This document describes the development of a one-stage implicit rational Runge-Kutta method for solving second-order ordinary differential equations. The method is derived by taking Taylor series expansions of the solution equations about the point (xn, yn, yn') and equating terms with the same powers of the step size h. This results in expressions for the parameters of the method in terms of the leading order truncation errors. The method is consistent, as the parameters satisfy constraints that ensure the method approaches the true solution in the limit as h approaches 0.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
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eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
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Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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Abstract : Here we consider first order autoregressive process with changing autoregressive coefficient at some point of time m. This is called change point inference problem. For Bayes estimation of m and autoregressive coefficient we used MHRW (Metropolis Hasting Random Walk) algorithm and Gibbs sampling. The effects of prior information on the Bayes estimates are also studied.
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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Development of implicit rational runge kutta schemes for second order ordinary differential equations
1. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
www.iiste.org
Development of Implicit Rational Runge-Kutta Schemes for
Second Order Ordinary Differential Equations
Usman Abdullahi S.1*, Mukhtar Hassan S.2
1.Department of Statistics, Federal Polytechnic Bali, Taraba State – Nigeria
2.GNS Department, Federal Polytechnic Bali, Taraba State – Nigeria
*E-Mail of corresponding author: usasmut@gmail.com
Abstract
In this paper, the development of One – Stage Implicit Rational Runge – Kutta methods are considered using
Taylor and Binomial series expansion for the direct solution of general second order initial value problems of
ordinary differential equations with constant step length. The basic properties of the developed method were
investigated and found to be consistent and convergent.
Keywords: Implicit Rational Runge Kutta scheme, Second Order Equations, Convergence and Consistent
1.
Introduction
Consider the numerical approximation first order initial value problems of the form,
y ′ = f (x, y ),
y ( x0 ) = y 0 a ≤ x ≤ b
,
(1.1)
A Runge-Kutta method is the most important family of implicit and explicit iterative method of
approximation of initial value problems of ordinary differential equations. So far many work and schemes have
been developed for solving problem (1). The numerical solution of (1.1) is.
yn+1 = yn + hφ (xn , yn , h )
(1.2)
where
s
φ (x, y, h ) = ∑ ci ki
i =1
r
k r = f x + hai , y + h ∑ bij k j ,
k1 = f (x, y ),
j =1
and
with constraints
r = 1(1)s
(1.3)
i
ai = ∑ bij , i = 1(1)s
j =1
The derivative of suitable parameters , and of higher order term involves a large amount of
tedious algebraic manipulations and functions evaluations which is both time consuming and error prone, Julyan
and Oreste (1992). The derivation of the Runge – Kutta methods is extensively discussed by Lambert (1973),
Butcher (1987), Fatunla (1987), According to Julyan and Oreste (1992) the minimum number of stages
necessary for an explicit method to attain order p is still an open problem. Therefore so many new schemes and
approximation formula have been derived this includes the work of Ababneh et al. (2009a), Ababneh et al.
(2009b) Faranak and Ismail (2010).
Since the stability function of the implicit Runge-Kutta scheme is a rational function, Butcher (2003);
Hong (1982) first proposes rational form of Runge-Kutta method (1.2), then Okunbor (1987) investigate rational
form and derived the explicit rational Runge-Kutta scheme:
r
yn+1 =
yn + h∑ wi K i
i =1
r
1 + hyn ∑ vi H i
i =1
where
=
and
where
+ ℎ,
=
in which
,
,
(1.4)
+
,
,
=−
ℎ,
!
+ℎ∑
+ℎ∑
,
,
,
and
=1 1
(1.5)
=1 1
=
(1.6)
"#
are arbitrary constants to be determined.
131
(1.7)
2. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
www.iiste.org
=∑
,
(1.8)
is imposed to ensure consistency of the method.
