demonstrated in (Figs. 1 and 2).

Application:

The new developed method is applied for solving physical problem occurs in ship dynamics. In particular, this problem has been studied and solved numerically by Twizell (1988) and Cortell (1993) which describes how the sinusoidal wave of frequency $Ω$ passes along a ship or offshore structure to lead to fourth order differential equation relates the fluids action with time x as below

(10)

y^{⁗} = - 3 y^{″} - y (2 + ∊ \cos (Ω x))

which is imposed to the following conditions:

y (0) = 1, y^{'} (0) = 0, y^{″} (0) = 0, y^{‴} (0) = 0

. where

∊ = 0

for the existence of the theoretical solution,

y (x) = 2 \cos x - \cos (x \sqrt{2})

. The theoretical solution is undefined when

∊ \neq 0

The comparison of the new method with Twizell (1988) and Cortell (1993) has been made at the end point $x = 15$ in terms of accuracy (refer to Table 5), where $h = 0.25$ and $h = 0.1$ .

Table 5 Comparison of the new method with Twizell (1988) and Cortell (1993) for solving Eq. (10).

h	Method	Error at x = 15.
	New method	$8.15 e^{- 8}$
0.25	Yap and Ismail (2015)	$5.2 e^{- 7}$
	Twizell	$1.9 e^{- 4}$

	New method	$3.6 e^{- 11}$
0.1	Yap and Ismail (2015)	$2.8 e^{- 10}$
	Cortell	$3.7 e^{- 5}$

The block method (7) is self-starting and applied without the use of predictors which can reduce the accuracy of the method as well as increase programming time.

5 Conclusion

We have developed a four-step block method with three generalized off step points for the solution of fourth order initial value problems which was applied without the use of predictors that reduce the accuracy of the method. The properties of the developed block method which include: zero stability, order, consistency and convergence are established. The existing four-step hybrid block methods only focus on the specific off step points. Since the new method works for any three off-step points, it is therefore more flexible and robust. The application of the developed method was then made to several fourth order initial value problems. In general, the benefit of using the four-step hybrid block method for solving IVPs of fourth order ODEs is obvious. The new method displays its superiority by producing less error if compared to the present methods as shown in Tables 1 and 2. In addition, the developed method does not require much computation when compared with predictor corrector methods.

Author Contributions

Both authors equally contributed to this work.

Conflicts of Interest

The authors declare no conflict of interest.

