40
J. Math. Fund. Sci., Vol. 50, No. 1, 2018, 40-58
New Generalized Algorithm for Developing k-Step Higher Derivative Block Methods for Solving Higher Order Ordinary Differential Equations Oluwaseun Adeyeye & Zurni Omar Department of Mathematics, School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, 06010, Kedah, Malaysia E-mail:
[email protected] Abstract. This article presents a new generalized algorithm for developing kstep 1 derivative block methods for solving order ordinary differential equations. This new algorithm utilizes the concept from the conventional Taylor series approach of developing linear multistep methods. Certain examples are given to show the simplicity involved in the usage of this new generalized algorithm. Keywords: block methods; generalized algorithm; higher derivative; higher order; kstep; Taylor series.
1
Introduction
The main focus of numerical methods for solving differential equations in recent times has been placed on presenting numerical methods with an improved level of accuracy, ranging from first order ordinary differential equations [1-3] to higher order ordinary differential equations [4-6]. Numerical methods adopted to solve such ODEs have been developed using two popular approaches, i.e. numerical integration and collocation. However, there is a need to introduce new approaches to develop these numerical methods so that they become easier and less burdensome compared to the conventional approaches. Referring to the work of [7], three basic methods can be highlighted for developing single linear multistep methods: numerical integration, interpolation and Taylor series expansion. Although the derivations originally involved just single multistep methods for first order ordinary differential equations, over time the numerical methods evolved into a family of block methods and a further improvement with the introduction of higher derivatives, sometimes referred to as Obrechkoff type [8,9]. The aim of this article is to present a novel algorithm that is convenient to adopt for developing new generalized algorithm k-step 1 derivative block order ordinary differential equations. This work is methods for solving any Received January 5th, 2017, 1st revision August 2nd, 2017, 2nd revision September 27th, 2017. Accepted for publication October 3rd, 2017. Copyright © 2018 Published by ITB Journal Publisher, ISSN: 2337-5760, DOI: 10.5614/j.math.fund.sci.2018.50.1.4
k-Step Higher Derivative Block for Solving Higher Order ODE 41
an extension of [10], which proposed a generalized algorithm for developing block methods without the introduction of higher derivatives. This article is organized as follows: Section 2 presents the generalized algorithm for developing k-step higher derivative block methods with sample second order and third order block methods developed in Sections 3 and 4 respectively. The basic properties of the block methods developed in Sections 3 and 4 are also highlighted. Certain numerical examples are considered in Section 5 and the article is concluded in Section 6.
2
Generalized Algorithm for k-Step Higher Derivative Block Methods
Considering the general form of
order ordinary differential equation
y ( m ) f x, y, y ', y '', , y m 1 , x a, b Extending the generalized algorithm presented by [10] for k-step block methods to accommodate the presence of higher derivatives, the following algorithm is presented for developing k-step 1 derivative block methods for solving order ODEs. m 1
yn
h
i 0
i
k
yn i f n i i g n i , 1, 2, , k i
i!
(1)
i 0
with corresponding derivatives obtained from a
yn
m a 1
i 0
h i!
i
yn
ia
k
ia f n i ia g n i , i 0
(2)
a 1 , 2 , , m 1 where
f n i f xn i , yn i , y 'n i , y ''n i , , yn i g n i
0
,
df n i f xn i , yn i , y 'n i , y ''n i , , yn i
and the condition unknown ∅, ,
m 1
m 1
dx
,
is set to avoid an underdetermined system. The and coefficients are obtained from
, 1 , , k , 0 , 1 , , k A1 B and T
42
Oluwaseun Adeyeye & Zurni Omar
0a
, 1a , , ka , 0 a , 1a , , ka A1 D ; T
A 2 k 2 2 k 2
1 0 0 0 . . . 0
1
1
1
0
0
0
h
2h
kh
1
1
1
h 2
2 h 2 2!
kh 2
2!
0
h
h
2 h
2!
2h
kh 3!
0
h
3!
2!
2!
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2 k 1
2 k 1
h 2 k 1!
3
2h 2 k 1!
3
2 k 1
kh 2 k 1!
2
2 k
h 2 k !
0
2
2 k
2h 2 k !
m 1
2
2k
m
m 1
m2
2 k 2 m 1
kh kh 2! . . . kh k 2 ! 1
m 1
h h m! m 1! h mh 1! m! h h m 2 ! m 1! , D k 2 2 1 . . . . . . h h 2 k 2 m 1! 2 k 2 m 2 ! m
B 2 k 2 1
2h
3!
3
0
(3)
2 k 2 m2
To ascertain that there are no linearly dependent columns or rows det 0 , the rank of the matrix is investigated when developing the block methods. Note that the approach in [7] for developing multistep methods using Taylor series expansion is adopted to expand individual terms in Eq. (1) and Eq. (2) to obtain the unknown coefficients. The conventional Taylor expansions concept is about as defined for
y
n
xn ah y n xn ahy n1 xn ah2!
2
y
n 2
xn
(4)
A more detailed explanation is given in the following subsections, where the new generalized algorithm is used to develop k-step 1 derivative block methods for , 2,2 and , 3,3 .
k-Step Higher Derivative Block for Solving Higher Order ODE 43
3
Development of Two-Step Third Derivative Block Methods for Second Order ODEs
Consider developing a two-step third derivative block method for second order ODEs, that is 2 and 2. Substituting and in Eq. (1) and Eq. (2) gives 1
yn i 0
h
i
2
yn i f n i i g n i , 1, 2 i
i!