In view of these inadequacies of the explicit schemes and the superior region of absolute stability associated with
implicit schemes, Ademuluyi and Babatola (2000) generate implicit rational Runge-Kutta and generates also the
parameters so that the resulting numerical approximation method shall be A-stable and will have low bound for
local truncation error. Since then many new rational Runge – Kutta schemes have been developed for the
solution of first order initial value problems and found out to give better estimates. Among these authors are:
Ademuluyi and Babatola (2001), Odekunle (2001), Odekunle et al. (2004), Bolarinwa (2005), Babatola et al.
(2007), Bolarinwa et al. (2012) and Abhulimen and Uluko (2012) The schemes are absolutely stable, consistent
and convergent and was used to approximate a variety of first order differential equations. However, the methods
are presently receiving more attention as efficient schemes for the solutions of various types of first order initial
value problems are considered.
2.
Derivation of the Scheme
Consider the second order initial value problems
y′′ = f (x, y, y′),
y (x0 ) = y0 , y ′(x0 ) = y0 a ≤ x ≤ b
,
(2.1)
The general $ − stage Runge-Kutta scheme for general second order initial value problems of ordinary
differential equations of the form (1) as defined by Jain (1984) is
s
yn+1 = yn + hy′ + ∑ cr k r
n
r =1
(2.2)
1 s
∑ cr′ kr
h r =1
(2.3)
and
′
yn+1 = y′ +
n
where
r
h2
1 r
′
Kr =
f xn + ci h, y n + hci yn + ∑ aij k j , y ′ + ∑ bij k j , i = 1(1)s
n
2
h j =1
j =1
(2.4)
i
1 i
ci = ∑ aij = ∑ bij , i(1)r
2 j =1
j =1
with
The rational form of (2.2) and (2.3) can be defined as
s
′
yn + hyn + ∑ wr K r
yn+1 =
r =1
s
′
1 + y n ∑ vr H r
r =1
y′ +1 =
n
yn +
1+
(2.5)
1 s
∑ wr′ K r
h r =1
1 ' s
y n ∑ v′ H r
r
h r =1
(2.6)
where
Kr =
Hr =
s
h2
1 s
′
′
f xn + ci h, y n + hci y n + ∑ aij K j , y n + ∑ bij K j , i = 1(1) s
2
h j =1
j =1
h2
2
s
1 s
g xn + d i h, zn + hd i z ′ + ∑ α ij H j , z ′ + ∑ β ij H j , i = 1(1) s
n
n
h j =1
j =1
with constraints
i
ci = ∑ aij =
j =1
1 i
∑ bij , i = 1(1)r
2 j
132
(2.7)
(2.8)
3. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
i
d i = ∑ α ij =
j =1
in which
,
, ′
=−
www.iiste.org
1 i
∑ β ij , i = 1(1)r
2 j
,
!
, ′
(A)
&
=
(B)
"#
The constraint equations are to ensure consistency of the method, ℎ is the step size and the parameters
, , , , ' , ( are constants called the parameters of the method.
Using Bobatola etal (2007), the following procedures are adapted.
K
H
( x , y , y′ )
i. Obtain the Taylor series expansion of r and r about the point n n n and binomial series
expansion of right side of (2.1) and (2.2).
iv.
Insert the Taylor series expansion into (2.1) and (2.2) respectively.
K
H
( x , y , y′ )
v.
Compare the final expansion of r and r about the point n n n to the Taylor series expansion of
yn+1
y′
( x , y , y′ )
and n+1 about n n n in the powers of h .
Normally the numbers of parameters exceed the number of equations, these parameters are chosen to ensure
that (one or more of the following conditions are satisfied.
iv.
Minimum bound of local truncation error exists.
v.
The method has maximized interval of absolute stability.
vi.
Minimized computer storage facilities are utilized.
To derive a One – stage scheme, we set s = 1 in equations (2.5), (2.6), (2.7) and (2.8) to have
′
y + hyn + w1K1
yn+1 = n
1 + y′ v1H1
n
(2.9)
and
y′ +1 =
n
yn +
1+
1
′
w1 K1
h
1 '
′
yn v1 H1
h
(2.10)
where
ℎ!