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Appendix A

$\begin{matrix} D_{11} = \frac{- h^{4} (s_{1} - 1) (s_{1} - 2) (s_{1} - 3)}{3991680 s_{2} s_{3}} (2310 s_{2} - 220 s_{1} + 2310 s_{3} + 3718 s_{1} s_{2} + 3718 s_{1} s_{3} \\ + 924 s_{2} s_{3} - 3377 s_{1}^{2} s_{2} - 3377 s_{1}^{2} s_{3} - 1430 s_{1}^{3} s_{2} - 1430 s_{1}^{3} s_{3} + 1375 s_{1}^{4} s_{2} + 1375 s_{1}^{4} s_{3} - 308 s_{1}^{5} s_{2} \\ - 308 s_{1}^{5} s_{3} + 22 s_{1}^{6} s_{2} + 22 s_{1}^{6} s_{3} + 1762 s_{1}^{2} + 1052 s_{1}^{3} + 130 s_{1}^{4} - 520 s_{1}^{5} + 148 s_{1}^{6} - 12 s_{1}^{7} \\ + 11550 s_{1}^{2} s_{2} s_{3} - 4400 s_{1}^{3} s_{2} s_{3} + 726 s_{1}^{4} s_{2} s_{3} - 44 s_{1}^{5} s_{2} s_{3} - 11000 s_{1} s_{2} s_{3} - 5040) \\ D_{12} = \frac{- h^{4} (s_{2} - 1) (s_{2} - 2) (s_{2} - 3)}{3991680 s_{1} s_{3}} (2310 s_{1} - 220 s_{2} + 2310 s_{3} + 3718 s_{1} s_{2} + 924 s_{1} s_{3} \\ + 3718 s_{2} s_{3} - 3377 s_{1} s_{2}^{2} - 1430 s_{1} s_{2}^{3} + 1375 s_{1} s_{2}^{4} - 3377 s_{2}^{2} s_{3} - 308 s_{1} s_{2}^{5} - 1430 s_{2}^{3} s_{3} \\ + 22 s_{1} s_{2}^{6} + 1375 s_{2}^{4} s_{3} - 308 s_{2}^{5} s_{3} + 22 s_{2}^{6} s_{3} + 1762 s_{2}^{2} + 1052 s_{2}^{3} + 130 s_{2}^{4} - 520 s_{2}^{5} \\ + 148 s_{2}^{6} - 12 s_{2}^{7} + 11550 s_{1} s_{2}^{2} s_{3} - 4400 s_{1} s_{2}^{3} s_{3} + 726 s_{1} s_{2}^{4} s_{3} - 44 s_{1} s_{2}^{5} s_{3} - 11000 s_{1} s_{2} s_{3} - 5040) \\ D_{13} = \frac{- h^{4} (s_{3} - 1) (s_{3} - 2) (s_{3} - 3)}{3991680 s_{1} s_{2}} (2310 s_{1} + 2310 s_{2} - 220 s_{3} + 924 s_{1} s_{2} + 3718 s_{1} s_{3} \\ + 3718 s_{2} s_{3} - 3377 s_{1} s_{3}^{2} - 1430 s_{1} s_{3}^{3} - 3377 s_{2} s_{3}^{2} + 1375 s_{1} s_{3}^{4} - 1430 s_{2} s_{3}^{3} - 308 s_{1} s_{3}^{5} + 1375 s_{2} s_{3}^{4} \\ + 22 s_{1} s_{3}^{6} - 308 s_{2} s_{3}^{5} + 22 s_{2} s_{3}^{6} + 1762 s_{3}^{2} + 1052 s_{3}^{3} + 130 s_{3}^{4} - 520 s_{3}^{5} + 148 s_{3}^{6} - 12 s_{3}^{7} \\ + 11550 s_{1} s_{2} s_{3}^{2} - 4400 s_{1} s_{2} s_{3}^{3} + 726 s_{1} s_{2} s_{3}^{4} - 44 s_{1} s_{2} s_{3}^{5} - 11000 s_{1} s_{2} s_{3} - 5040) \\ D_{14} = \frac{h^{4} (50 s_{1} + 50 s_{2} + 50 s_{3} - 7 s_{1} s_{2} s_{3} - 200)}{5040 s_{1} s_{2} s_{3}} \\ D_{15} = \frac{h^{3} (770 s_{1} + 770 s_{2} + 770 s_{3} + 308 s_{1} s_{2} + 308 s_{1} s_{3} + 308 s_{2} s_{3} - 2123 s_{1} s_{2} s_{3} - 1680)}{221760 s_{1} s_{2} s_{3}} \\ D_{16} = \frac{- h^{2} (262 s_{1} + 262 s_{2} + 262 s_{3} + 1150 s_{1} s_{2} + 1150 s_{1} s_{3} + 1150 s_{2} s_{3} - 4463 s_{1} s_{2} s_{3} - 1220)}{60480 s_{1} s_{2} s_{3}} \\ D_{17} = \frac{- h (850 s_{1} + 850 s_{2} + 850 s_{3} - 4415 s_{1} s_{2} - 4415 s_{1} s_{3} - 4415 s_{2} s_{3} + 20034 s_{1} s_{2} s_{3} + 782)}{60480 s_{1} s_{2} s_{3}} \\ E_{11} = \frac{- h^{4}}{((166320 s_{1} - 665280) (s_{1} - s_{2}) (s_{1} - s_{3}))} (2530 s_{1} - 2310 s_{2} - 2310 s_{3} - 2794 s_{1} s_{2} \\ - 2794 s_{1} s_{3} - 924 s_{2} s_{3} - 1254 s_{1}^{2} s_{2} - 1254 s_{1}^{2} s_{3} + 110 s_{1}^{3} s_{2} + 110 s_{1}^{3} s_{3} + 990 s_{1}^{4} s_{2} + 990 s_{1}^{4} s_{3} \\ - 352 s_{1}^{5} s_{2} - 352 s_{1}^{5} s_{3} + 33 s_{1}^{6} s_{2} + 33 s_{1}^{6} s_{3} + 1032 s_{1}^{2} + 202 s_{1}^{3} - 240 s_{1}^{4} - 470 s_{1}^{5} + 204 s_{1}^{6} - 21 s_{1}^{7} \\ + 1320 s_{1}^{2} s_{2} s_{3} - 2365 s_{1}^{3} s_{2} s_{3} + 660 s_{1}^{4} s_{2} s_{3} - 55 s_{1}^{5} s_{2} s_{3} + 4631 s_{1} s_{2} s_{3} + 5040) \\ E_{12} = \frac{- (h^{4} s_{1} (s_{1} - 2) (s_{1} - 3))}{((997920 s_{2} - 997920) (s_{3} - 1))} (24684 s_{1} s_{2} - 31251 s_{2} - 31251 s_{3} - 24464 s_{1} \\ + 24684 s_{1} s_{3} + 28941 s_{2} s_{3} + 2684 s_{1}^{2} s_{2} + 2684 s_{1}^{2} s_{3} - 1188 s_{1}^{3} s_{2} - 1188 s_{1}^{3} s_{3} - 748 s_{1}^{4} s_{2} - 748 s_{1}^{4} s_{3} \\ + 264 s_{1}^{5} s_{2} + 264 s_{1}^{5} s_{3} - 22 s_{1}^{6} s_{2} - 22 s_{1}^{6} s_{3} - 4446 s_{1}^{2} + 136 s_{1}^{3} + 618 s_{1}^{4} + 256 s_{1}^{5} - 126 s_{1}^{6} + 12 s_{1}^{7} \\ + 693 s_{1}^{2} s_{2} s_{3} + 2618 s_{1}^{3} s_{2} s_{3} - 627 s_{1}^{4} s_{2} s_{3} + 44 s_{1}^{5} s_{2} s_{3} - 28402 s_{1} s_{2} s_{3} + 36291) \end{matrix}$ $\begin{matrix} E_{13} = \frac{- (h^{4} s_{1} (s_{1} - 1) (s_{1} - 2) (s_{1} - 3))}{(166320 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} (2310 s_{3} - 220 s_{1} \\ + 3718 s_{1} s_{3} - 3377 s_{1}^{2} s_{3} - 1430 s_{1}^{3} s_{3} + 1375 s_{1}^{4} s_{3} - 308 s_{1}^{5} s_{3} + 22 s_{1}^{6} s_{3} + 1762 s_{1}^{2} \\ + 1052 s_{1}^{3} + 130 s_{1}^{4} - 520 s_{1}^{5} + 148 s_{1}^{6} - 12 s_{1}^{7} - 5040)) \\ E_{14} = \frac{(h^{4} s_{1} (s_{1} - 1) (s_{1} - 3))}{((665280 s_{2} - 1330560) (s_{3} - 2))} (11000 s_{1} + 27126 s_{2} + 27126 s_{3} - 5390 s_{1} s_{2} \\ - 5390 s_{1} s_{3} - 14718 s_{2} s_{3} - 2959 s_{1}^{2} s_{2} - 2959 s_{1}^{2} s_{3} - 1298 s_{1}^{3} s_{2} - 1298 s_{1}^{3} s_{3} - 319 s_{1}^{4} s_{2} - 319 s_{1}^{4} s_{3} \\ + 220 s_{1}^{5} s_{2} + 220 s_{1}^{5} s_{3} - 22 s_{1}^{6} s_{2} - 22 s_{1}^{6} s_{3} + 4156 s_{1}^{2} + 1544 s_{1}^{3} + 508 s_{1}^{4} + 80 s_{1}^{5} - 104 s_{1}^{6} \\ + 12 s_{1}^{7} + 3168 s_{1}^{2} s_{2} s_{3} + 