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A Simplified Derivation and Analysis of Fourth Order Runge Kutta Method

Authors:
  • Universiti Putra Malaysia and Umaru Musa Yar'adua University Katsina

Abstract

The derivation of fourth order Runge-Kutta method involves tedious computation of many unknowns and the detailed step by step derivation and analysis can hardly be found in many literatures. Due to the vital role played by the method in the field of computation and applied science/engineering, we simplify and further reduce the complexity of its derivation and analysis by exploring some possibly well-known works and propose a step by step derivation of the method. We have also shown the stability region graphically
International Journal of Computer Applications (0975 8887)
Volume 9 No.8, November 2010
51
A Simplified Derivation and Analysis of Fourth Order
Runge Kutta Method
Musa H.
Department of Mathematics and
Computer Science
UMYU
Ibrahim Saidu
Faculty of Computer Science and
Information Technology
University Putra,Malaysia.
M. Y. Waziri
Faculty of Science, Department of
Mathematics
University Putra,Malaysia
ABSTRACT
The derivation of fourth order Runge-Kutta method involves
tedious computation of many unknowns and the detailed step by
step derivation and analysis can hardly be found in many
literatures. Due to the vital role played by the method in the
field of computation and applied science/engineering, we
simplify and further reduce the complexity of its derivation and
analysis by exploring some possibly well-known works and
propose a step by step derivation of the method. We have also
shown the stability region graphically
Keywords: Fourth order Runge Kutta Method, Derivation,
Stability Analysis
1. INTRODUCTION
Runge-Kutta formulas are among the oldest and best understood
schemes in numerical analysis. However, despite the evolution of
a vast and comprehensive body of knowledge, it continues to be a
source of active research [7]. Runge-Kutta methods provide a
popular way to solve the initial value problem for a system of
ordinary differential equations [11]:
with a given step length h through the interval ,
successively producing approximations to . We deal
exclusively with the step by step derivation and the stability
analysis of the fourth order Runge-Kutta Method. For a thorough
coverage of the derivation and analysis the reader is referred to
[1,2,3,4,5].
The paper has the following structure: section 2 presents
mathematical formulation and derivation, Section 3 presents the
analysis and section 4 presents the conclusion.
2. MATHEMATICAL FORMULATION
AND DERIVATION
We begin by defining the function as in [1,2,3,4,5 and 6]
1( , , )
nn
y y h x y h

Where
1
1
1
1
1
1
( , , )
( , )
( , ), 2,3,..., 1
s
ii
i
i
i i n ij j
j
i
i ij
j
x y h b k
k f x y
k f x c h y h a k i i
ca
 
1 1 1 2 2 3 3 4 4
1
2 2 21 1
3 3 31 1 32 2
4 4 41 1 42 2 43 3
()
( , )
( , )
( , ( ))
( , ( ))
nn
n
n
n
y y h b k b k b k b k
k f x y
k f x c h y ha k
k f x c h y h a k a k
k f x c h y h a k a k a k
 
 
 
 
The functions are expanded using a Taylor series expansion for
function of two variables. To get the unknowns, we use the
fourth order coefficients of order 4
(1)
1
(2)
1
(3) 2
1
(3)
2
(4) 3
1
(2)
2
(4) 2
3
(4)
4
1
1
2
11
26
1
6
11
6 24
1
8
11
2 24
1
24
i
i
ii
i
ii
i
i ij j
ij
ii
i
i i ij j
ij
i ij j
ij
i ij jk k
ij
b
bc
bc
ba c
bc
b c a c
ba c
ba a c








Setting the coefficients to zero, we have
1 2 3 4 1b b b b  
(1)
International Journal of Computer Applications (0975 8887)
Volume 9 No.8, November 2010
52
2 2 3 3 4 4 1
2
b c b c b c 
(2)
2 2 2
2 2 3 3 4 4 1
3
b c b c b c
(3)
(4)
3 3 3
2 2 3 3 4 4 1
4
b c b c b c 
(5)
3 3 32 2 4 4 42 2 4 4 43 3 1
8
b c a c b c a c b c a c  
(6)
2 2 2
3 32 2 4 42 2 4 43 3 1
12
b a c b a c b a c  
(7)
4 43 32 2 1
24
b a a c
(8)
We use the simplifying assumptions by Butcher:
1(1 ), 2,3,4
s
i ij i j
iba b c j
 