1
=
* + ℎ, + ℎ ′ +
+, ′ +
+ ,, = 1 1 $
2
ℎ
and
ℎ!
1
=
* + ℎ, + ℎ
′ +'
, ′ + (
,, = 1 1 $
2
ℎ
with constraints
=
=
&
=' = (
!
!
where ,
,
, , ' , ( , . , .′ , / and /′ are all constants to be determined.
Equation (2.9) can be written as
−1
'
yn+1 = yn + hyn + w1k1 (1 + yn v1H1 )
(
)
Expanding the bracket and neglecting 2nd and higher orders gives
'
yn+1 = yn + hyn + w1k1 (1 − yn v1 H1 )
(
)
Expanding (2.14) and re-arranging, gives
+ ℎ ′ − !/ + ℎ
0 =
Equation (2.10) can be written as
′
0
=
′ + .′ + 31 +
2
2
/
1
′ /′
+ . −
4
′ 1 ′ 1 ′ ′
′
yn+1 = yn + w1k1 1 − yn v1H1
h
h
Expanding the binomial and re-arranging also gives
′ 0 = ′ + .′ + − 3 ′! /′ + 5 ′ .′ /′ + 4
2
2
2
Now, the Taylor’s series expansion of
yn +1 about xn is given as
133
2.11
2.11
(2.12)
(2.13)
(2.14)
/
. +
(2.15)
(2.16)
(2.17)
4. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
0
and
′
=
0
where
11
=
111
=
1:
=
1:
+ℎ ′ +
ℎ! ′′ ℎ7 ′′′ ℎ9 :
+
+
+ …
2!
3!
4!
= ′ + ℎ ′′ +
, , ′ =
+ ′ " + "1 = ∆
!
!
?? + ′ "" +
"1"1 + 2
(2.20)
= ∆! + "1 ∆ + "
∆=
B
B?
2.18
ℎ! ′′′ ℎ7 :
+
+ …
2!
3!
?
Since
www.iiste.org
+ ′
B
B"
1
"" A
2.19
+2
?"1
+
"1 ∆
B
+
(2.21)
B"1
Using the Taylor’s series of the function of three variables we have from
2
1
+ = +* ℎ ?+ ℎ ′ +
+
+
"1 ,
!
ℎ
ℎ
1
1
+ C ℎ ! ?? + 2 ℎ ℎ ′ +
+ ?" + 2 ℎ *
+,
2!
ℎ
!
1
1
+2 ℎ ′ +
+ *
+ , ""1 + *
+ , "1"1 D + …
ℎ
ℎ
Simplifying further and arranging the equation in powers of ℎ gives,
2G
2
+ = E
!
+
"1
+
+!
""1 F
+
25
!
E
+
+
2H
"
+
+
?"1
+
?"1
!
+ ℎ
+!
""
′ +
+
1
+
+
!
""
""1 F
+
1
1
E ?+
+ ?" +
+ "" F + E ! ?? + ! 1 ?" + ! ′! "" F + 0 ℎJ
(2.22)
" +
9
Equation (2.22) is implicit; one cannot proceed by successive substitution. Following Lambert (1973), we can
assume that the solution for + may be express in the form
+ = ℎK + ℎ! L + ℎ7 M + ℎ9 N + 0 ℎJ
(2.23)
Substituting equation (2.23) into (2.22) gives
ℎ
+ = E
ℎK + ℎ! L + ℎ7 M "1 +
ℎK + ℎ! L ! ""1 F
2
ℎ!