(5)
i 0
and 2
yn1 yn1 i1 f n i i1 g n i , 1, 2
(6)
i 0
1.
where Therefore,
yn 1 yn hyn 10 f n 11 f n 1 12 f n 2 10 g n 11 g n 1 12 g n 2 , 1
yn 2 yn 2hyn 20 f n 21 f n 1 22 f n 2 20 g n 21 g n 1 22 g n 2 , 1
yn 1 yn 101 f n 111 f n 1 121 f n 2 101 g n 111 g n 1 121 g n 2 , 1
1
(7)
yn 2 yn 201 f n 211 f n 1 221 f n 2 201 g n 211 g n 1 221 g n 2 1
1
Expanding yn 1 using Taylor series yields:
y xn hy xn
h 2
1
2!
y
2
xn h3!
3
y
3
xn h4!
4
y
4
xn
xn h6! y 6 xn h7! y 7 xn y xn 1 2 2 3 hy xn {10 y xn 11 ( y xn hy xn
h
5
y
5
h 2
y
4
2 h 2
5!
2!
y
2!
hy
4
2hy
6
xn h3!
4
2
y
xn 22!h
2
5
xn ) 12 ( y 2 xn 2hy 3 xn
y
5
y
xn 23!h
xn h2!
4
3
7
3
5
xn ) 10 y 3 xn 11 ( y 3 xn
xn h3!
y
5
3
y
xn 23!h
3
6
xn ) 12 ( y 3 xn
y
6
xn )}
on both the left- and right-hand sides of the Equating coefficients of equation gives the following system of equations: h 2 2!
10 11 12
44
Oluwaseun Adeyeye & Zurni Omar
h 3 3!
h
4
4!
h 5 5!
h
6
6!
h 7 7!
h11 2h12 10 11 12
h 2
h 3
h
h 5
2!
3! 4
4!
5!
11 2! 12 h 11 2h 12 2h
2
11 3! 12 2! 11 2! 12 2h
3
h
2
2h
2
(8)
11 4! 12 3! 11 3! 12 2h
4
h
3
2h
3
11 5! 12 4! 11 4! 12 2h
5
h
4
2h
4
where truncation is made such that the number of unknowns equals the number gives: of equations. Rewriting Eq. (8) in matrix form
1 0 0 0 0 0
1 0 0 where 0 0 0
1
1
0
0
h
2h
1
1
h 2
2 h 2
2!
2!
0
h
h 3
2 h 3 3!
0
h 2
3!
h 4
2 h 4 4!
0
h 3
4!
h 5
2 h 5 5!
0
h 4
5!
0
h
2h
1
1
h 2
2 h 2
2!
2!
0
h
h
2 h
0
h 2
0
h 3
3!
3!
h 4
2 h 4
4!
4!
h 5
2 h 5
5!
5!
2
5
3
6
4
4!
0
3
4
3!
1
2
3
2!
1
3
h 2! 10 h 1 11 3! h 2h 12 4! h 2h 2! 10 5! 2 h h 11 3! 6! 2 h 12 h 4! 7! 0
7
(9)
0
2h 2 h has rank = 6, which implies that there 2! 2h 3! 2h 4! 1
2
2!
3
3!
h 4
4
0 4! are no linearly dependent columns or rows and the inverse exists. These matrices correspond to the definitions assigned in Eq. (3) above and to obtain the unknown coefficients, the matrix inverse approach is adopted to obtain
10 , 11 , 12 ,10 ,11 ,12 T
2
Following the same approach for as:
20 , 21 , 22 , 20 , 21 , 22 T
59 h 11h , h6 , h42 , 1680 , 8105h , 1680 .
2
13 h 42
2
3
3
3
, the unknown coefficients are obtained
2
79 h 105
4h , 1615h , 19105h , 221h , 16105h , 105 . 2
2
3
3
3
k-Step Higher Derivative Block for Solving Higher Order ODE 45
Furthermore, in the case of obtaining the coefficients for , the same approach is followed as for using Taylor series expansion to obtain: y xn hy 1
h
5
5!
y
6
2
xn h2!
y
2
xn h6!
6
y
3
xn h3!
2
xn hy 3 xn h2!
121 ( y
2
xn 2hy 3 xn 22!h
h 3
2 h 2
3!
2!
y
3
6
y
y
4
xn h4!
4
y
5
xn
xn y 1 xn {101 y 2 xn
7
111 ( y
101 y
3
2
y 2
xn h3!
4
y
4
3
y
xn 23!h
xn 111 ( y 3 xn hy 4 xn h2!
2
5
3
xn )
y y
5
5
xn )
xn
xn ) 121 ( y 3 xn 2hy 4 xn
5
xn 23!h
3
y
6
xn )}
on both the left- and right-hand side of the Equating coefficients of equation gives the following system of equations: h 101 111 121 h 2 2!
h 3 3!
h 4 4!
h 5 5!
h 6 6!
h111 2h121 101 111 121
h 2
h 3
h 4
h 5
2!
3!
4!
5!