1364 s_{1}^{3} s_{2} s_{3} - 528 s_{1}^{4} s_{2} s_{3} + 44 s_{1}^{5} s_{2} s_{3} \\ + 836 s_{1} s_{2} s_{3} - 49212) \\ E_{15} = \frac{(h^{4} s_{1} (s_{1} - 1) (s_{1} - 2) (s_{1} - 3))}{(166320 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} (2310 s_{2} - 220 s_{1} \\ + 3718 s_{1} s_{2} - 3377 s_{1}^{2} s_{2} - 1430 s_{1}^{3} s_{2} + 1375 s_{1}^{4} s_{2} - 308 s_{1}^{5} s_{2} + 22 s_{1}^{6} s_{2} + 1762 s_{1}^{2} + 1052 s_{1}^{3} + 130 s_{1}^{4} \\ - 520 s_{1}^{5} + 148 s_{1}^{6} - 12 s_{1}^{7} - 5040) \\ E_{16} = \frac{(h^{4} s_{1} (s_{1} - 1) (s_{1} - 2))}{((997920 s_{2} - 2993760) (s_{3} - 3))} (484 s_{1} s_{2} - 2541 s_{2} - 2541 s_{3} - 1672 s_{1} + 484 s_{1} s_{3} \\ + 1617 s_{2} s_{3} + 880 s_{1}^{2} s_{2} + 880 s_{1}^{2} s_{3} + 484 s_{1}^{3} s_{2} + 484 s_{1}^{3} s_{3} + 88 s_{1}^{4} s_{2} + 88 s_{1}^{4} s_{3} - 176 s_{1}^{5} s_{2} \\ - 176 s_{1}^{5} s_{3} + 22 s_{1}^{6} s_{2} + 22 s_{1}^{6} s_{3} - 878 s_{1}^{2} - 400 s_{1}^{3} - 134 s_{1}^{4} + 8 s_{1}^{5} + 82 s_{1}^{6} - 12 s_{1}^{7} \\ - 1419 s_{1}^{2} s_{2} s_{3} - 638 s_{1}^{3} s_{2} s_{3} + 429 s_{1}^{4} s_{2} s_{3} - 44 s_{1}^{5} s_{2} s_{3} + 1078 s_{1} s_{2} s_{3} + 2583) \\ E_{17} = \frac{- (h^{4} s_{1} (s_{1} - 1) (s_{1} - 2) (s_{1} - 3))}{((3991680 s_{2} - 15966720) (s_{1} - 4) (s_{3} - 4))} (198 s_{1} s_{2} - 1914 s_{2} - 1914 s_{3} - 1012 s_{1} \\ + 198 s_{1} s_{3} + 1056 s_{2} s_{3} + 583 s_{1}^{2} s_{2} + 583 s_{1}^{2} s_{3} + 330 s_{1}^{3} s_{2} + 330 s_{1}^{3} s_{3} + 55 s_{1}^{4} s_{2} + 55 s_{1}^{4} s_{3} - 132 s_{1}^{5} s_{2} \\ - 132 s_{1}^{5} s_{3} + 22 s_{1}^{6} s_{2} + 22 s_{1}^{6} s_{3} - 570 s_{1}^{2} - 268 s_{1}^{3} - 90 s_{1}^{4} + 8 s_{1}^{5} + 60 s_{1}^{6} - 12 s_{1}^{7} - 990 s_{1}^{2} s_{2} s_{3} \\ - 440 s_{1}^{3} s_{2} s_{3} + 330 s_{1}^{4} s_{2} s_{3} - 44 s_{1}^{5} s_{2} s_{3} + 880 s_{1} s_{2} s_{3} + 2616) \\ E_{21} = \frac{(h^{4} s_{2} (s_{2} - 1) (s_{2} - 2) (s_{2} - 3))}{(166320 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} (2310 s_{3} - 220 s_{2} \\ + 3718 s_{2} s_{3} - 3377 s_{2}^{2} s_{3} - 1430 s_{2}^{3} s_{3} + 1375 s_{2}^{4} s_{3} - 308 s_{2}^{5} s_{3} + 22 s_{2}^{6} s_{3} + 1762 s_{2}^{2} \\ + 1052 s_{2}^{3} + 130 s_{2}^{4} - 520 s_{2}^{5} + 148 s_{2}^{6} - 12 s_{2}^{7} - 5040) \\ E_{22} = \frac{- (h^{4} s_{2} (s_{2} - 2) (s_{2} - 3))}{((997920 s_{1} - 997920) (s_{3} - 1))} (24684 s_{1} s_{2} - 24464 s_{2} - 31251 s_{3} - 31251 s_{1} + 28941 s_{1} s_{3} \\ + 24684 s_{2} s_{3} + 2684 s_{1} s_{2}^{2} - 1188 s_{1} s_{2}^{3} - 748 s_{1} s_{2}^{4} + 2684 s_{2}^{2} s_{3} + 264 s_{1} s_{2}^{5} - 1188 s_{2}^{3} s_{3} - 22 s_{1} s_{2}^{6} \\ - 748 s_{2}^{4} s_{3} + 264 s_{2}^{5} s_{3} - 22 s_{2}^{6} s_{3} - 4446 s_{2}^{2} + 136 s_{2}^{3} + 618 s_{2}^{4} + 256 s_{2}^{5} - 126 s_{2}^{6} \\ + 12 s_{2}^{7} + 693 s_{1} s_{2}^{2} s_{3} + 2618 s_{1} s_{2}^{3} s_{3} - 627 s_{1} s_{2}^{4} s_{3} + 44 s_{1} s_{2}^{5} s_{3} - 28402 s_{1} s_{2} s_{3} + 36291) \\ E_{23} = \frac{- (h^{4})}{((166320 s_{2} - 665280) (s_{1} - s_{2}) (s_{2} - s_{3}))} (2310 s_{1} - 2530 s_{2} + 2310 s_{3} + 2794 s_{1} s_{2} \\ + 924 s_{1} s_{3} + 2794 s_{2} s_{3} + 1254 s_{1} s_{2}^{2} - 110 s_{1} s_{2}^{3} - 990 s_{1} s_{2}^{4} + 1254 s_{2}^{2} s_{3} + 352 s_{1} s_{2}^{5} - 110 s_{2}^{3} s_{3} \\ - 33 s_{1} s_{2}^{6} - 990 s_{2}^{4} s_{3} + 352 s_{2}^{5} s_{3} - 33 s_{2}^{6} s_{3} - 1032 s_{2}^{2} - 202 s_{2}^{3} + 240 s_{2}^{4} + 470 s_{2}^{5} - 204 s_{2}^{6} \\ + 21 s_{2}^{7} - 1320 s_{1} s_{2}^{2} s_{3} + 2365 s_{1} s_{2}^{3} s_{3} - 660 s_{1} s_{2}^{4} s_{3} + 55 s_{1} s_{2}^{5} s_{3} - 4631 s_{1} s_{2} s_{3} - 5040) \end{matrix}$ $\begin{matrix} E_{24} = \frac{(h^{4} s_{2} (s_{2} - 1) (s_{2} - 3))}{((665280 s_{1} - 1330560) (s_{3} - 2))} (27126 s_{1} + 11000 s_{2} + 27126 s_{3} - 5390 s_{1} s_{2} \\ - 14718 s_{1} s_{3} - 5390 s_{2} s_{3} - 2959 s_{1} s_{2}^{2} - 1298 s_{1} s_{2}^{3} - 319 s_{1} s_{2}^{4} - 2959 s_{2}^{2} s_{3} + 220 s_{1} s_{2}^{5} - 1298 s_{2}^{3} s_{3} \\ - 22 s_{1} s_{2}^{6} - 319 s_{2}^{4} s_{3} + 220 s_{2}^{5} s_{3} - 22 s_{2}^{6} s_{3} + 4156 s_{2}^{2} + 1544 s_{2}^{3} + 508 s_{2}^{4} + 80 s_{2}^{5} - 104 s_{2}^{6} + 12 s_{2}^{7} \\ + 3168 s_{1} s_{2}^{2} s_{3} + 1364 s_{1} s_{2}^{3} s_{3} - 528 s_{1} s_{2}^{4} s_{3} + 44 s_{1} s_{2}^{5} s_{3} + 836 s_{1} s_{2} s_{3} - 49212) \\ E_{25} = \frac{(h^{4} s_{2} (s_{2} - 1) (s_{2} - 2) (s_{2} - 3))}{(166320 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} (2310 s_{1} - 220 s_{2} \\ + 3718 s_{1} s_{2} - 3377 s_{1} s_{2}^{2} - 1430 s_{1} s_{2}^{3} + 1375 s_{1} s_{2}^{4} - 308 s_{1} s_{2}^{5} \\ + 22 s_{1} s_{2}^{6} + 1762 s_{2}^{2} + 1052 s_{2}^{3} + 130 s_{2}^{4} - 520 s_{2}^{5} + 148 s_{2}^{6} - 12 s_{2}^{7} - 5040) \\ E_{26} = \frac{(h^{4} s_{2} (s_{2} - 1) (s_{2} - 2))}{((997920 s_{1} - 2993760) (s_{3} - 3))} (484 s_{1} s_{2} - 1672 s_{2} - 2541 s_{3} - 2541 s_{1} + 1617 s_{1} s_{3} \\ + 484 s_{2} s_{3} + 880 s_{1} s_{2}^{2} + 484 s_{1} s_{2}^{3} + 88 s_{1} s_{2}^{4} + 880 s_{2}^{2} s_{3} - 176 s_{1} s_{2}^{5} + 