+ E +
ℎK + ℎ! L " +
ℎK + ℎ! L ?"1 + ! ℎK ! ""
2
ℎ7
1
1
1
+
ℎK + ℎ! L ""1 F + E ? +
ℎK ?" +
ℎK "" F
" +
2
9
ℎ
+ E ! ?? + ! 1 ?" + ! ′! "" F + 0 ℎJ
2.24
4
On equating powers of ℎ from equation (2.22) and (2.23), gives
1
K = 0, L =
, M =
∆ , since =
? +
" + 1/2
"1 =
!
!
!
!
! !
N =
∆
+
∆ "1 +
"
9
Substituting K , L , M and N into (2.23) gives.
25
2G
2H
!
(2.25)
! !
+ =
+
∆ +
∆
+
∆ "1 +
"
!
!
!
(2.26)
Similarly, expanding in Taylor’s series about
, , 1 , from (2.11b), we have
!
ℎ
ℎ
!
!
!
1
P +'
= P(
(
(
Q1 + ' (
QQ1 R +
Q +
?Q1 + '
QQ +
QQ1 R
2
2
7
9
ℎ
ℎ !
1
! 1
! !
P
+ P
'
' 1
′ QQ R
? +
Q +
?Q +
QQ R +
?? +
?Q +
2
4
J
+0 ℎ
2.27
Equation (2.27) is also implicit which cannot be proceed by successive substitution. Assuming a solution of the
equation is of the form
= ℎT + ℎ! U + ℎ7 V + ℎ9 W + 0 ℎJ
2.28
Substituting the values of
in (2.28) into equation (2.27) and equating powers of ℎ of the equation, we can get
the following after substitutions:
134
5. Mathematical Theory and Modeling
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Vol.3, No.11, 2013
T = 0,
1
2
U =
,
V =
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1
2
∆
& W =
1
4
∆
! !
+( ∆
Q1
+'
Q
(2.29
)
Substituting equation (2.29) into equation (2.28) gives
ℎ!
ℎ7
ℎ9 ! !
=
+
∆ +
∆
+( ∆
Q1 + '
Q
2
2
2
Using equations (2.23) and (2.28) into equations (2.15) and (2.17) respectively gives
1
+ ℎ 1 − !/ + ℎ
/ ℎ! U + ℎ7 V + ℎ9 W
0 =
+ P. − / . ℎ! U + ℎ7 V + ℎ9 W R ℎ! L + ℎ7 M + ℎ9 N
Expanding the brackets and re-arranging in powers of ℎ gives
1
+ ℎ 1 + ℎ! . L − ! / U + ℎ7 . M − ! / V −
/ U + 0 ℎ9
0 =
1
Also for
0
2.30
(2.31)
gives
1
+ . 1 ℎ! L + ℎ7 M + ℎ9 N
ℎ
1
1
− X ′! / 1 + ! 1 . 1 / 1 ℎ ! L + ℎ 7 M + ℎ 9 N Y ℎ ! U + ℎ 7 V + ℎ 9 W
ℎ
ℎ
Expanding the brackets and re-arrange in powers of ℎ gives
1
1
+ ℎ . 1 L − ′! / 1 U + ℎ! . 1 M − ′! / 1 V − 1 . 1 / 1 L U
0 =
+ ℎ7 . 1 N − ′! / 1 W − 1 . 1 / 1 L V − 1 . 1 / 1 M U + 0 ℎ9
2.32
Comparing the corresponding powers in ℎ of equations (2.31) and (2.32) with equations (2.18) and (2.19) we
obtain
. − 1/
=
1
!
!
!
.. ∆
!
.1 ∆
!
.1
−
=
0
−
!
!
−
1
1!
!
/
/
1! 1
!
/
∆
−
=
1! 1
∆
−
!
1
!
1
/
.1 /1
= ∆
Z
= ∆
!
!
(By using the equations in (2.25) and (2.29))
Since from equation (1.7)
=−
∆
[#
5
"#
,
?