111 2! 121 h111 2h121 2h
2
2h
3
h
2
2h
1
2
111 4! 121 3! 111 3! 121 2h
4
h
3
2h
3
111 5! 121 4! 111 4! 121 2h
5
h
4
2h
Rewriting Eq. (10) in matrix form
1 0 0 0 0 0
(10)
111 3! 121 2! 111 2! 121
1
h
2h
h 2
2 h 2
2!
2!
h 3
2 h 3
3!
3!
h 4
2 h 4
4!
4!
h 5
2 h 5
5!
5!
0
0
1
1
0
h
0
h 2
0
h 3
0
h 4
2!
3!
4!
4
gives:
h 101 h 1 111 2! 2h h3! 121 h 2 h 101 2! 4! 2 h h 111 3! 5! 2 h 121 h 4! 6! 0
2
3
2
4
3
5
4
6
(11)
46
Oluwaseun Adeyeye & Zurni Omar
which also corresponds to the definitions assigned in Eq. (3) above. The matrix inverse approach is adopted to obtain the unknown coefficients as:
101 , 111 , 121 , 101 , 111 , 121 T
101h 240
7h 15
2
2
2
are obtained as:
Similarly, the unknown coefficients for
201 , 211 , 221 , 201 , 211 , 221 T
h , 815h , 11240h , 13240h , h6 , 80
h h , 1615h , 715h , 15 , 0, 15 2
2
Substituting all obtained coefficients back in Eq. (7) gives the two-step third derivative block method for solving second order ordinary differential equations as: h yn 1 yn hyn h42 13 f n 7 f n 1 f n 2 1680 59 g n 128 g n 1 11g n 2 , 2
1
3
h yn 2 yn 2hyn 105 79 f n 112 f n 1 19 f n 2 105h 10 g n 16 g n 1 4 g n 2 1
2
3
h 101 f n 128 f n 1 11 f n 2 240 13 g n 40 g n 1 3 g n 2 , 1 1 h yn 2 yn 15h 7 f n 16 f n 1 7 f n 2 15 gn gn2
yn 1 yn 1
1
2
h 240
2
4
Properties of the Two-Step Third-Derivative Block Method
The following properties of the two-step third-derivative block method are discussed: order, zero-stability, consistency and convergence. Following the definition of Equation (11), the individual terms of the two-step third derivative block method are expanded using Taylor series expansions about . The order of the two-step third derivative block method is 6. obtained to be Secondly, to analyze the two-step third derivative block method for zerostability, the modulus of the roots of its first characteristic polynomial is expected to be simple or less than one. Thus, the correctors of the two-step third derivative block method are normalized to give the first characteristic 0 1 polynomial as with roots satisfying 1. 0 1 The two-step third derivative block method is consistent if it has order 1 as satisfied in the previous paragraphs. Therefore, the two-step third derivative block method is convergent [12].
k-Step Higher Derivative Block for Solving Higher Order ODE 47
5
Development of Three-Step Fourth Derivative Block Methods for Third Order ODEs
Consider developing a three-step higher derivative block method for third order 3 and 3. ODEs, i.e. Substituting k and m in Eq. (1) and Eq. (2) gives: 2
y n i 0
h
i
3
yn i f n i i g n i , 1, 2, 3 i
i!
(12)
i 0
and 1
yn
1
i0
2
yn
0
i0
h
i
i!
h
3
i 1
yn
i1
f n i i1 g n i , 1, 2, 3
i 0
i
i 2
yn
i!
(13)
3
i2
f n i i 2 g n i , 1, 2, 3
i 0
1,2.
where
yn 1 yn hyn 1
h 2 2!
2
yn 2 yn 2hyn
2 h 2
yn 3 yn 3hyn
3 h 2
1
1
10 f n 11 f n 1 12 f n 2 13 f n 3 , 10 g n 11 g n 1 12 g n 2 13 g n 3
yn
2!
2!
20 f n 21 f n 1 22 f n 2 23 f n 3 , 20 g n 21 g n 1 22 g n 2 23 g n 3
yn 2
30 f n 31 f n 1 32 f n 2 33 f n 3 , 30 g n 31 g n 1 32 g n 2 33 g n 3
yn 2
101 f n 111 f n 1 121 f n 2 131 f n 3 , 101 g n 111 g n 1 121 g n 2 131 g n 3
yn 1 yn hyn 1
1
2
201 f n 211 f n 1 221 f n 2 231 f n 2 yn12 yn1 2hyn 2 , 201 g n 211 g n 1 221 g n 2 231 g n 3 301 f n 311 f n 1 321 f n 2 331 f n 2 yn13 yn1 3hyn 2 , 301 g n 311 g n 1 321 g n 2 331 g n 3 102 f n 112 f n 1 122 f n 2 132 f n 3 yn 21 yn 2 , 102 g n 112 g n 1 122 g n 2 132 g n 3
48
Oluwaseun Adeyeye & Zurni Omar
202 f n 212 f n 1 222 f n 2 232 f n 3 , 202 g n 212 g n 1 222 g n 2 232 g n 3
yn 2 yn 2
2
2
2
yn 3 y n For
(14)
302 f n 312 f n 1 322 f n 2 332 f n 3 302 g n 312 g n 1 322 g n 2 332 g n 3
: y xn hy xn
h 2
1
h 5
h 9
5!