484 s_{2}^{3} s_{3} + 22 s_{1} s_{2}^{6} + 88 s_{2}^{4} s_{3} \\ - 176 s_{2}^{5} s_{3} + 22 s_{2}^{6} s_{3} - 878 s_{2}^{2} - 400 s_{2}^{3} - 134 s_{2}^{4} + 8 s_{2}^{5} + 82 s_{2}^{6} - 12 s_{2}^{7} \\ - 1419 s_{1} s_{2}^{2} s_{3} - 638 s_{1} s_{2}^{3} s_{3} + 429 s_{1} s_{2}^{4} s_{3} - 44 s_{1} s_{2}^{5} s_{3} + 1078 s_{1} s_{2} s_{3} + 2583) \\ E_{27} = \frac{- (h^{4} s_{2} (s_{2} - 1) (s_{2} - 2) (s_{2} - 3))}{((3991680 s_{2} - 15966720) (s_{1} - 4) (s_{3} - 4))} (198 s_{1} s_{2} - 1012 s_{2} - 1914 s_{3} - 1914 s_{1} \\ + 1056 s_{1} s_{3} + 198 s_{2} s_{3} + 583 s_{1} s_{2}^{2} + 330 s_{1} s_{2}^{3} + 55 s_{1} s_{2}^{4} + 583 s_{2}^{2} s_{3} - 132 s_{1} s_{2}^{5} + 330 s_{2}^{3} s_{3} \\ + 22 s_{1} s_{2}^{6} + 55 s_{2}^{4} s_{3} - 132 s_{2}^{5} s_{3} + 22 s_{2}^{6} s_{3} - 570 s_{2}^{2} - 268 s_{2}^{3} - 90 s_{2}^{4} + 8 s_{2}^{5} + 60 s_{2}^{6} - 12 s_{2}^{7} \\ - 990 s_{1} s_{2}^{2} s_{3} - 440 s_{1} s_{2}^{3} s_{3} + 330 s_{1} s_{2}^{4} s_{3} - 44 s_{1} s_{2}^{5} s_{3} + 880 s_{1} s_{2} s_{3} + 2616) \\ E_{31} = \frac{(h^{4} s_{3} (s_{3} - 1) (s_{3} - 2) (s_{3} - 3))}{(166320 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} (2310 s_{2} - 220 s_{3} \\ + 3718 s_{2} s_{3} - 3377 s_{2} s_{3}^{2} - 1430 s_{2} s_{3}^{3} + 1375 s_{2} s_{3}^{4} - 308 s_{2} s_{3}^{5} + 22 s_{2} s_{3}^{6} + 1762 s_{3}^{2} + 1052 s_{3}^{3} \\ + 130 s_{3}^{4} - 520 s_{3}^{5} + 148 s_{3}^{6} - 12 s_{3}^{7} - 5040) \\ E_{32} = \frac{- (h^{4} s_{3} (s_{3} - 2) (s_{3} - 3))}{((997920 s_{1} - 997920) (s_{2} - 1))} (28941 s_{1} s_{2} - 31251 s_{2} - 24464 s_{3} - 31251 s_{1} \\ + 24684 s_{1} s_{3} + 24684 s_{2} s_{3} + 2684 s_{1} s_{3}^{2} - 1188 s_{1} s_{3}^{3} + 2684 s_{2} s_{3}^{2} - 748 s_{1} s_{3}^{4} - 1188 s_{2} s_{3}^{3} + 264 s_{1} s_{3}^{5} \\ - 748 s_{2} s_{3}^{4} - 22 s_{1} s_{3}^{6} + 264 s_{2} s_{3}^{5} - 22 s_{2} s_{3}^{6} - 4446 s_{3}^{2} + 136 s_{3}^{3} + 618 s_{3}^{4} + 256 s_{3}^{5} - 126 s_{3}^{6} \\ + 12 s_{3}^{7} + 693 s_{1} s_{2} s_{3}^{2} + 2618 s_{1} s_{2} s_{3}^{3} - 627 s_{1} s_{2} s_{3}^{4} + 44 s_{1} s_{2} s_{3}^{5} - 28402 s_{1} s_{2} s_{3} + 36291) \\ E_{33} = \frac{- (h^{4} s_{3} (s_{3} - 1) (s_{3} - 2) (s_{3} - 3))}{(166320 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} (2310 s_{1} - 220 s_{3} \\ + 3718 s_{1} s_{3} - 3377 s_{1} s_{3}^{2} - 1430 s_{1} s_{3}^{3} + 1375 s_{1} s_{3}^{4} - 308 s_{1} s_{3}^{5} + 22 s_{1} s_{3}^{6} + 1762 s_{3}^{2} + 1052 s_{3}^{3} \\ + 130 s_{3}^{4} - 520 s_{3}^{5} + 148 s_{3}^{6} - 12 s_{3}^{7} - 5040) \\ E_{34} = \frac{(h^{4} s_{3} (s_{3} - 1) (s_{3} - 3))}{((665280 s_{1} - 1330560) (s_{2} - 2))} (27126 s_{1} + 27126 s_{2} + 11000 s_{3} - 14718 s_{1} s_{2} - 5390 s_{1} s_{3} \\ - 5390 s_{2} s_{3} - 2959 s_{1} s_{3}^{2} - 1298 s_{1} s_{3}^{3} - 2959 s_{2} s_{3}^{2} - 319 s_{1} s_{3}^{4} - 1298 s_{2} s_{3}^{3} \\ + 220 s_{1} s_{3}^{5} - 319 s_{2} s_{3}^{4} - 22 s_{1} s_{3}^{6} + 220 s_{2} s_{3}^{5} - 22 s_{2} s_{3}^{6} + 4156 s_{3}^{2} + 1544 s_{3}^{3} + 508 s_{3}^{4} + 80 s_{3}^{5} - 104 s_{3}^{6} \\ + 12 s_{3}^{7} + 3168 s_{1} s_{2} s_{3}^{2} + 1364 s_{1} s_{2} s_{3}^{3} - 528 s_{1} s_{2} s_{3}^{4} + 44 s_{1} s_{2} s_{3}^{5} + 836 s_{1} s_{2} s_{3} - 49212) \end{matrix}$ $\begin{matrix} E_{35} = \frac{(h^{4})}{((166320 s_{3} - 665280) (s_{1} - s_{3}) (s_{2} - s_{3}))} (2310 s_{1} + 2310 s_{2} - 2530 s_{3} + 924 s_{1} s_{2} \\ + 2794 s_{1} s_{3} + 2794 s_{2} s_{3} + 1254 s_{1} s_{3}^{2} - 110 s_{1} s_{3}^{3} + 1254 s_{2} s_{3}^{2} \\ - 990 s_{1} s_{3}^{4} - 110 s_{2} s_{3}^{3} + 352 s_{1} s_{3}^{5} - 990 s_{2} s_{3}^{4} - 33 s_{1} s_{3}^{6} + 352 s_{2} s_{3}^{5} - 33 s_{2} s_{3}^{6} \\ - 1032 s_{3}^{2} - 202 s_{3}^{3} + 240 s_{3}^{4} + 470 s_{3}^{5} - 204 s_{3}^{6} + 21 s_{3}^{7} - 1320 s_{1} s_{2} s_{3}^{2} + 2365 s_{1} s_{2} s_{3}^{3} \\ - 660 s_{1} s_{2} s_{3}^{4} + 55 s_{1} s_{2} s_{3}^{5} - 4631 s_{1} s_{2} s_{3} - 5040) \\ E_{36} = \frac{(h^{4} s_{3} (s_{3} - 1) (s_{3} - 2))}{((997920 s_{1} - 2993760) (s_{2} - 3))} (1617 s_{1} s_{2} - 2541 s_{2} - 1672 s_{3} - 2541 s_{1} + 484 s_{1} s_{3} \\ + 484 s_{2} s_{3} + 880 s_{1} s_{3}^{2} + 484 s_{1} s_{3}^{3} + 880 s_{2} s_{3}^{2} + 88 s_{1} s_{3}^{4} + 484 s_{2} s_{3}^{3} - 176 s_{1} s_{3}^{5} + 88 s_{2} s_{3}^{4} \\ + 22 s_{1} s_{3}^{6} - 176 s_{2} s_{3}^{5} + 22 s_{2} s_{3}^{6} - 878 s_{3}^{2} - 400 s_{3}^{3} - 134 s_{3}^{4} + 8 s_{3}^{5} + 82 s_{3}^{6} \\ - 12 s_{3}^{7} - 1419 s_{1} s_{2} s_{3}^{2} - 638 s_{1} s_{2} s_{3}^{3} + 429 s_{1} s_{2} s_{3}^{4} - 44 s_{1} s_{2} s_{3}^{5} + 1078 s_{1} s_{2} s_{3} + 2583) \\ E_{37} = \frac{- (h^{4} s_{3} (s_{3} - 1) (s_{3} - 2) (s_{3} - 3))}{((3991680 s_{3} - 15966720) (s_{1} - 4) (s_{2} - 4))} (1056 s_{1} s_{2} - 1914 s_{2} - 1012 s_{3} - 1914 s_{1} \\ + 198 s_{1} s_{3} + 198 s_{2} s_{3} + 583 s_{1} s_{3}^{2} + 330 s_{1} s_{3}^{3} + 583 s_{2} s_{3}^{2} + 55 s_{1} s_{3}^{4} + 330 s_{2} s_{3}^{3} \\ - 132 s_{1} s_{3}^{5} + 55 s_{2} s_{3}^{4} + 22 s_{1} s_{3}^{6} - 132 s_{2} s_{3}^{5} + 22 s_{2} s_{3}^{6} - 570 s_{3}^{2} - 268 