=−
[
5
"#
,
Q
= −2
[#
"#
(2.33)
+
"
,
Q1
= −2
[#
"#
+
"1
,
1
=−
A
"#
5
"#
and
=
+ 1 Q+
Q1
Using those equations into equation (2.33), we get the following simultaneous equations
. +/ = 1
=
. +/
7
1
1
. +/ = 2
.1 + /1 = 1
Equation (2.35) has (4) equations with (6) unknowns; there will not be a unique solution for
There will be a family of one-stage scheme of order four.
i.
Choosing the parameters
1
2
1
′
′
w1 = , v1 = , c1 = a11 = b11 = 0, w1 = 0, v1 = 2, d1 = α11 = , β11 = 1
3
3
2
arbitrarily the
following scheme is obtain.
1
hy′ + K1
n
3
yn+1 =
2
1 + yn H1
3
and
y′
n
′
yn +1 =
2
1 + H1
h
135
(2.34)
(2.35)
(2.35).
(2.36)
(2.37)
6. Mathematical Theory and Modeling
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Vol.3, No.11, 2013
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where
h2
f ( xn , y n , y ′ )
n
2
h2
1
1
1
1
′
H1 =
f x n + h, z n + hz ′ + H 1 , z n + H 1
n
2
2
2
2
h , since
=' = (
!
Choosing the parameters
1
1
1
1
′
′
w1 = v1 = , c1 = a11 = , d1 = α11 = , w1 = 2, v1 = 0, b11 = 1, β11 =
2
2
6
3
From (2.35) setting
Then,
1
y n + hy ′ + K 1
n
2
y n +1 =
1
1 + yn H 1
2
K1 =
ii.
(2.38)
and
′
′
yn+1 = yn +
2
K1
h
(2.39)
where
K1 =
h2
1
1
1
1
1
′
′
f xn + h, yn + hyn + K1 , yn + K1
c =a = b
2
2
2
2
h , since 1 11 2 11
H1 =
h2
1
1
1
1
1
′
′
g xn + h, zn + hzn + H1 , zn + H1
2
6
6
6
3h , since d 1= α11 = 2 β11
and
3.
CONVERGENCE
A numerical method is said to be convergent if the numerical solution approaches the exact solution as
the step size tends to zero.
Convergent = lim2→a |
− 0 |
0
In other words, if the discretiation error at 0 tends to zero as ℎ → ∞, i.e if
− 0 |→0
d 0 =|
as & → ∞
(3.1)
0
From equation (2.5),
1 s
′
yn + ∑ wr K r
h r =1
′
yn+1 =
1 ' s
′
1 + y n ∑ vr H r
h r =1
(3.2)
1
seems to satisfy the equation of the form
0
1 s
′
y ( xn ) + ∑ wr K r
h r =1
y ′(xn+1 ) =
+ Tn+1
s
1
′
1 + y′( xn )∑ vr H r
h
r =1
while the exact solution
Where e 0 is a local truncation error.