(9)
Which affect the expression for
(3) (4)
23
,

and
(4)
4
. i.e.
(3) (2) (3)
2 1 1
2
 

(4) (3) (4)
3 1 1
3
 

(4) (2) (3) (4)
4 1 1 2
2
 
 
Now using equation (9) for
2,3 and 4j
we have:
3 32 4 42 2 2
(1 )b a b a b c  
(i)
4 43 3 3
(1 )b a b c
(ii)
44
0 (1 )bc
respectively. (iii)
Now when
4j
in (iii),
41c
and
40b
for a four stage
method.
We substitute
41c
in equations 2, 3 and 5 and solve for
2
b
,
3
b
and
4
b
simultaneously. Therefore equations 2, 3 and 5
becomes
2 2 3 3 4 1
2
b c b c b  
22
2 2 3 3 4 1
3
b c b c b
33
2 2 3 3 4 1
4
b c b c b  
Using crammer’s rule, we first find the determinant of the
coefficient matrix
23
22
2 3 2 3 2 2 3 3
33
23
1
1 ( 1)( )( 1)
1
cc
D c c c c c c c c
cc
 
To solve for
2
b
2
3
23 3 3
3
3
3
11
2( 1)(2 1)
11
3 12
11
4
b
c
c c c
Dc
c
 

23 3 3 3
2 2 3 2 2 3 3 2 2 3 2
( 1)(2 1) 1 2
( 1)( )( 1)
12 12 (1 )( )
b
Dc c c c
b c c c c c c
D c c c c
 
    
To solve for
3
b
3
2
22 2 2
2
3
2
11
2( 1)(2 1)
11
3 12
11
4
b
c
c c c
Dc
c


32 2 2 2
3 2 3 2 2 3 3 3 3 2 3
( 1)(2 1) 1 2
( 1)( )( 1)
12 12 ( )(1 )
b
Dc c c c
b c c c c c c
D c c c c
 
    
To solve for
4
b
4
23
22 2 3 2 3 2 3 2 3
23
33
23
1
2( )(3 4 4 6 )
1
3 12
1
4
b
cc
c c c c c c c c
D c c
cc
  

42 3 2 3 2 3 2 3 2 3 2 3
4 2 3 2 2 3 3 23
( )(3 4 4 6 ) 6 4( ) 3
( 1)( )( 1)
12 12(1 )(1 )
b
Dc c c c c c c c c c c c
b c c c c c c
D c c
 
    
Now to solve for
43
a
, we use equation (ii) i.e. when j=3
Hence, we have
3 3 2 3
2
43 3
4 3 3 2 3 2 3 2 3
(1 ) 12(1 )(1 )
12 (1 )
12 ( )(1 ) 6 4( ) 3
b c c c
c
ac
b c c c c c c c c
 
 
 
International Journal of Computer Applications (0975 8887)
Volume 9 No.8, November 2010
53
2 2 3
3 2 3 2 3 3 2
(1 )(2 1)(1 )
( )(6 4( )) 3
c c c
c c c c c c c
 
 
To solve for
32
a
and
42
a
, we use equations (i) (when j=2) and
(8) i.e.
3 32 4 42 2 2
(1 )b a b a b c  
(i)
4 43 32 2 1
24
b a a c
(8)
From equation (8) above,
2 3 3 2 3 2 3 3 2
32 2 4 43 2 2 3 3 2 2 2 3
12(1 )(1 ) ( )(6 4( ) 3)
1 1 1 1
24 24 6 4( ) 3 (1 )(2 1(1 )
c c c c c c c c c
ac b a c c c c c c c c
 
   
 
3 2 3
22
()
2 (2 1)
c c c
cc
Substituting this value into (i), we have
2 2 3 32
42 4
3 3 2 3 2 3
2
2
2 2 3 2 3 3 3 2 2 2 2 3 2 3
2 3 3 2 3
2 2 3 2 3 2 3
(1 )
1 2 ( ) 12(1 )(1 )
12
(1 )
12 (1 )( ) 12 (1 )( ) 2 (2 1) 6 4( ) 3
(1 ){2(1 )(1 2 ) ( )}
2 ( ){6 4( ) 3)}
b c b a
ab
c c c c c c
c
c
c c c c c c c c c c c c c c
c c c c c
c c c c c c c


 
 

 

 
 