s_{3}^{3} - 90 s_{3}^{4} \\ + 8 s_{3}^{5} + 60 s_{3}^{6} - 12 s_{3}^{7} - 990 s_{1} s_{2} s_{3}^{2} - 440 s_{1} s_{2} s_{3}^{3} + 330 s_{1} s_{2} s_{3}^{4} - 44 s_{1} s_{2} s_{3}^{5} + 880 s_{1} s_{2} s_{3} + 2616) \\ E_{41} = \frac{- (5 h^{4} (s_{2} + s_{3} - 4))}{(21 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} \\ E_{42} = \frac{(h^{4} (167 s_{1} + 167 s_{2} + 167 s_{3} - 217 s_{1} s_{2} - 217 s_{1} s_{3} - 217 s_{2} s_{3} + 217 s_{1} s_{2} s_{3} + 33))}{((1260 s_{3} - 1260) (s_{1} - 1) (s_{2} - 1))} \\ E_{43} = \frac{(5 h^{4} (s_{1} + s_{3} - 4))}{(21 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} \\ E_{44} = \frac{(h^{4} (2262 s_{1} + 2262 s_{2} + 2262 s_{3} - 1106 s_{1} s_{2} - 1106 s_{1} s_{3} - 1106 s_{2} s_{3} + 553 s_{1} s_{2} s_{3} - 4724))}{((840 s_{3} - 1680) (s_{1} - 2) (s_{2} - 2))} \\ E_{45} = \frac{- (5 h^{4} (s_{1} + s_{2} - 4))}{(21 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} \\ E_{46} = \frac{(h^{4} (1903 s_{1} + 1903 s_{2} + 1903 s_{3} - 651 s_{1} s_{2} - 651 s_{1} s_{3} - 651 s_{2} s_{3} + 217 s_{1} s_{2} s_{3} - 5509))}{((1260 s_{3} - 3780) (s_{1} - 3) (s_{2} - 3))} \\ E_{47} = \frac{- (h^{4} (62 s_{1} + 62 s_{2} + 62 s_{3} - 28 s_{1} s_{2} - 28 s_{1} s_{3} - 28 s_{2} s_{3} + 7 s_{1} s_{2} s_{3} - 48))}{((5040 s_{3} - 20160) (s_{1} - 4) (s_{2} - 4))} \\ E_{51} = \frac{- (h^{3} (55 s_{2} + 55 s_{3} + 22 s_{2} s_{3} - 120))}{(660 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} \\ E_{52} = - (h^{3} (10417 s_{1} + 10417 s_{2} + 10417 s_{3} - 9647 s_{1} s_{2} - 9647 s_{1} s_{3} - 9647 s_{2} s_{3} \\ + 9955 s_{1} s_{2} s_{3} - 12097)) / ((55440 s_{1} - 55440) (s_{2} - 1) (s_{3} - 1)) \end{matrix}$ $\begin{matrix} E_{53} = \frac{(h^{3} (55 s_{1} + 55 s_{3} + 22 s_{1} s_{3} - 120))}{(660 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} \\ E_{54} = - (h^{3} (9042 s_{1} + 9042 s_{2} + 9042 s_{3} - 4906 s_{1} s_{2} - 4906 s_{1} s_{3} - 4906 s_{2} s_{3} \\ + 2299 s_{1} s_{2} s_{3} - 16404)) ((36960 s_{1} - 73920) (s_{2} - 2) (s_{3} - 2)) \\ E_{55} = \frac{- (h^{3} (55 s_{1} + 55 s_{2} + 22 s_{1} s_{2} - 120))}{(660 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} \\ E_{56} = \frac{(h^{3} (121 s_{1} + 121 s_{2} + 121 s_{3} - 77 s_{1} s_{2} - 77 s_{1} s_{3} - 77 s_{2} s_{3} + 11 s_{1} s_{2} s_{3} - 123))}{((7920 s_{1} - 23760) (s_{2} - 3) (s_{3} - 3))} \\ E_{57} = \frac{- (h^{3} (638 s_{1} + 638 s_{2} + 638 s_{3} - 352 s_{1} s_{2} - 352 s_{1} s_{3} - 352 s_{2} s_{3} + 11 s_{1} s_{2} s_{3} - 872))}{((221760 s_{1} - 887040) (s_{2} - 4) (s_{3} - 4))} \\ E_{61} = \frac{(h^{2} (131 s_{2} + 131 s_{3} + 575 s_{2} s_{3} - 610))}{(1260 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} \\ E_{62} = \frac{(h^{2} (9643 s_{1} + 9643 s_{2} + 9643 s_{3} - 9381 s_{1} s_{2} - 9381 s_{1} s_{3} - 9381 s_{2} s_{3} + 10531 s_{1} s_{2} s_{3} - 10863))}{((15120 s_{1} - 15120) (s_{2} - 1) (s_{3} - 1))} \\ E_{63} = \frac{- (h^{2} (131 s_{1} + 131 s_{3} + 575 s_{1} s_{3} - 610))}{(1260 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} \\ E_{64} = \frac{(h^{2} (7346 s_{1} + 7346 s_{2} + 7346 s_{3} - 3804 s_{1} s_{2} - 3804 s_{1} s_{3} - 3804 s_{2} s_{3} + 1327 s_{1} s_{2} s_{3} - 13472))}{((10080 s_{1} - 20160) (s_{2} - 2) (s_{3} - 2))} \\ E_{65} = \frac{(h^{2} (131 s_{1} + 131 s_{2} + 575 s_{1} s_{2} - 610))}{(1260 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} \\ E_{66} = \frac{(h^{2} (301 s_{1} s_{2} - 641 s_{2} - 641 s_{3} - 641 s_{1} + 301 s_{1} s_{3} + 301 s_{2} s_{3} + 283 s_{1} s_{2} s_{3} + 703))}{((15120 s_{1} - 45360) (s_{2} - 3) (s_{3} - 3))} \\ E_{67} = \frac{- (h^{2} (186 s_{1} s_{2} - 482 s_{2} - 482 s_{3} - 482 s_{1} + 186 s_{1} s_{3} + 186 s_{2} s_{3} + 241 s_{1} s_{2} s_{3} + 708))}{((60480 s_{1} - 241920) (s_{2} - 4) (s_{3} - 4))} \\ E_{71} = \frac{(h (850 s_{2} + 850 s_{3} - 4415 s_{2} s_{3} + 782))}{(2520 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} \\ E_{72} = - (h (12123 s_{1} + 12123 s_{2} + 12123 s_{3} - 12973 s_{1} s_{2} - 12973 s_{1} s_{3} - 12973 s_{2} s_{3} \\ + 17388 s_{1} s_{2} s_{3} - 12905)) / ((15120 s_{1} - 15120) (s_{2} - 1) (s_{3} - 1)) \\ E_{73} = \frac{- (h (850 s_{1} + 850 s_{3} - 4415 s_{1} s_{3} + 782))}{(2520 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} \\ E_{74} = \frac{(h (3155 s_{1} s_{2} - 7160 s_{2} - 7160 s_{3} - 7160 s_{1} + 3155 s_{1} s_{3} + 3155 s_{2} s_{3} + 630 s_{1} s_{2} s_{3} + 13538))}{((10080 s_{1} - 20160) (s_{2} - 2) (s_{3} - 2))} \\ E_{75} = \frac{(h (850 s_{1} + 850 s_{2} - 4415 s_{1} s_{2} + 782))}{(2520 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} \\ E_{76} = \frac{(h (487 s_{1} + 487 s_{2} + 487 s_{3} + 121 s_{1} s_{2} + 121 s_{1} s_{3} + 121 s_{2} s_{3} - 1512 s_{1} s_{2} s_{3} - 679))}{((15120 s_{1} - 45360) (s_{2} - 3) (s_{3} - 3))} \\ E_{77} = \frac{- (h (366 s_{1} + 366 s_{2} + 366 s_{3} + 121 s_{1} s_{2} + 121 s_{1} s_{3} + 121 s_{2} s_{3} - 1134 s_{1} s_{2} s_{3} - 682))}{((60480 s_{1} - 241920) (s_{2} - 4) (s_{3} - 4))} \end{matrix}$