Subtracting equation (3.3) from (3.2) gives
1 s
1 s
yn + ∑ w′ K r
y (xn ) + ∑ w′ K r
r
r
h r =1
h r =1
y′ +1 − y′( xn+1 ) =
−
+ Tn+1
n
s
1 ' s
1
′
′
1 + y n ∑ vr H r
1 + y′(xn )∑ vr H r
h r =1
h
r =1
Adopting equation (3.4) gives
136
(3.3)
(3.4)
7. Mathematical Theory and Modeling
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s
1 s
1 s
1
1 ' s
′
1 + y ′(xn )∑ v′ H r yn + ∑ w′ K r − 1 + yn ∑ v′ H r y (xn ) + ∑ wr K r
r
r
r
h
h r =1
h r =1
r =1
h r =1
+T
en+1 =
n +1
s
s
1 '
1
1 + yn ∑ v′ H r 1 + y ′(xn )∑ v′ H r
r
r
r =1
h r =1
h
Expanding the brackets and re-arranging gives
s
s
1
en + 2 ( y′ − y′(xn )) ∑ w′ K r ∑ v′ H r
n
r
r
h
r =1
r =1
en+1 =
+ Tn+1
s
s
1 '
1
′
′
1 + yn ∑ vr H r 1 + y′(xn )∑ vr H r
r =1
h r =1
h
This implies that
s
1 s
′
′
en + en 2 ∑ wr K r ∑ vr H r
h r =1
r =1
en+1 =
+ Tn+1
s
s
1 '
1
′
1 + yn ∑ vr H r 1 + y′(xn )∑ v′ H r
r
r =1
h r =1
h
s
1 s
′
′
en 1 + 2 ∑ wr K r ∑ vr H r
r =1
h r =1
en+1 =
+ Tn+1
s
s
1 '
1
′ H r 1 + y′(xn )∑ vr H r
′
1 + yn ∑ vr
r =1
h r =1
h
From equations (3.7), setting
s
1 s
′
′
An = 1 + 2 ∑ wr K r ∑ vr H r
r =1
h r =1
and e
Then
=e
0
d
0
=
f#
g# h#
(3.6)
(3.7)
s
1 ' s
1
′
Bn = 1 + yn ∑ v′ H r Cn = 1 + y′(xn )∑ vr H r
r
r =1
h r =1
,
h
,
d +e
Let L = max L > 0, M = max M > 0 and K = max K < 0 then (3.8) becomes,
f
d 0 ≤ d +e
f
(3.5)
(3.8)
gh
= + < 1, then
gh
d 0 ≤ +d + e
(3.9)
If & = 0, then from (3.9),
d = +da + e
d! = +d + e = + ! da + +e + e by substituting the value of d
d7 = +d! + e = + 7 da + + ! e + e
Continuing in this manner, we get the following
d 0 = + 0 da + ∑n 0a + n e
(3.10)
f
Since = + < 1, then one can see that as & → ∞, d 0 → 0. This proves that the scheme converges.
gh
7. CONSISTENCY
A scheme is said to be consistent, if the difference equation of the computation formula exactly
approximates the differential equation it intends to solve as the step size ends to zero. To prove if equation (2.5)
is consistent, subtract from both side of (2.5), then
Set
s
y n+1 − yn =
′
y n + hyn + ∑ wr K r
r =1
s
1 + y n ∑ vr H r
− yn
r =1
(4.1)
s
yn+1 − yn =
y n + hy′ + ∑ wr K r − yn − y
n
r =1
2
n
s
∑ vr H r
r =1
s
1 + y n ∑ vr H r
r =1
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8. Mathematical Theory and Modeling
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Vol.3, No.11, 2013
s
y n+1 − y n =
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s
2
′
hy n + ∑ wr K r − y n ∑ vr H r
r =1
r =1
s
1 + y n ∑ vr H r
r =1
(4.2)
but
Kr =
s
h2
1 s
′
′
f x n + c i h, y n + hc i y n + ∑ a ij K j , y n + ∑ bij K j
2
h j =1
j =1
and
s
1 s
′
′
g x n + d i h, z n + hd i z n + ∑ α ij H j , z n + ∑ β ij H j
h j =1
j =1
Then (4.2) becomes
Hr =
h2
2
s
hy ′ + ∑ wr
n
y n +1 − y n =
r =1
s
s
s
1 s
1 s
h2
h2
2
′
′
′
f xn + ci h, y n + hci y n + ∑ aij K j , y n + ∑ bij K j − y n ∑ v r
g xn + d i h, z n + hd i z n + ∑ α ij H j , z ′ + ∑ β ij H j
n
2
2
h j =1
h j =1
j =1
r =1
j =1
2
s
s
1 s
h
xn + d i h, z n + hd i z ′ + ∑ α ij H j , z ′ + ∑ β ij H j