9!
y 5 xn y
h 3
2 h 2
3!
y
3hy
10 y
h
2 h 2
3
3!
y
2!
3hy
y 6 xn
h 7 7!
3
y
3
xn h4!
4
5
y 7 xn
3
y
2
8!
xn h4!
y
xn 32!h
2
3
y
5
xn
6
5
xn 24!h
xn 33!h
3
4
y
y 6
7
xn ) 13 ( y 3 xn
xn 34!h
4
y
y
6
7
xn )
xn
8
xn ) 12 ( y 4 xn 2hy 5 xn
y
7
y
xn 23!h
xn
y
xn 11 ( y 4 xn hy 5 xn h2!
6
2
xn ) 12 ( y 3 xn 2hy 4 xn
2
4
xn
y 8 xn
h 8
y
4
7
y
xn 23!h
y
4
xn 11 ( y 3 xn hy 4 xn h2!
xn h4!
4
y 5
3
7
xn h3!
2
xn 32!h
4
2
2
6
y
2!
6!
y
xn y xn hy 1 xn h2!
9
{10 y
h 6
2!
6
xn 24!h
xn 33!h
3
4
y
y 7
8
xn ) 13 ( y 4 xn
xn 34!h
4
y
8
xn )}
on both the left- and right-hand side of the Equating coefficients of equation gives the following system of equations: h 3 3!
h 4 4!
h
5
5!
h 6 6!
h 7 7!
h 8 8!
10 11 12 13 h11 2h12 3h13 10 11 12 13
h 2
h 3
h 4
h 5
2!
3!
4!
5!
11 2! 12 2! 13 h 11 2h 12 3h 13 2h
2
3h
2
11 3! 12 3! 13 2! 11 2! 12 2! 13 2h
3
3h
3
h
2
2h
2
3h
2
11 4! 12 4! 13 3! 11 3! 12 3! 13 2h
4
3h
4
h
3
2h
3
3h
3
11 5! 12 5! 13 4! 11 4! 12 4! 13 2h
5
3h
5
h
4
2h
4
3h
4
k-Step Higher Derivative Block for Solving Higher Order ODE 49
h 9 9!
h 10 10!
h 6
h 7
6!
7!
11 6! 12 6! 13 5! 11 5! 12 5! 13 2h
6
3h
6
1 0 0 0 where 0 0 0 0
2h
2h
7
3h
7
h
6
2h
3h
5
1
0
0
0
h
2h
3h
1
1
1
h 2
2 h 2
3 h 2
2!
2!
2!
0
h
2h
h 3
2 h 3
3 h 3
2 h 2
3!
3!
0
h 2
3!
2!
2!
h 4
2 h 4
3 h 4
2 h 3
4!
4!
0
h 3
4!
3!
3!
h 5
2 h 5
3 h 5
2 h 4
5!
5!
0
h 4
5!
4!
4!
h 6
2 h 6
3 h 6
2 h 5
6!
6!
0
h 5
6!
5!
5!
h 7
2 h 7
3 h 7
2 h 6
7!
7!
0
h 6
7!
6!
6!
1
1
1
0
0
0
h
2h
3h
1
1
1
h 2
2 h 2
3 h 2
2!
2!
2!
0
h
h
2h
0
h
2!
2!
0
h 3
2 h 3
3!
3!
0
h 4
2 h 4
4!
4!
0
h 5
2 h 5
5!
5!
h 6
2 h 6
3h
3
3!
3!
3!
h 4
2 h 4
3 h 4
4!
4!
4!
h 5
2 h 5
3 h 5
5!
5!
5!
h 6
2 h 6
3 h 6
6!
6!
6!
h 7
2 h 7
3 h 7
7!
7!
7!
3h
(15)
6
3
4
5
2
6
3
7
4
8
5
9
6
10
(16)
0
3h 3h 2! 3 h has rank = 8, which 3! 3h 4! 3h 5! 3h 6! 1
2h 2
h 3! 1 10 h 4! 11 h 3h 12 5! 3h h 2! 13 6! 3h 10 h 3! 7! 3h 11 h 4! 8! 3h 12 h 5! 9! 3 h 13 h 6! 10! 0
1
3
6
gives:
1
3
5
11 7! 12 7! 13 6! 11 6! 12 6! 13
Rewriting Eq. (15) in matrix form
1 0 0 0 0 0 0 0
5
h
2 h 2
2
3
4
5
6
0 6! 6! implies that there are also no linearly dependent columns or rows and the inverse exists. These matrices likewise correspond to the definitions in Eq. (3). The unknown coefficients are obtained below:
10 , 11 , 12 , 13 , 10 , 11 , 12 , 13
T
3
62387 h 544320
17 h h h , 89 , 439 h , 1031h , 1879 h , 359 , 13960h , 18144 3360 20160 272160 181440 10080 3
3
3
Next the unknown coefficients for
4
4
and
4
4
are obtained as:
50
Oluwaseun Adeyeye & Zurni Omar
20 , 21 , 22 , 23 , 20 , 21 , 22 , 23
5048 h 8505
T
h h h , 164315h , 421h , 244 , 172 , 415h , 34315h , 4567 8505 2835
3
3
3
3
4
4
4
4
30 , 31 , 32 , 33 , 30 , 31 , 32 , 33 T
657 h 448
h h h h h h h , 2187 , 2187 , 117 , 351 , 729 , 729 , 27 1120 2240 1120 2240 1120 2240 1120
3
3
3
3
4
4
In obtaining the coefficients for gives:
y xn hy 1
xn h2!