This solution assumes that
2 3 2 3 2 1
0,1, 0,1, , 2
c c c c c  
We choose two free parameters
21
3
c
and
32
3
c
Substituting these values into
4
b
,
3
b
and
2
b
we have:
4
1 2 2 1 4
6 4 3 11
3 3 3 3 38
12 8
12 1 1 3
33
b
  
 
  
  
 
  

  
  
3
11
12 3
33
8
2 2 2 1 8
12 1 9
3 3 3 3
b



 
 

 
 
2
21
12 3
33
8
1 1 1 2 8
12 1 9
3 3 3 3
b



 
 

 
 
Using equation (1)
1 2 3 4
1 2 3 4
1
1
3 3 1 1
1 8 8 8 8
b b b b
b b b b
  
 
  
Also
2 21 1
3
ca
Using equation (ii) (when j=3),
4 43 3 3
(1 )b a b c
33
43 4
(1 ) 3 2 8
11
8 3 1
bc
ab

 


Also
2 3 3 2 3
42 2 2 3 2 3 2 3
(1 ){2(1 )(1 2 ) ( )}
2 ( ){6 4( ) 3)}
1 2 2 1 2
1 {2(1 )(1 2 ) ( )}
3 3 3 3 3
1
1 1 2 1 2 1 2
2 ( ){6 4( ) 3)}
3 3 3 3 3 3 3
c c c c c
ac c c c c c c
 
 

 


 
 
Using equation (2) we can obtain
4
c
as
4 4 2 2 3 3
4
1
21 3 1 3 2
2 8 3 8 3 1
1
8
b c b c b c
c
 
 
   
 
 
 
Hence,
4 41 42 43
41 4 42 43 1 ( 1) 1 1
c a a a
a c a a
  
   
Also
3 2 3
32 22
2 1 2 2
()
()3 3 3 9 1
1 1 2
2 (2 1) 2 (2 1)
3 3 9
c c c
acc
 
 
From
3 31 32
c a a
31 3 32 21
1
33
a c a
 
Finally, we know that
1 11 0ca
.
We have therefore determined all the unknowns in the method
and the method can be written in Butcher’s Tableu [3] as
0 0 0 0 0
1/3 1/3 0 0 0
2/3 -1/3 1 0 0
1/8 3/8 3/8 1/8
Which has the form
1 1 2 3 4
( 3 3 )
8
nn
h
y y k k k k
 
International Journal of Computer Applications (0975 8887)
Volume 9 No.8, November 2010
54
1
1
2 2 21 1
3 3 31 1 32 2 1 2
4 4 41 1 42 2 43 3 1 2 3
( , )
( , ) ( , )
33
21
( , ( )) ( , ( )
33
( , ( ) ( , ( )
nn
n n n n
n n n n
n n n n
k f x y
hk
h
k f x c h y ha k f x y
k f x c h y h a k a k f x h y h k k
k x c h y h a k a k a k f x h y h k k k
   
   
 
3. ANALYSIS OF THE METHOD
The stability polynomial is given by
1
( ) 1 ( )
T
R h hb I hA e
 
and it is required that
( ) 1Rh
for absolute stability see [6]. Now for the Runge Kutta forth
order method,
1 1 2 3 4
( 3 3 )
8
nn
h
y y k k k k
 
The Butcher’s Tableu is
0 0 0 0 0
1/3 1/3 0 0 0
2/3 -1/3 1 0 0
1/8 3/8 3/8 1/8
0 0 0 0
10 0 0
311 0 0
3
1 1 1 0
A









,
1 0 0 0
1 0 0
3
10
3
1
h
I hA hh
h h h











,
1 3 3 1 3 3
8 8 8 8 8 8 8 8
Th h h h
hb h 






1
( ) 1 ( )
T
R h hb I hA e
 
1
1 0 0 0
1
1 0 0 1
33 3
11
8 8 8 8 10 1
3
1
h
h h h h
hh
h h h






 
 
 








2
23 2
1 0 0 0
1
1 0 0
31
33
11
8 8 8 8 10
33 1
21
33
h
h h h h hh h
hh
h h h h



















 


2 2 2 3
22
2
32
8 8 8 3 3 8 3 3
1
33 1
8 8 8
11
31
88
8
h h h h h h h h
h
h h h hh
hh
h

 

 
 
 

 




 













2 2 3 2 3 4
2 2 3
2
3 3 2
8 8 24 24 8 24 24 1
33 1
8 8 8 8
11
31
88
8
h h h h h h h
h h h h
hh
h


   





 