Appendix B

$\begin{matrix} y_{n + s_{1}} = y_{n} + \frac{h^{2} s_{1}^{2} y_{n}^{″}}{2} + \frac{h^{3} s_{1}^{3} y_{n}^{‴}}{6} + {hs}_{1} y_{n}^{'} \\ - \frac{h^{4} s_{1}^{4}}{(3991680 s_{2} s_{3})} (22176 s_{1} s_{2} + 22176 s_{1} s_{3} - 133056 s_{2} s_{3} - 13200 s_{1}^{2} s_{2} - 13200 s_{1}^{2} s_{3} + 3465 s_{1}^{3} s_{2} \\ + 3465 s_{1}^{3} s_{3} - 440 s_{1}^{4} s_{2} - 440 s_{1}^{4} s_{3} + 22 s_{1}^{5} s_{2} + 22 s_{1}^{5} s_{3} - 6336 s_{1}^{2} + 4950 s_{1}^{3} - 1540 s_{1}^{4} \\ - 9240 s_{1}^{2} s_{2} s_{3} + 46200 s_{1} s_{2} s_{3} - 12 s_{1}^{6} + 990 s_{1}^{3} s_{2} s_{3} - 44 s_{1}^{4} s_{2} s_{3} + 220 s_{1}^{5}) f_{n} \\ + \frac{h^{4} s_{1}^{6}}{(3991680 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} (1584 s_{1} s_{2} + 1584 s_{1} s_{3} - 5544 s_{2} s_{3} - 1089 s_{1}^{2} s_{2} - 1089 s_{1}^{2} s_{3} \\ + 264 s_{1}^{3} s_{2} + 264 s_{1}^{3} s_{3} - 22 s_{1}^{4} s_{2} - 22 s_{1}^{4} s_{3} - 594 s_{1}^{2} + 484 s_{1}^{3} - 132 s_{1}^{4} + 12 s_{1}^{5} - 594 s_{1}^{2} s_{2} s_{3} \\ + 44 s_{1}^{3} s_{2} s_{3} + 2904 s_{1} s_{2} s_{3}) f_{n + 4} \\ - \frac{h^{4} s_{1}^{6}}{(997920 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} (2112 s_{1} s_{2} + 2112 s_{1} s_{3} - 7392 s_{2} s_{3} - 1386 s_{1}^{2} s_{2} - 1386 s_{1}^{2} s_{3} \\ + 308 s_{1}^{3} s_{2} + 308 s_{1}^{3} s_{3} - 22 s_{1}^{4} s_{2} - 22 s_{1}^{4} s_{3} - 792 s_{1}^{2} + 616 s_{1}^{3} - 154 s_{1}^{4} + 12 s_{1}^{5} \\ - 693 s_{1}^{2} s_{2} s_{3} + 44 s_{1}^{3} s_{2} s_{3} + 3696 s_{1} s_{2} s_{3}) f_{n + 3} \\ + \frac{h^{4} s_{1}^{6}}{(665280 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} (3168 s_{1} s_{2} + 3168 s_{1} s_{3} - 11088 s_{2} s_{3} - 1881 s_{1}^{2} s_{2} - 1881 s_{1}^{2} s_{3} \\ + 352 s_{1}^{3} s_{2} + 352 s_{1}^{3} s_{3} - 22 s_{1}^{4} s_{2} - 22 s_{1}^{4} s_{3} - 1188 s_{1}^{2} + 836 s_{1}^{3} - 176 s_{1}^{4} + 12 s_{1}^{5} \\ - 792 s_{1}^{2} s_{2} s_{3} + 44 s_{1}^{3} s_{2} s_{3} + 5016 s_{1} s_{2} s_{3}) f_{n + 2} \\ - \frac{h^{4} s_{1}^{6}}{(997920 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} (6336 s_{1} s_{2} + 6336 s_{1} s_{3} - 22176 s_{2} s_{3} - 2574 s_{1}^{2} s_{2} - 2574 s_{1}^{2} s_{3} \\ + 396 s_{1}^{3} s_{2} + 396 s_{1}^{3} s_{3} - 22 s_{1}^{4} s_{2} - 22 s_{1}^{4} s_{3} - 2376 s_{1}^{2} + 1144 s_{1}^{3} - 198 s_{1}^{4} + 12 s_{1}^{5} - 891 s_{1}^{2} s_{2} s_{3} \\ + 44 s_{1}^{3} s_{2} s_{3} + 6864 s_{1} s_{2} s_{3}) f_{n + 1} \\ - \frac{h^{4} s_{1}^{4}}{(166320 (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} (11088 s_{1} s_{2} + 11088 s_{1} s_{3} \\ - 33264 s_{2} s_{3} - 21 s_{1}^{6} - 9900 s_{1}^{2} s_{2} - 9900 s_{1}^{2} s_{3} + 3465 s_{1}^{3} s_{2} + 3465 s_{1}^{3} s_{3} - 550 s_{1}^{4} s_{2} \\ - 550 s_{1}^{4} s_{3} + 33 s_{1}^{5} s_{2} + 33 s_{1}^{5} s_{3} - 4752 s_{1}^{2} + 4950 s_{1}^{3} - 1925 s_{1}^{4} + 330 s_{1}^{5} - 6930 s_{1}^{2} s_{2} s_{3} \\ + 990 s_{1}^{3} s_{2} s_{3} - 55 s_{1}^{4} s_{2} s_{3} + 23100 s_{1} s_{2} s_{3}) f_{n + s_{1}} \\ + h^{4} s_{1}^{6} (6336 s_{1} - 22176 s_{3} + 13200 s_{1} s_{3} - 3465 s_{1}^{2} s_{3} + 440 s_{1}^{3} s_{3} - 22 s_{1}^{4} s_{3} - 4950 s_{1}^{2} + 1540 s_{1}^{3} \\ - 220 s_{1}^{4} + 12 s_{1}^{5}) / (166320 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4)) f_{n + s_{2}} \\ - h^{4} s_{1}^{6} (6336 s_{1} - 22176 s_{2} + 13200 s_{1} s_{2} - 3465 s_{1}^{2} s_{2} + 440 s_{1}^{3} s_{2} - 22 s_{1}^{4} s_{2} - 4950 s_{1}^{2} \\ + 1540 s_{1}^{3} - 220 s_{1}^{4} + 12 s_{1}^{5}) / (166320 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4)) f_{n + s_{3}} \end{matrix}$ $\begin{matrix} y_{n + 1} = y_{n} + + {hy}_{n}^{'} + \frac{h^{2}}{2} y_{n}^{″} + \frac{h^{3}}{6} y_{n}^{‴} \\ - \frac{h^{4} (4136 s_{1} + 4136 s_{2} - 16159 s_{1} s_{2} - 1418)}{(166320 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} f_{n + s_{3}} \\ + \frac{h^{4} (4136 s_{1} + 4136 s_{2} + 4136 s_{3} - 16159 s_{1} s_{2} - 16159 s_{1} s_{3} - 16159 s_{2} s_{3} + 111309 s_{1} s_{2} s_{3} - 1418)}{(3991680 s_{1} s_{2} s_{3}} f_{n} \\ - \frac{h^{4} (396 s_{1} + 396 s_{2} + 396 s_{3} - 1133 s_{1} s_{2} - 1133 s_{1} s_{3} - 1133 s_{2} s_{3} + 4323 s_{1} s_{2} s_{3} - 166)}{(3991680 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} f_{n + 4} \\ + \frac{h^{4} (550 s_{1} + 550 s_{2} + 550 s_{3} - 1562 s_{1} s_{2} - 1562 s_{1} s_{3} - 1562 s_{2} s_{3} + 5907 s_{1} s_{2} s_{3} - 232)}{(997920 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} f_{n + 3} \\ - \frac{h^{4} (902 s_{1} + 902 s_{2} + 902 s_{3} - 2519 s_{1} s_{2} - 2519 s_{1} s_{3} - 2519 s_{2} s_{3} + 9339 s_{1} s_{2} s_{3} - 386)}{(665280 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} f_{n + 2} \\ + \frac{h^{4} (2640 s_{1} + 2640 s_{2} + 2640 s_{3} - 6776 s_{1} s_{2} - 6776 s_{1} s_{3} - 6776 s_{2} s_{3} + 22935 s_{1} s_{2} s_{3} - 1222)}{(997920 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} f_{n + 1} \\ - \frac{h^{4} (4136 s_{2} + 4136 s_{3} - 16159 s_{2} s_{3} - 1418)}{(166320 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} f_{n + s_{1}} \\ + \frac{h^{4} (4136 s_{1} + 4136 s_{3} - 16159 s_{1} s_{3} - 1418)}{(166320 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} f_{n + s_{2}} \\ y_{n + s_{2}} = y_{n} + {hs}_{2} y_{n}^{'} + \frac{h^{2} s_{2}^{2}}{2 y_{n}^{″}} + \frac{h^{3} s_{2}^{3} w 0}{6} y_{n}^{‴} \\ - \frac{h^{4} s_{2}^{4}}{(3991680 s_{1} s_{3})} (22176 s_{1} s_{2} - 133056 s_{1} s_{3} + 22176 s_{2} s_{3} - 13200 s_{1} s_{2}^{2} + 3465 s_{1} s_{2}^{3} - 440 s_{1} s_{2}^{4} \\ - 13200 s_{2}^{2} s_{3} + 22 s_{1} s_{2}^{5} + 3465 s_{2}^{3} s_{3} - 440 s_{2}^{4} s_{3} + 22 s_{2}^{5} s_{3} - 6336 s_{2}^{2} + 4950 s_{2}^{3} \\ - 1540 s_{2}^{4} + 220 s_{2}^{5} - 12 s_{2}^{6} - 9240 s_{1} s_{2}^{2} s_{3} + 990 s_{1} s_{2}^{3} s_{3} - 44 s_{1} s_{2}^{4} s_{3} + 46200 s_{1} s_{2} s_{3}) f_{n} \\ + \frac{h^{4} s_{2}^{6}}{(3991680 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} (1584 s_{1} s_{2} - 5544 s_{1} s_{3} + 1584 s_{2} s_{3} - 1089 s_{1} s_{2}^{2} \\ + 264 s_{1} s_{2}^{3} - 22 s_{1} s_{2}^{4} - 1089 s_{2}^{2} s_{3} + 264 s_{2}^{3} s_{3} - 22 s_{2}^{4} s_{3} - 594 s_{2}^{2} + 484 s_{2}^{3} \\ - 132 s_{2}^{4} + 12 s_{2}^{5} - 594 s_{1} s_{2}^{2} s_{3} + 44 s_{1} s_{2}^{3} s_{3} + 2904 s_{1} s_{2} s_{3})) f_{n + 4} \\ - \frac{h^{4} s_{2}^{6}}{(997920 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} (2112 s_{1} s_{2} - 7392 s_{1} s_{3} + 2112 s_{2} s_{3} - 1386 s_{1} s_{2}^{2} \\ + 308 s_{1} s_{2}^{3} - 22 s_{1} s_{2}^{4} - 1386 s_{2}^{2} s_{3} + 308 s_{2}^{3} s_{3} - 22 s_{2}^{4} s_{3} - 792 s_{2}^{2} + 616 s_{2}^{3} - 154 s_{2}^{4} \\ + 12 s_{2}^{5} - 693 s_{1} s_{2}^{2} s_{3} + 44 s_{1} s_{2}^{3} s_{3} + 3696 s_{1} s_{2} s_{3}) f_{n + 3} \\ + \frac{h^{4} s_{2}^{6}}{(665280 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} (3168 s_{1} s_{2} - 11088 s_{1} s_{3} + 3168 s_{2} s_{3} - 1881 s_{1} s_{2}^{2} \\ + 352 s_{1} s_{2}^{3} - 22 s_{1} s_{2}^{4} - 1881 s_{2}^{2} s_{3} + 352 s_{2}^{3} s_{3} - 22 s_{2}^{4} s_{3} - 1188 s_{2}^{2} + 836 s_{2}^{3} - 176 s_{2}^{4} + 12 s_{2}^{5} \\ - 792 s_{1} s_{2}^{2} s_{3} + 44 s_{1} s_{2}^{3} s_{3} + 5016 s_{1} s_{2} s_{3}) f_{n + 2} \\ - \frac{h^{4} s_{2}^{6}}{(997920 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} (6336 s_{1} s_{2} - 22176 s_{1} s_{3} + 6336 s_{2} s_{3} - 2574 s_{1} s_{2}^{2} \end{matrix}$ $\begin{matrix} + 396 s_{1} s_{2}^{3} - 22 s_{1} s_{2}^{4} - 2574 s_{2}^{2} s_{3} + 396 s_{2}^{3} s_{3} - 22 s_{2}^{4} s_{3} - 2376 s_{2}^{2} + 1144 s_{2}^{3} - 198 s_{2}^{4} + 12 s_{2}^{5} \\ - 891 s_{1} s_{2}^{2} s_{3} + 44 s_{1} s_{2}^{3} s_{3} + 6864 s_{1} s_{2} s_{3}) f_{n + 1} \\ + \frac{h^{4} s_{2}^{4}}{(166320 (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} (11088 s_{1} s_{2} - 33264 s_{1} s_{3} \\ + 11088 s_{2} s_{3} - 9900 s_{1} s_{2}^{2} + 3465 s_{1} s_{2}^{3} - 550 s_{1} s_{2}^{4} - 9900 s_{2}^{2} s_{3} + 33 s_{1} s_{2}^{5} + 3465 s_{2}^{3} s_{3} \\ - 550 s_{2}^{4} s_{3} + 33 s_{2}^{5} s_{3} - 4752 s_{2}^{2} + 4950 s_{2}^{3} - 1925 s_{2}^{4} + 330 s_{2}^{5} - 21 s_{2}^{6} - 6930 s_{1} s_{2}^{2} s_{3} + 990 s_{1} s_{2}^{3} s_{3} \\ - 55 s_{1} s_{2}^{4} s_{3} + 23100 s_{1} s_{2} s_{3}) f_{n + s_{2}} \\ - (h^{4} s_{2}^{6} (6336 s_{2} - 22176 s_{3} + 13200 s_{2} s_{3} - 3465 s_{2}^{2} s_{3} + 440 s_{2}^{3} s_{3} - 22 s_{2}^{4} s_{3} - 4950 s_{2}^{2} + 1540 s_{2}^{3} \\ - 220 s_{2}^{4} + 12 s_{2}^{5})) / (166320 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4)) f_{n + s_{1}} \\ + (h^{4} s_{2}^{6} (22176 s_{1} - 6336 s_{2} - 13200 s_{1} s_{2} + 3465 s_{1} s_{2}^{2} - 440 s_{1} s_{2}^{3} + 22 s_{1} s_{2}^{4} + 4950 s_{2}^{2} - 1540 s_{2}^{3} \\ + 220 s_{2}^{4} - 12 s_{2}^{5})) / (166320 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4)) f_{n + s_{3}} \\ y_{n + 2} = y_{n} + + 2 {hy}_{n}^{'} + 2 h^{2} y_{n}^{″} + \frac{4 h^{3}}{3 y_{n}^{‴}} \\ + \frac{2 h^{4} (319 s_{1} + 319 s_{2} + 319 s_{3} - 968 s_{1} s_{2} - 968 s_{1} s_{3} - 968 s_{2} s_{3} + 4884 s_{1} s_{2} s_{3} - 124)}{(31185 s_{1} s_{2} s_{3}} f_{n} \\ - \frac{2 h^{4} (33 s_{1} + 33 s_{2} + 33 s_{3} - 88 s_{1} s_{2} - 88 s_{1} s_{3} - 88 s_{2} s_{3} + 264 s_{1} s_{2} s_{3} - 8)}{(31185 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} f_{n + 4} \\ + \frac{8 h^{4} (44 s_{1} + 44 s_{2} + 44 s_{3} - 121 s_{1} s_{2} - 121 s_{1} s_{3} - 121 s_{2} s_{3} + 363 s_{1} s_{2} s_{3} - 8)}{(31185 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} f_{n + 3} \\ - \frac{2 h^{4} (110 s_{1} + 110 s_{2} + 110 s_{3} - 374 s_{1} s_{2} - 374 s_{1} s_{3} - 374 s_{2} s_{3} + 1155 s_{1} s_{2} s_{3} + 28)}{(10395 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} f_{n + 2} \\ + \frac{8 h^{4} (660 s_{1} + 660 s_{2} + 660 s_{3} - 979 s_{1} s_{2} - 979 s_{1} s_{3} - 979 s_{2} s_{3} + 1947 s_{1} s_{2} s_{3} - 536)}{(31185 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} f_{n + 1} \\ - \frac{16 {fns}_{1} h^{4} (319 s_{2} + 319 s_{3} - 968 s_{2} s_{3} - 124)}{(10395 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} f_{n + s_{1}} \\ + \frac{16 h^{4} (319 s_{1} + 319 s_{3} - 968 s_{1} s_{3} - 124)}{(10395 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} f_{n + s_{2}} \\ - \frac{16 h^{4} (319 s_{1} + 319 s_{2} - 968 s_{1} s_{2} - 124)}{(10395 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} f_{n + s_{3}} \end{matrix}$ $\begin{matrix} y_{n + s_{3}} = y_{n} + {hs}_{3} y_{n}^{'} + \frac{h^{2} s_{3}^{2}}{2} y_{n}^{″} + \frac{h^{3} s_{3}^{3}}{6} y_{n}^{‴} \\ + \frac{h^{4} s_{3}^{4}}{(3991680 s_{1} s_{2}} (133056 s_{1} s_{2} - 22176 s_{1} s_{3} - 22176 s_{2} s_{3} + 13200 s_{1} s_{3}^{2} - 3465 s_{1} s_{3}^{3} \\ + 13200 s_{2} s_{3}^{2} + 440 s_{1} s_{3}^{4} - 3465 s_{2} s_{3}^{3} - 22 s_{1} s_{3}^{5} + 440 s_{2} s_{3}^{4} - 22 s_{2} s_{3}^{5} + 6336 s_{3}^{2} - 4950 s_{3}^{3} + 1540 s_{3}^{4} \\ - 220 s_{3}^{5} + 12 s_{3}^{6} + 9240 s_{1} s_{2} s_{3}^{2} - 990 s_{1} s_{2} s_{3}^{3} + 44 s_{1} s_{2} s_{3}^{4} - 46200 s_{1} s_{2} s_{3}) f_{n} \\ - \frac{h^{4} s_{3}^{6}}{(3991680 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} (5544 s_{1} s_{2} - 1584 s_{1} s_{3} - 1584 s_{2} s_{3} + 1089 s_{1} s_{3}^{2} \\ - 264 s_{1} s_{3}^{3} + 1089 s_{2} s_{3}^{2} + 22 s_{1} s_{3}^{4} - 264 s_{2} s_{3}^{3} + 22 s_{2} s_{3}^{4} + 594 s_{3}^{2} - 484 s_{3}^{3} + 132 