2
4
4
, Taylor series expansion is used, which
2
y
xn h3!
3
3
y
xn h4!
4
y
4
5
xn
xn h6! y 7 xn h7! y 8 xn h8! y 9 xn 1 2 3 3 y xn hy xn {101 y xn 111 ( y xn
h 5
y
5!
hy
4
6
xn h2!
2
2 h 4
3 h 3
y
3!
hy
5
5
xn h3!
2 h 4
3h
4! 3
3!
2!
h 3 3!
h
4
4!
h 5 5!
h 6 6!
xn h4!
4
y
7
xn 34!h 2
y
4
6
y
7
2
y
5
xn 23!h
xn )
6
3
y
6
xn h3!
3
y
2h 2!
7
2
xn h4!
y
y 6 xn
2 h
4
8 3
3!
xn 34!h
8
y
xn
5
y 7 xn
7
y
2
xn )
xn ) 131 ( y 4 xn 3hy 5 xn 32!h 4
xn
xn ) 101 y 4 xn 111 ( y 4 xn
8
y
2
y
6
xn
xn )} gives:
Equating coefficients of h 2
6
xn ) 131 ( y 3 xn 3hy 4 xn 32!h
xn h2!
y
y
7
121 ( y 4 xn 2hy 5 xn
3
8
xn 2hy 4 xn 22!h
y
4!
y
7
3
121 ( y
6
101 111 121 131 h111 2h121 3h131 101 111 121 131
h 2
h 3
h 4
2!
3!
4!
111 2! 121 2! 131 h111 2h121 3h131 2h
2
3h
2
111 3! 121 3! 131 2! 111 2! 121 2! 131 2h
3
3h
3
h
2
2h
2
3h
2
111 4! 121 4! 131 3! 111 3! 121 3! 131 2h
4
3h
4
h
3
2h
3
3h
3
k-Step Higher Derivative Block for Solving Higher Order ODE 51
h 7 7!
h 8 8!
h 9 9!
h 5
h 6
h 7
5!
6!
7!
111 5! 121 5! 131 4! 111 4! 121 4! 131 5
2h
3h
5
4
2h
4
3h
111 6! 121 6! 131 5! 111 5! 121 5! 131 6
2h
3h
6
5
h
5
2h
5
3h
(17)
111 7! 121 7! 131 6! 111 6! 121 6! 131 7
2h
3h
7
6
h
Rewriting Eq. (17) in matrix form
1 0 0 0 0 0 0 0
4
h
6
2h
6
3h
gives:
h 2! 1 101 h 3! 111 h 3h 121 4! 3 h h 2! 131 5! 3 h 101 h 3! 6! 3h 111 h 4! 7! 3h 121 h 5! 8! 3 h 131 h 6! 9!
2
0
1
1
1
0
0
0
h
2h
3h
1
1
1
h 2
2 h 2
3 h 2
2!
2!
2!
0
h
2h
h 3
2 h 3
3 h 3
2 h 2
3!
3!
0
h 2
3!
2!
2!
h 4
2 h 4
3 h 4
h 3
2 h 3
4!
4!
4!
0
3!
3!
h 5
2 h 5
3 h 5
2 h 4
5!
5!
0
h 4
5!
4!
4!
h 6
2 h 6
3 h 6
h 5
2 h 5
6!
6!
6!
0
5!
5!
h 7
2 h 7
3 h 7
2 h 6
7!
7!
0
h 6
7!
6!
6!
3
4
2
5
3
6
4
7
5
8
6
9
(18)
corresponding to the definitions in Eq. (3). The values of the unknown coefficients are given below:
101 , 111 , 121 , 131 , 101 , 111 , 121 , 131
T
19519 h 68040
2
h h h h h h , 1301 , 181 , 3329 , 371h , 313 , 892016 , 137 10080 2520 272160 12960 2520 45360 2
2
2
3
3
Similarly, the unknown coefficients for
3
3
and
are obtained as:
201 , 211 , 221 , 231 , 201 , 211 , 221 , 231
T
5731h 8505
2
h h h h , 296 , 109315h , 344 , 2606 , 2063h , 52315h , 4405 315 8505 2835 2
2
2
3
3
3
3
301 , 311 , 321 , 331 , 301 , 311 , 321 , 331 T
603 h 560
2
h h h h h , 2187 , 729 , 27160h , 27224h , 243 , 243 , 9280 1120 560 560 1120 2
while the coefficients for
2
2
2
,
3
3
and
3
are obtained as:
102 , 112 , 122 , 132 , 102 , 112 , 122 , 132
T
6893 h 18144
h 89 h 397 h 1283 h h h h , 313 , 672 , 18144 , 30240 , 851 , 269 , 163 672 3360 3360 30240 2
2
2
2
52
Oluwaseun Adeyeye & Zurni Omar
202 , 212 , 222 , 232 , 202 , 212 , 222 , 232
T
223 h 567
h 43 h h , 2021h , 1321h , 20 , 945 , 16105h , 19105h , 8945 567 2
2
2
2
302 , 312 , 322 , 332 , 302 , 312 , 322 , 332 T
93 h 224
81h 81h h 243 h 93 h 57 h h , 243 , 224 , 224 , 1120 , 1120 , 1120 , 57 224 1120 2
2
2
2
Substituting all obtained coefficients back in Eq. (14) gives the three-step fourth derivative block method for solving third order ordinary differential equations as: yn 1 yn hyn 1
h 2 2!