2 3 4 2 3 2
33
18 8 24 24 8 4 8 8 8 8
h h h h h h h h h h
       
2 3 4
12 6 24
h h h
h 
For absolute stability
2 3 4
1 1 1
1 1 1
2 6 24
h h h h 
Taking the RHS
2 3 4
1 1 1
11
2 6 24
h h h h 
2 3 4
1 1 1 0
2 6 24
h h h h  
Using Mathematica we get the roots as
NSolve[h+h*h/2+h*h*h/6+h*h*h*h/24==0,h]
{{h-2.78529},{h-0.607353-2.8719 },{h-
0.607353+2.8719 },{h0.}}
We consider 3 cases as it can be found in [1]
Case 1
When
is real and
0
,
The roots are -2.785 and 0
International Journal of Computer Applications (0975 8887)
Volume 9 No.8, November 2010
55
Hence the stability interval is
( 2.785,0)h
.
Case 2
2 3 4
23
2
8 8 24 24 1
31
8 4 8
11
31
88
8
h h h h
h h h
hh
h


  



















When
h
is pure and
imaginary,
We set
iy
in the stability polynomial to get
3
24
( ) ( ) ( )
1 ( ) 1
2 6 24
yh yh yh
i yh i 
3
24
( ) ( ) ( )
1 ( ) 1
2 24 6
yh yh yh
iyh i 
Let
t yh
and take the magnitude
2
2 4 3
11
2 24 6
t t t
t
 
   
 
 
2 4 2 4 6 4 6 8 4 4 6
2
11
2 24 2 4 48 24 48 578 6 6 36
t t t t t t t t t t t
t
 
       
 
 
Simplifying, we get
68
11
72 576
tt
 
68
0
72 576
tt
 
Using Mathematica to find the roots we have
NSolve[(-(t^6)/72)+((t^8)/576)0,t]
{{t-
.82843},{t0.},{t0.},{t0.},{t0.},{t0.},{
t0.},{t2.82843}}
The equation is satisfied for
2.82843t
i.e.
22t
Hence the stability interval is
0 2 2h
. i.e.
(0,2 2)h
Case 3 When
is complex with
Re( ) 0
, we set
x iy