s_{3}^{4} - 12 s_{3}^{5} \\ + 594 s_{1} s_{2} s_{3}^{2} - 44 s_{1} s_{2} s_{3}^{3} - 2904 s_{1} s_{2} s_{3}) f_{n + 4} \\ + \frac{h^{4} s_{3}^{6}}{(997920 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} (7392 s_{1} s_{2} - 2112 s_{1} s_{3} - 2112 s_{2} s_{3} + 1386 s_{1} s_{3}^{2} \\ - 308 s_{1} s_{3}^{3} + 1386 s_{2} s_{3}^{2} + 22 s_{1} s_{3}^{4} - 308 s_{2} s_{3}^{3} + 22 s_{2} s_{3}^{4} + 792 s_{3}^{2} - 616 s_{3}^{3} + 154 s_{3}^{4} - 12 s_{3}^{5} \\ + 693 s_{1} s_{2} s_{3}^{2} - 44 s_{1} s_{2} s_{3}^{3} - 3696 s_{1} s_{2} s_{3}) f_{n + 3} \\ - \frac{h^{4} s_{3}^{6}}{(665280 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} (11088 s_{1} s_{2} - 3168 s_{1} s_{3} - 3168 s_{2} s_{3} + 1881 s_{1} s_{3}^{2} \\ - 352 s_{1} s_{3}^{3} + 1881 s_{2} s_{3}^{2} + 22 s_{1} s_{3}^{4} - 352 s_{2} s_{3}^{3} + 22 s_{2} s_{3}^{4} + 1188 s_{3}^{2} - 836 s_{3}^{3} + 176 s_{3}^{4} - 12 s_{3}^{5} \\ + 792 s_{1} s_{2} s_{3}^{2} - 44 s_{1} s_{2} s_{3}^{3} - 5016 s_{1} s_{2} s_{3}) f_{n + 2} \\ + \frac{h^{4} s_{3}^{6}}{(997920 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} (22176 s_{1} s_{2} - 6336 s_{1} s_{3} - 6336 s_{2} s_{3} + 2574 s_{1} s_{3}^{2} \\ - 396 s_{1} s_{3}^{3} + 2574 s_{2} s_{3}^{2} + 22 s_{1} s_{3}^{4} - 396 s_{2} s_{3}^{3} + 22 s_{2} s_{3}^{4} + 2376 s_{3}^{2} - 1144 s_{3}^{3} + 198 s_{3}^{4} - 12 s_{3}^{5} \\ + 891 s_{1} s_{2} s_{3}^{2} - 44 s_{1} s_{2} s_{3}^{3} - 6864 s_{1} s_{2} s_{3}) f_{n + 1} \\ + \frac{h^{4} s_{3}^{4}}{(166320 (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} (33264 s_{1} s_{2} - 11088 s_{1} s_{3} \\ - 11088 s_{2} s_{3} + 9900 s_{1} s_{3}^{2} - 3465 s_{1} s_{3}^{3} + 9900 s_{2} s_{3}^{2} + 550 s_{1} s_{3}^{4} - 3465 s_{2} s_{3}^{3} - 33 s_{1} s_{3}^{5} + 550 s_{2} s_{3}^{4} \\ - 33 s_{2} s_{3}^{5} + 4752 s_{3}^{2} - 4950 s_{3}^{3} + 1925 s_{3}^{4} - 330 s_{3}^{5} + 21 s_{3}^{6} + 6930 s_{1} s_{2} s_{3}^{2} - 990 s_{1} s_{2} s_{3}^{3} \\ + 55 s_{1} s_{2} s_{3}^{4} - 23100 s_{1} s_{2} s_{3}) f_{n + s_{3}} \\ + (h^{4} s_{3}^{6} (22176 s_{2} - 6336 s_{3} - 13200 s_{2} s_{3} + 3465 s_{2} s_{3}^{2} - 440 s_{2} s_{3}^{3} + 22 s_{2} s_{3}^{4} + 4950 s_{3}^{2} - 1540 s_{3}^{3} \\ + 220 s_{3}^{4} - 12 s_{3}^{5}) / (166320 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4)) f_{n + s_{1}} \\ - (h^{4} s_{3}^{6} (22176 s_{1} - 6336 s_{3} - 13200 s_{1} s_{3} + 3465 s_{1} s_{3}^{2} - 440 s_{1} s_{3}^{3} + 22 s_{1} s_{3}^{4} + 4950 s_{3}^{2} - 1540 s_{3}^{3} \\ + 220 s_{3}^{4} - 12 s_{3}^{5}) / (166320 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4)) f_{n + s_{2}} \end{matrix}$ $\begin{matrix} y_{n + 3} = y_{n} + 3 {hy}_{n}^{'} + \frac{9 h^{2}}{2} y_{n}^{″} + \frac{9 h^{3}}{2} y_{n}^{‴} \\ + \frac{27 h^{4} (33 s_{1} s_{2} + 33 s_{1} s_{3} + 33 s_{2} s_{3} - 121 s_{1} s_{2} s_{3} - 18)}{(49280 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} f_{n + 4} \\ - \frac{27 h^{4} (44 s_{1} s_{2} + 44 s_{1} s_{3} + 44 s_{2} s_{3} - 165 s_{1} s_{2} s_{3} - 18)}{(12320 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} f_{n + 3} \\ + \frac{27 h^{4} (132 s_{1} + 132 s_{2} + 132 s_{3} - 451 s_{1} s_{2} - 451 s_{1} s_{3} - 451 s_{2} s_{3} + 2167 s_{1} s_{2} s_{3} - 18)}{(49280 s_{1} s_{2} s_{3}} f_{n} \\ - \frac{81 h^{4} (33 s_{1} s_{2} - 198 s_{2} - 198 s_{3} - 198 s_{1} + 33 s_{1} s_{3} + 33 s_{2} s_{3} + 209 s_{1} s_{2} s_{3} + 414)}{(24640 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} f_{n + 2} \\ + \frac{27 h^{4} (594 s_{1} + 594 s_{2} + 594 s_{3} - 726 s_{1} s_{2} - 726 s_{1} s_{3} - 726 s_{2} s_{3} + 1177 s_{1} s_{2} s_{3} - 576)}{(12320 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} f_{n + 1} \\ - \frac{81 {fns}_{1} h^{4} (132 s_{2} + 132 s_{3} - 451 s_{2} s_{3} - 18)}{(6160 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} f_{n + S_{1}} \\ + \frac{81 h^{4} (132 s_{1} + 132 s_{3} - 451 s_{1} s_{3} - 18)}{(6160 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} f_{n + s_{2}} \\ - \frac{81 h^{4} (132 s_{1} + 132 s_{2} - 451 s_{1} s_{2} - 18)}{(6160 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} f_{n + s_{3}} \\ y_{n + 4} = y_{n} + 4 {hy}_{n}^{'} + 8 h^{2} y_{n}^{″} + \frac{32 h^{3}}{3} y_{n}^{‴} \\ + \frac{32 h^{4} (44 s_{1} s_{2} + 44 s_{1} s_{3} + 44 s_{2} s_{3} - 165 s_{1} s_{2} s_{3} - 32)}{(31185 (s_{1} - 4) (s_{2} - 4) (s_{3} - 4))} f_{n + 4} \\ + \frac{256 h^{4} (22 s_{1} + 22 s_{2} + 22 s_{3} - 77 s_{1} s_{2} - 77 s_{1} s_{3} - 77 s_{2} s_{3} + 363 s_{1} s_{2} s_{3} - 4)}{(31185 s_{1} s_{2} s_{3}} f_{n} \\ + \frac{1024 h^{4} (44 s_{1} + 44 s_{2} + 44 s_{3} - 22 s_{1} s_{2} - 22 s_{1} s_{3} - 22 s_{2} s_{3} + 33 s_{1} s_{2} s_{3} - 128)}{(31185 (s_{1} - 3) (s_{2} - 3) (s_{3} - 3))} f_{n + 3} \\ + \frac{1024 h^{4} (132 s_{1} + 132 s_{2} + 132 s_{3} - 154 s_{1} s_{2} - 154 s_{1} s_{3} - 154 s_{2} s_{3} + 231 s_{1} s_{2} s_{3} - 128)}{(31185 (s_{1} - 1) (s_{2} - 1) (s_{3} - 1))} f_{n + 1} \\ - \frac{256 h^{4} (88 s_{1} s_{2} - 220 s_{2} - 220 s_{3} - 220 s_{1} + 88 s_{1} s_{3} + 88 s_{2} s_{3} + 33 s_{1} s_{2} s_{3} + 448)}{(10395 (s_{1} - 2) (s_{2} - 2) (s_{3} - 2))} f_{n + 2} \\ - \frac{2048 h^{4} (22 s_{2} + 22 s_{3} - 77 s_{2} s_{3} - 4)}{(10395 s_{1} (s_{1} - s_{2}) (s_{1} - s_{3}) (s_{1} - 1) (s_{1} - 2) (s_{1} - 3) (s_{1} - 4))} f_{n + s_{1}} \\ + \frac{2048 h^{4} (22 s_{1} + 22 s_{3} - 77 s_{1} s_{3} - 4)}{(10395 s_{2} (s_{1} - s_{2}) (s_{2} - s_{3}) (s_{2} - 1) (s_{2} - 2) (s_{2} - 3) (s_{2} - 4))} f_{n + s_{3}} \\ - \frac{2048 h^{4} (22 s_{1} + 22 s_{2} - 77 s_{1} s_{2} - 4)}{(10395 s_{3} (s_{1} - s_{3}) (s_{2} - s_{3}) (s_{3} - 1) (s_{3} - 2) (s_{3} - 3) (s_{3} - 4))} f_{n + s_{3}} \end{matrix}$