h yn 544630 62387 f n 14418 f n 1 11853 f n 2 2062 f n 3 3
2
h 544320 5637 g n 19386 g n 1 7371g n 2 510 g n 3 , 4
yn 2 yn 2hyn 1
2 h 2 2!
h yn 8505 5048 f n 4428 f n 1 1620 f n 2 244 f n 3 2
3
h 544320 516 g n 2268 g n 1 918 g n 2 60 g n 3 , 4
3285 f n 4374 f n 1 2187 f n 2 234 f n 3 h 544320 351g n 1458 g n 1 729 g n 2 54 g n 3 , 1 2 1 h yn 1 yn hyn 272160 78076 f n 35127 f n 1 19548 f n 2 3329 f n 3 h 272160 7791g n 33804 g n 1 12015 g n 2 822 g n 3 , h yn12 yn1 2hyn 2 8505 5731 f n 7992 f n 1 2943 f n 2 344 f n 3 h 272160 618 g n 2700 g n 1 1404 g n 2 84 g n 3 , h yn13 yn1 3hyn 2 1120 1206 f n 2187 f n 1 1458 f n 2 189 f n 3 h 272160 135 g n 486 g n 1 243g n 2 36 g n 3 2 2 h yn 1 yn 90720 34465 f n 422 f n 1 19548 f n 2 3329 f n 3 h 272160 7791g n 33804 g n 1 12015 g n 2 822 g n 3 , h yn 22 yn 2 8505 5731 f n 7992 f n 1 2943 f n 2 344 f n 3 h 272160 618 g n 2700 g n 1 1404 g n 2 84 g n 3 , 2 2 h yn 3 yn 1120 1206 f n 2187 f n 1 1458 f n 2 189 f n 3 h 272160 135 g n 486 g n 1 243g n 2 36 g n 3 yn 3 yn 3hyn 1
3 h 2 2!
4
2
3
2
3
2
3
2
2
2
yn 2
3
h 2240
k-Step Higher Derivative Block for Solving Higher Order ODE 53
6
Properties of the Three-Step Fourth-Derivative Block Method
The following properties of the three-step fourth derivative block method are discussed: order, zero-stability, consistency and convergence. Following the same approach as the two-step third derivative block method, the correctors of the three-step fourth derivative block method are also expanded . The order of the three-step fourth derivative using Taylor series about block method is obtained to be 8. Secondly, to analyze the three-step fourth derivative block method for zerostability, the modulus of the roots of its first characteristic polynomial is expected to be simple or less than one. Thus, the correctors of the three-step fourth derivative block method are normalized to give the first characteristic 0 0 1 1. polynomial as 0 0 1 with roots satisfying 0 0 1 1 The three-step fourth derivative block method is consistent if it has order as satisfied in the previous paragraphs. Therefore, the three-step fourth derivative block method is convergent [12].
7
Numerical Examples
This section considers certain linear and non-linear second and third order ordinary differential equations. The developed methods in Sections 3 and 4 above are adopted to solve these ODEs (encompassing both initial and boundary value problems) and a comparison is made with previous works. Problem 1. Consider the non-linear second order initial value problem
y '' x y ' 0, y 0 1, y ' 0 12 , x 0,1 2
with exact solution:
1
ln
.
This initial value problem was compared with [9], where a hybrid order six as presented in Table 1. The common block method was presented with ground for comparison is in terms of the order of the block methods since the two-step third-derivative block method and the hybrid block method presented by [9] are both of order six. Problem 2. Consider the linear second order initial value problem
54
Oluwaseun Adeyeye & Zurni Omar
y '' 6x y '
6 x
2
y 0, y 1 1, y ' 1 1, x 0, hN
with exact solution: The number of steps considered for this initial value problem is 10 . with respect to the selected step-size . The results as displayed in Table 2 were compared to the solution obtained by order six hybrid block method presented by [13]. The block method compared to [13] is the order six two-step third-derivative block method. Problem 3. Consider the non-linear second order boundary value problem
y '' 18 32 2 x3 yy ' , y 1 17, y 3
43 3
, x 1,3
with exact solution: y x 2 16x The boundary value problem defined here was solved by [14] using an order six 0.1. Thus, since the basis for comparison is self-starting block method and the equal order of the methods, the suitable block method is the two-step thirdderivative block method. The details of the results obtained can be seen in Table 3. Problem 4. Consider the special third order initial value problem y ''' y , y 0 1, y ' 0 1, y '' 0 1, x 0,1
with exact solution:
.
This third order IVP was compared with [4] in which the authors present an 0.1. Table 4 shows the comparison of the order eight block method with method in [9] with the order eight three-step fourth-derivative block method. Problem 5. Consider the non-linear third order boundary value problem
y ''' 2e3 y 4 1 x , y 0 0, y ' 0 1, y 1 ln 2, x 0,1 3
with exact solution:
ln 1
.