in
24
3
( ) ( ) ( )
11
2 6 24
h h h
h
  
 
and plot the boundary of the region by plotting the real and
imaginary parts.
The stability region is plotted using Maple as follows
Re( )h
4. CONCLUSION
In this paper, we have simplified the existing derivation and
analysis of the fourth order Runge-Kutta Method for easy
reference to students and plot the stability region. We also
reduced the complexity of the method by proposing a step by
step derivation approach for better understanding to students.
5. REFERENCES
[1] M.K. Jain, S.R.K. Iyengar, R.K. Jain, (2007), Numerical
Methods for Scientific and Engineering Computing.
[2] J. D. Lambert, (1991), Numerical Methods for Ordinary
Differential Systems, the initial value Problem, John Wiley &
Sons Ltd.
[3] J.C. Butcher, (2003), Numerical Methods for Ordinary
Differential Equations, John Wiley & Sons Ltd.
[4] John R. Dorman, (1996), Numerical Methods for
Differential Equations, a Computational Approach, CRC Press,
Inc.
[5] J. D. Lambert, (1973), Computational Methods in
Ordinary Differential Equations, John Wiley & Sons Ltd
[6] Fudzia Ismail, (2010), Lecture Notes on Numerical
Methods (unpublished), University Putra Malaysia
[7] G. Byrne and Hindmarsh, (1990), RK Methods prove
popular at IMA Conference on Numerical ODE’s,SIAM
News,23/2 pp.14-15.
[8] Lawrence F. Shampine, (1985),Interpolation for Runge-
Kutta Methods.SIAM Journal of numerical
analysis,22/5,pp.1014-1027.
... Basically the Euler method is geometric intuition, taking it by small steps is the way to solve it for exampleℎ > 0, and approaching ( + ℎ) ≈ ( ) + ℎ ( , ( )) (Edalat et al., 2020). In 1768 the first time Leonhard Euler, Beethoven of mathematics, reported on the use of basic difference methods to obtain approximate solutions to differential equations or initial value problems (Biswas et al., 2013;Musa et al., 2010). Basically, Euler method uses its slope at a point to find the numerical solution near that point. ...
... The Runge-Kutta method has similarities with the Euler method, which is close to the number at each point. However, the Runge-Kutta has a larger number of slope weights at each time, so it is more accurate than the Euler method (Molthrop, 2018).In numerical analysis, the Runge-Kutta formula is one of the oldest and best understood designs (Musa et al., 2010;Vaidyanathan et al., 2017;Vaidyanathan et al., 2018). ...
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Coronavirus Disease 2019 has become global pandemic in the world. Since its appearance, many researchers in world try to understand the disease, including mathematics researchers. In mathematics, many approaches are developed to study the disease. One of them is to understand the spreading of the disease by constructing an epidemiology model. In this approach, a system of differential equations is formed to understand the spread of the disease from a population. This is achieved by using the SIR model to solve the system, two numerical methods are used, namely Euler Method and 4th order Runge-Kutta. In this paper, we study the performance and comparison of both methods in solving the model. The result in this paper that in the running process of solving it turns out that using the euler method is faster than using the 4th order Runge-Kutta method and the differences of solutions between the two methods are large.
... An explicit Adams-Bashforth scheme of order four (AB4) is employed for marching forward-in-time, similarly to other FC-based solvers [22,26,28,21]. Other explicit timesteppers that provide adequate regions of stability [36,37] can also be employed, although timesteppers such as the fourth-order Runge- Kutta method may entail four evaluations of the right-hand-side F (x, t) that is given (from Equation (1)) ...
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Dye experimentation is a widely used method in experimental fluid mechanics for flow analysis or for the study of the transport of particles within a fluid. This technique is particularly useful in biomedical diagnostic applications ranging from hemodynamic analysis of cardiovascular systems to ocular circulation. However, simulating dyes governed by convection-diffusion partial differential equations (PDEs) can also be a useful post-processing analysis approach for computational fluid dynamics (CFD) applications. Such simulations can be used to identify the relative significance of different spatial subregions in particular time intervals of interest in an unsteady flow field. Additionally, dye evolution is closely related to non-discrete particle residence time (PRT) calculations that are governed by similar PDEs. PRT is a widely used metric for various fluid dynamics applications (e.g., environmental fluids, biological flows) and is a well-accepted biomarker for cardiovascular diseases since it is linked to thrombus formation. This contribution introduces a pseudo-spectral method based on Fourier continuation (FC) for conducting dye simulations and non-discrete particle residence time calculations without numerical diffusion errors. Convergence and error analyses are performed with both manufactured and analytical solutions. The methodology is applied to three distinct physical/physiological cases: 1) flow over a two-dimensional (2D) cavity; 2) pulsatile flow in a simplified partially-grafted aortic dissection model; and 3) non-Newtonian blood flow in a Fontan graft. Although velocity data is provided in this work by numerical simulation, the proposed approach can also be applied to velocity data collected through experimental techniques such as from particle image velocimetry.
... Point kinetic equations are solved using an explicit, fourth order Runge-Kutta solver with a fixed step size (Saidu and Waziri, 2010). One equation is evaluated to propagate the neutron population N t ð Þ; and another six equations are used to evaluate the neutron concentration for each individual group C i : ...
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A real-time research reactor simulator was developed at the Jožef Stefan Institute for education and training of students and future reactor operators, especially in countries with limited or no access to a research reactor. The research reactor simulator (RRS) simulates time behaviour of reactor power, fuel temperature and reactivity by using the 6-group point kinetics equation with feedback. It features temperature feedback mechanisms and xenon poisoning. Using graphics acceleration to render simulation results and a simple integration scheme with a short simulation step to propagate physics achieves low latency between a user’s input and display of simulation results, which is important for research reactors that can perform fast transients. The simulator targets TRIGA-type reactors, but all physical parameters can be changed on the fly. Thus, the simulator can be easily adapted to simulate other reactors.