1 + y n ∑ vr
g
n
n
2
h j =1
r =1
j =1
Dividing the above equation throughout by h and taking the limit as h tends to zero on both sides gives
y − yn
lim n+1
= y′
n
h→0
h
y′
Again recall that from (2.6), subtracting n on both sides gives
1 s
y ′ + ∑ w′ K r
n
r
h r =1
′
′
y n +1 − y ′ =
− yn
n
s
1 '
′ ∑ v′ H r
1 + yn
r
h r =1
1 s
1 ' s
y ′ + ∑ w′ K r − y ′ − y ′ 2 n ∑ v′ H r
n
r
n
r
h r =1
h
r =1
y ′ +1 − y ′ =
n
n
s
1
′
1 + y n' ∑ v′ H r
r
h r =1
Simplify further gives
1 s
1 ' s
∑ wr′ K r − h y′ 2 n ∑ vr′ H r
r =1
′ h
y ′ +1 − y n = r =1
n
1 ' s
′
′
1 + y n ∑ vr H r
h r =1
(4.3)
(4.4)
Substituting the values of Kr and Hr (4.4) becomes
s
s
1 ' s
1 s
h2
1 s
h2
1 s
′
∑ w′r K r 2 f xn + ci h, yn + hci yn + ∑ aij K j , y′n + h ∑ bij K j − h y′ 2 n ∑ v′r H r 2 g xn + di h, zn + hdi z′n + ∑α ij H j , z′n + h ∑ β ij H j
h r =1
j =1
j =1
r =1
j =1
j =1
y ′ +1 − y ′ =
n
n
2
s
s
s
1 '
1
h
1 + y′ ∑ v′
g xn + d i h, z n + hd i z ′ + ∑α ij H j , z ′ + ∑ β ij H j
n
r
n
n
h r =1 2
h j =1
j =1
Dividing all through by ℎ and taking the limit as ℎ tends to zero on both sides gives
s
s
1 ' s
1 s
h2
1 s
h2
1 s
′
∑ w′r K r 2 f xn + ci h, yn + hci y′n + ∑ aij K j , yn + h ∑ bij K j − h y′ 2 n ∑ v′r H r 2 g xn + di h, zn + hdi z′n + ∑α ij H j , z′n + h ∑ β ij H j
h r =1
j =1
j =1
r =1
j =1
j =1
y ′ +1 − y ′ =
n
n
s
1 ' s
1 s
h2
1 + y ′ ∑ v′
g xn + d i h, z n + hdi z ′ + ∑α ij H j , z ′ + ∑ β ij H j
n
r
n
n
h r =1 2
h j =1
j =1
lim
h→0
yn+1 − yn
=
h
1
2
s
s
' 1
1 s
1 s
′
f xn + ci h, yn + hci yn + ∑aij K j , y′ + ∑bij K j − y′2 n g xn + di h, zn + hdi z′ + ∑αij H j , z′ + ∑ βij H j
n
n
n
h j =1
2
h j=1
j =1
j =1
s
1
1 s
1 + y′' g xn + di h, zn + hdi z′ + ∑αij H j , z′ + ∑βij H j
n
n
n
2
h j =1
j =1
but by definition
138
9. Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.11, 2013
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′
f n = y ′ 2 g ( xn , z n , z n )
n
hence the above equation becomes
y − yn
lim n+1
= fn
h→0
h
Hence, the numerical method is consistent.
Conclusion
The new numerical schemes derived follows the techniques of rational form of Runge – Kutta methods proposed
by Hong (1982) which was adopted by Okunbor (1987) and Ademiluyi and Babatola (2000) by using Taylor and
Binomial expansion in stages evaluation. The order condition obtained in this research is up to five (5) and the
stage is up to three (3). This is an improvement on the work of earlier authors.
Due it convergence and consistency of the new schemes, the scheme will be of high accuracy for direct
numerical solution of general second order ordinary differential equations. The steps to the derivation of the new
schemes are presented in the methodology while the analysis of the schemes proved to be consistent, convergent.
The implementation of the schemes will be highlighted in the forthcoming paper.
References
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139
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