The third order BVP defined here was solved by [15] using an order eight block method and compared with the three-step fourth-derivative method also of order eight. The step-size h for this problem varied with respect to the number of steps (N) as shown in Table 5. The maximum error (MAXE) at the boundary 1 was recorded for 7,14,28,56.
k-Step Higher Derivative Block for Solving Higher Order ODE 55
Table 1 Comparison of the Two-Step Third Derivative Block Method with [9] for Solving Problem 1.
x Exact Solution Computed Solution 0.1 1.0500417292784914 1.0550555592082120 0.2 1.1003353477310756 1.1003353477310753 1.1511404359364668 0.3 1.1511404359364668 0.4 1.2027325540540823 1.2027325540540821 0.5 1.2554128118829952 1.2554128118829952 1.3095196042031116 0.6 1.3095196042031119 0.7 1.3654437542713964 1.3654437542713957 0.8 1.4236489301936019 1.4236489301936006 0.9 1.4847002785940520 1.4847002785940489 1.0 1.5493061443340550 1.5493061443340488 CPU Time Note: TSTD: Two-step third derivative block method.
Error [9] 6.661338E-16 1.332268E-15 4.440892E-16 1.332268E-15 3.774758E-15 1.065814E-14 2.642331E-14 5.861978E-14 1.265654E-13 2.711165E-13 N/A
Error (TSTD) 2.220446E-16 2.220446E-16 0.000000E+00 2.220446E-16 0.000000E+00 2.220446E-16 6.661338E-16 1.332268E-15 3.108624E-15 6.217249E-15 ≈ 0.59 secs
Table 2 Comparison of the Two-Step Third Derivative Block Method with [13] for Solving Problem 2.
x Exact Solution Computed Solution 1.003125 1.0030765258576961 1.0030765258576961 1.006250 1.0060575030835164 1.0060575030835164 1.009375 1.0089449950888376 1.0089449950888376 1.012500 1.0117410181679887 1.0117410181679887 1.015625 1.0144475426864139 1.0144475426864139 1.018750 1.0170664942356729 1.0170664942356726 1.021875 1.0195997547562881 1.0195997547562876 1.025000 1.0220491636294322 1.0220491636294318 1.028125 1.0244165187384029 1.0244165187384029 1.031250 1.0267035775008062 1.0267035775008062 CPU Time Note: TSTD: Two-step third derivative block method.
Error [13] N/A N/A 9.661126E-08 9.425732E-08 9.197108E-08 8.975049E-08 8.759359E-08 8.549846E-08 8.346327E-08 8.148622E-08 N/A
Error (TSTD) 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 2.220446E-16 4.440892E-16 4.440892E-16 0.000000E+00 0.000000E+00 ≈ 0.06 secs
Table 3 Comparison of the Two-Step Third Derivative Block Method with [14] for Solving Problem 3.
x 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5
Exact Solution 15.755454545454546 14.773333333333333 13.997692307692306 13.388571428571428 12.916666666666666 12.559999999999999 12.301764705882352 12.128888888888888 12.031052631578946 12.000000000000000 12.029047619047619 12.112727272727273 12.246521739130435 12.426666666666666 12.650000000000000
Computed Solution 15.755455083760621 14.773334073676603 13.997693050796402 13.388572125044842 12.916667287517622 12.560000544356562 12.301765175115765 12.128889289986821 12.031052970831665 12.000000284503839 12.029047854947255 12.112727465928481 12.246521894771508 12.426666789477615 12.650000094140001
Error [14] 1.50E-06 3.97E-07 8.12E-08 1.57E-06 1.83E-06 1.79E-06 1.58E-06 7.93E-07 1.07E-06 1.07E-06 6.42E-07 1.27E-06 2.18E-07 1.84E-06 3.82E-07
Error (TSTD) 5.383061E-07 7.403433E-07 7.431041E-07 6.964734E-07 6.208510E-07 5.443566E-07 4.692334E-07 4.010979E-07 3.392527E-07 2.845038E-07 2.358996E-07 1.932012E-07 1.556411E-07 1.228109E-07 9.414000E-08
56
Oluwaseun Adeyeye & Zurni Omar
x Exact Solution Computed Solution 2.6 12.913846153846155 12.913846223094138 2.7 13.215925925925927 13.215925973633468 2.8 13.554285714285715 13.554285743482653 2.9 13.927241379310345 13.927241392694905 3.0 14.333333333333336 14.333333333333336 CPU Time Note: TSTD: Two-step third derivative block method.
Error [14] 1.86E-06 1.35E-06 1.22E-06 2.75E-06 0.000000 N/A
Error (TSTD) 6.924798E-08 4.770754E-08 2.919694E-08 1.338456E-08 0.000000E+00 ≈ 0.12 secs
Table 4 Comparison of the Three-Step Fourth Derivative Block Method with [4] for Solving Problem 4.
x Exact Solution Computed Solution 0.1 0.90483741803595952 0.90483741803595963 0.2 0.81873075307798182 0.81873075307798204 0.3 0.74081822068171777 0.74081822068171832 0.4 0.67032004603563933 0.67032004603564022 0.5 0.60653065971263342 0.60653065971263520 0.6 0.54881163609402639 0.54881163609402939 0.7 0.49658530379140947 0.49658530379141408 0.8 0.44932896411722156 0.44932896411722822 0.9 0.40656965974059905 0.40656965974060832 1.0 0.36787944117144222 0.36787944117145466 CPU Time TSFD: Three-step fourth derivative block method.