... The space step (space grid) is equal to h = 0.001cm. Then, the ODE system was carried out by the 4th-order Runge-Kutta method [31], [35]- [38], which reads ...
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The super multi-armed and segmented (SMAS) spiral pattern has been observed in nature, such as sunflower inflorescence, spiral aloe, pine cone, ball cactus and Roman broccoli, which is characterized by several segmented spiral arms sharing the same spiral tip. The mechanism for the emergency of the SMAS spiral pattern has not been found. In this article, we observed the emergence of the SMAS spiral pattern by the simulations of a reaction-diffusion model. Additionally, our theoretical analysis found that the instability of concentrations in spiral arms leads to the emergence of this pattern. This study provides an alternative explanation for the formation of this type of pattern in nature and sheds light on the dynamics of pattern formation.
... Taylor series method has a major disadvantage ,it requires evaluation of partial derivatives of higher orders manually and this is not possible in any practical application, therefore we seek for improved methods which do not need evaluation for repeated differentiation of the differential equation, Although the famous paper by Runge, and subsequently developed by Heun and Kutta , still the explicit Runge-Kutta of the 4 th order method have been widely used and the most popular version is the classical 4 th order, the Runge paper is now recognized as the starting point for modern one-step methods with multivalued and multistage, construction of this method needs the derivation and solution of many nonlinear algebraic system of equations called the order conditions , thus the calculation process were performed using Mathematica program, this system of nonlinear algebraic equations has many solutions, therefore we must guess the integration step size using trial and error to estimate the truncation error ,this is the main drawback of Runge-Kutta methods. a simplified derivation of fourth order Runge-Kutta method given in [14], the derivation of fifth order method were introduce by Kutta [4] and corrected by Nystrom [6], and the sixth order with eight stages founded by Huta [3], this is a short brief history of the method also we find that it is impossible to present a general formula to the order conditions for all families of Runge-Kutta methods but there is a connection can be presented in this paper between the several sequences orders. Complications arise specially at the conditions for order five and more. ...
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The initial value problems (IVPs) in ordinary differential equations are numerically solved by one step explicit methods for different order , the behavior of Runge- Kutta of third order method is obtained by Heun [4], Kutta found the complete classification and derivation of fourth and fifth order methods in [6], the derivation of sixth order method was found by Huta in [5] but in eight stages, Butcher in ([1], [2], [3]) presented the relation between the order conditions and the rooted trees up to sixth order. the main aim of this paper is to exhibit a new more simplest representation for the trees and the derivation of Runge- Kutta method of order six with seven stages including ( rooted trees, order condition and stability region), symbolically computations are used in the study to simplify the method, finally example illustrate the method are presented
... Taylor series method has a major disadvantage ,it requires evaluation of partial derivatives of higher orders manually and this is not possible in any practical application, therefore we seek for improved methods which do not need evaluation for repeated differentiation of the differential equation, Although the famous paper by Runge, and subsequently developed by Heun and Kutta , still the explicit Runge-Kutta of the 4 th order method have been widely used and the most popular version is the classical 4 th order, the Runge paper is now recognized as the starting point for modern one-step methods with multivalued and multistage, construction of this method needs the derivation and solution of many nonlinear algebraic system of equations called the order conditions , thus the calculation process were performed using Mathematica program, this system of nonlinear algebraic equations has many solutions, therefore we must guess the integration step size using trial and error to estimate the truncation error ,this is the main drawback of Runge-Kutta methods. a simplified derivation of fourth order Runge-Kutta method given in [14], the derivation of fifth order method were introduce by Kutta [4] and corrected by Nystrom [6], and the sixth order with eight stages founded by Huta [3], this is a short brief history of the method also we find that it is impossible to present a general formula to the order conditions for all families of Runge-Kutta methods but there is a connection can be presented in this paper between the several sequences orders. Complications arise specially at the conditions for order five and more. ...
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Full-text available
The initial value problems (IVPs) in ordinary differential equations are numerically solved by one step explicit methods for different order , the behavior of Runge-Kutta of third order method is obtained by Heun [4], Kutta found the complete classification and derivation of fourth and fifth order methods in [6], the derivation of sixth order method was found by Huta in [5] but in eight stages, Butcher in ([1], [2], [3]) presented the relation between the order conditions and the rooted trees up to sixth order. the main aim of this paper is to exhibit a new more simplest representation for the trees and the derivation of Runge-Kutta method of order six with seven stages including (rooted trees, order condition and stability region), symbolically computations are used in the study to simplify the method, finally example illustrate the method are presented.
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Puncture robot can improve the accuracy and efficiency of puncture surgery, such as thoracoabdominal and liver puncture. However, as soft tissue is deformed and shifted under respiratory motion and during the puncture process, the needle is pulled, resulting in the needle’s bending and deformation, which increases the risks and sufferings of the patient, a robotic puncture system with optical and mechanical feedback is necessary. Therefore, this paper proposes a muti‐information sensing “guide‐clamp” end effector for puncture surgery to accurately detect the posture and force on the puncture needle in real‐time. And gravity bias method with trajectory planning and the compensational controling model are also proposed to offset the interference of self‐weight and achieve zero‐force following. This system is evaluted by the experiments of robot controling and human tissue simulation and the results prove the excellent robustness of the system, which meet the clinical requirement. This article is protected by copyright. All rights reserved.
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Families’ influence on romantic relationship is investigated from theoretical and practical perspectives. Theoretically, an ordinary differential equation based model is proposed to describe the romantic relationship between two partners, where the influence received by each individual from its family is taken into account. The introduction of the families’ opinions lead to rich and interesting structure in the dynamical process. With the decreasing response of a partner to its own family, two bifurcations occur, separating the dynamical behavior into three types, i.e., the transition from damping oscillation to one of four stable states, two of stable states, and limit cycles. These findings are explained with the stability analysis of equilibrium points. Practically, for each individual the opinion from its partner’s family is an interesting but hidden variable. The reservoir computing is adopted to discover the hidden variable from the activities of the individual, its family, and its partner. The model and the discovering method can be extended easily to investigate the relationship between two social groups such as the lateral negotiation, where the two representatives play game under the guidance from their own groups each.
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A high-order numerical algorithm is proposed for the solution of one-dimensional arterial pulse wave propagation problems based on use of an accelerated “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions. The solver provides high-order accuracy, mild Courant-Friedrichs-Lewy (CFL) constraints on the time discretization and, importantly, results that are essentially free of spatial dispersion errors—enabling fast and accurate resolution of clinically-relevant problems requiring simulation of many cardiac cycles or vascular segments. The left ventricle-arterial model that is employed presents a particularly challenging case of ordinary differential equation (ODE)-governed boundary conditions that include a hybrid ODE-Dirichlet model for the left ventricle and an ODE-based Windkessel model for truncated vasculature. Results from FC-based simulations are shown to capture the important physiological features of pressure and flow waveforms in the systemic circulation. The robustness of the proposed solver is demonstrated through a number of numerical examples that include performance studies and a physiologically-accurate case study of the coupled left ventricle-arterial system. The results of this paper imply that the FC-based methodology is straightforwardly applicable to other biological and physical phenomena that are governed by similar hyperbolic partial differential equations (PDEs) and ODE-based time-dependent boundaries.
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This paper presents a review of the role played by trees in the theory of Runge–Kutta methods. The use of trees is in contrast to early publications on numerical methods, in which a deceptively simpler approach was used. This earlier approach is not only non-rigorous, but also incorrect. It is now known, for example, that methods can have different orders when applied to a single equation and when applied to a system of equations; the earlier approach cannot show this. Trees have a central role in the theory of Runge–Kutta methods and they also have applications to more general methods, involving multiple values and multiple stages.
Chapter
This introduction expresses commonly understood ideas in the style that will be used for the rest of this book. The aim is to understand, as much as possible, what is expected to happen to a quantity that satisfies a differential equation. The chapter discusses differential equation problems, differential equation theory, evolutionary problems, difference equation theory and location of polynomial zeros. It emphasizes numerical methods, and often discusses problems with known solutions mainly to illustrate qualitative and numerical behaviour. Differential and difference equations belong together as a unified theory and as related areas of applicable mathematics. Furthermore, each is used to approximate the other. The link between the smooth and the discrete is not only the numerical approximation process but, in the reverse direction, it is an interpolation process, aimed at finding values of the smooth function from the discrete values. Many properties of differential equation solutions have discrete counterparts.
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Runge-Kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations. In contrast to the Adams methods, there is no natural way to approximate the solution between mesh points. A way to accomplish this is proposed which is applicable to some important formulas. Its theoretical support is much better than that of interpolation in the popular variable order, variable step Adams codes.
Chapter
Numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons. The first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of Bashforth and Adams for linear multistep methods and Runge for Runge-Kutta methods. Other reasons, which of course apply to numerical analysis in general, are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics. This survey paper follows many of the main strands in the developments of these methods, both for general problems, stiff systems, and for many of the special problem types that have been gaining in significance as the century draws to an end.
RK Methods prove popular at IMA Conference on Numerical ODE's,SIAM News
  • G Byrne
G. Byrne and Hindmarsh, (1990), RK Methods prove popular at IMA Conference on Numerical ODE's,SIAM News,23/2 pp.14-15.
  • Fudzia Ismail
Fudzia Ismail, (2010), Lecture Notes on Numerical Methods (unpublished), University Putra Malaysia