Error [4] 2.138401E-12 6.055156E-13 7.395751E-12 2.158163E-12 1.484579E-11 1.098521E-11 3.142886E-11 2.309530E-11 5.154149E-11 8.200535E-11 N/A
Error (TSFD) 1.110223E-16 2.220446E-16 5.551115E-16 8.881784E-16 1.776357E-15 2.997602E-15 4.607426E-15 6.661338E-15 9.270362E-15 1.243450E-14 ≈ 0.06 secs
Table 5 Comparison of the MAXE for the Three-Step Fourth Derivative Block Method with [15] for Solving Problem 5. Computed Solution MAXE CPU Time MAXE (TSFD) (TSFD) [15] 7 0.69314718055994529 0.69314718055994529 ≈ 0.10 secs 0.000000E+00 5.24E-09 14 0.69314718055994529 0.69314718055994529 ≈ 0.15 secs 0.000000E+00 2.39E-11 28 0.69314718055994529 0.69314718055994529 ≈ 0.30 secs 0.000000E+00 9.50E-14 56 0.69314718055994529 0.69314718055994529 ≈ 0.61 secs 0.000000E+00 3.62E-16 Note: TSTD: Two-step third derivative block method, TSFD: Three-step fourth derivative block method, N: Number of steps, MAXE: Maximum error, N/A: Not available. N
8
Exact Solution
Conclusion
This article presented a new approach to developing block methods for solving order ordinary differential equations with the presence of 1 . The generalized approach is seen to be quite flexible as the algorithm can simultaneously produce block methods of step length k for solving any order m of ordinary differential equations. The sample methods developed using this new approach were seen to satisfy the basic properties to ensure convergence and their accuracy is also displayed (see Tables 1-5). Thus, this new generalized approach is quite suitable for developing block methods for solving higher order ODEs.
k-Step Higher Derivative Block for Solving Higher Order ODE 57
References [1] Badmus, A.M., Yahaya, Y.A. & Pam, Y.C., Adams Type Hybrid Block Methods Associated with Chebyshev Polynomial for the Solution of Ordinary Differential Equations, British Journal of Mathematics & Computer Science, 6(6), pp. 464-474, 2015. [2] Ibrahim, Z.B., Othman, K.I. & Suleiman, M., Implicit r-point Block Backward Differentiation Formula for Solving First-order Stiff ODEs, Applied Mathematics and Computation, 186(1), pp. 558-565, 2007. [3] Jator, S.N. & Li, J., A Self-Starting Linear Multistep Method for A Direct Solution of the General Second-order Initial Value Problem, International Journal of Computer Mathematics, 86(5), pp. 827-836, 2009. [4] Kuboye, J.O. & Omar, Z., Numerical Solution of Third Order Ordinary Differential Equations Using A Seven-step Block Method, International Journal of Mathematical Analysis, 9(15), pp. 743-745, 2015. [5] Lambert, J.D., Computational Methods in Ordinary Differential Equations, Wiley, London, 1973. [6] Majid, Z.A., See, P.P. & Suleiman, M., Solving Directly Two Point Non Linear Boundary Value Problems using Direct Adams Moulton Method, Journal of Mathematics and Statistics, 7(2), pp. 124-128, 2011. [7] Okuonghae, R.I. & Ikhile, M.N.O., On the Construction of High Order A (α)-stable Hybrid Linear Multistep Methods for Stiff IVPs in ODEs, Numerical Analysis and Applications, 5(3), pp. 231-241, 2012. [8] Omar, Z. & Adeyeye, O., K-step Block Method Algorithm for Arbitrary Order Ordinary Differential Equations, International Journal of Mathematical Analysis, 10(11), pp. 545-552, 2016. [9] Omar, Z. & Alkasassbeh, M.F., Generalized One-Step Third Derivative Implicit Hybrid Block Method for the Direct Solution of Second Order Ordinary Differential Equation, Applied Mathematical Sciences, 10(9), pp. 417-430, 2016. [10] Sunday, J., Odekunle, M.R., Adesanya, A.O. & James, A.A., Extended Block Integrator for First-order Stiff and Oscillatory Differential Equations, American Journal of Computational and Applied Mathematics, 3, pp. 283-290, 2013. [11] Butcher, J.C., Numerical Methods for Ordinary Differential Equations, Wiley, West Sussex, 2008. [12] Fatunla, S.O., Numerical Methods for Initial Value Problems in Ordinary Differential Equation, Academic Press, New York, 1988. [13] Kayode, S.J. & Obarhua, F.O., Continuous y-function Hybrid Methods for Direct Solution of Differential Equations, International Journal of Differential Equations and Applications, 12(1), pp. 37-48, 2013. [14] Jator, S.N. & Li, J., Boundary Value Methods via A Multistep Method with Variable Coefficients for Second Order Initial and Boundary Value
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Problems, International Journal of Pure and Applied Mathematics, 50(3), pp. 403-420, 2009. [15] Sahi, R.K., Jator, S.N. & Khan, N.A., Continuous Fourth Derivative Method for Third Order Boundary Value Problems, International Journal of Pure and Applied Mathematics, 85(2), pp. 907-923, 2013.
