Singly Diagonally Implicit Runge-Kutta Method For The Solution Of The Linear And Non-Linear

409

Singly Diagonally Implicit Runge-Kutta Method

For The Solution Of The Linear And Non-Linear

Delayed-Action Oscillator Equation K.Ponnammal, R.Sayeelakshmi

Abstract: This paper presents continuous extension singly diagonally implicit Runge – kutta (CESDIRK) method based in continuous extension polynomial cubic spline polynomial interpolation, cubic hermite polynomial interpolation for solving Delay Differential Equation. The method modeled climate model which applied to linear and non-linear Delay Action Oscillator Delay Differential Equation. Errors of numerical results compared with the solution MATLAB solver DDE23. It is observed that CESDIRK polynomial given better results than other polynomial.

Index Terms: DDE, Runge-Kutta method, Natural Continuous Extension, Cubic interpolation, Hermite Interpolation.

————————————————————

1

INTRODUCTION

El Niño–Southern Oscillation (ENSO) is a major driver of climate variability in many parts of the world. ENSO is an irregularly regular variation in winds and ocean surface temperatures across the tropical eastern Pacific Ocean, affecting much of the tropics and subtropics environment. The sea temperature warming stage is known as El Niño and La Niña is known as the cooling phase. The Southern Oscillation is the atmospheric component occurred with the changes in sea temperature.[8]The linear Delayed Action Oscillator (DAO) has the form

= 𝐚 − 𝐛 (𝐭 − 𝛕) (1) where y is an index for ENSO variability (Sea Surface

Temperature) at the time t, a, b, and τ are all real parameters y(t-τ) is a delay term. The rate of change of y is determined by a balance between a positive feedback term ‗ay‘ and delayed feedback term by ‗(t-τ)‘. The ENSO pattern drives major changes in rainfall, river flow, agricultural production, disease and ecosystems. Therefore, it is important to investigate ENSO delay differential equation. Researchers explored the numerical solution of DAO equation.

The non-linear DAO has the form

= − − 𝛂 (𝐭 − 𝛕) (2) Literature Review

Max J. Suarez and Paul S. Schopf [8] modeled the El Niño phenomenon and solved a delay differential equation numerically using Maple's software. The construction of dense formulae for RK pairs has been explored by several authors including Horn (1983) [6], Shampine (1985, 1986) [10], Enright et al. (1986)[5], Bellen and Zennaro [1] stated the existence of NCEs for all RK processes and presented the NCEs for some of the most popularexplicit RK processes. Wright [14] compared relationship between Implicit Runge-Kutta and collocation method. Bellen [2] proved that some IRK methods are equivalent to collocation methods and NCE gives the approximate solution, without anyextra evaluations of the function.

2

CONTINUOUS

RK

METHOD

FOR

ODE

An instance of a continuous extension is a discrete variable method. However, even in the earliest computations there was a need for an approximation to y(t) between mesh points



t ,t,....t n,....t N tf



Δ  _{0 1}  . A continuous extension on [tn, tn+1] is

a polynomial Pn(t) that approximates y(t) accurately not just at end of the step where Pn(tn+1) = yn+1, but all the way through.Given a meshΔ



t_0,t₁,....t _n,....t _N  t_f



a v-stage R-K method for the numerical solution of the ODE has the form

 



 



 

0 f

0 0

y t g t , y t , t t t y t y

    

 



 ₍₃₎

For i=1,2,..v,

v

j j

i

y y_n h_{n 1} a_{i j}g t , y

n 1 n 1 n 1

j 1

 

    

 _   _





(4)

= + 𝐡 ∑𝐢 𝐛𝐢𝐠(𝐭 𝐢 , 𝐢 ) (5)

Where t = t + ch , i=1,……v, hn+1=tn+1-tn and v is referred to as the number of stages. The bi‘s are called weights of the quadrature formula and c′s ∈ [0,1]are called abscissa. The one - step interpolants of the RK method given in Eq. (4), and Eq.(5) are formed step by step by making use of information from the underlying mesh interval [tn,tn+1] only, possibly by including some additional stages, that is by some extra evaluations of the right-hand-side function g(t,y) in Eq. (3). Interpolants derived using no extra stages are called interpolants of the first class. The value obtained from continuous extension  t is defined in each sub interval of

the mesh, by a one-step continuous quadrature rule of the form

  v  



i i



n n 1 n n 1 i n 1 n 1

i 1

η t θ h _ y h _ b θ g t _ ,y _ 0 θ 1



  

_

 

  v   i i i

η t_n θ h y_n h_{n 1} b_i θ f t Y ,Y , 0 θ 1 n 1 , n 1 n 1

i 1

 

   _{     }  

 





( 6) (or) in the K-Notation

  v   i

n n 1 n n 1 i n 1

i 1

η t θ h _ y h _ b θ K _ 0 θ 1



  

_

 

( 7)

where the b_i

 

θ s are polynomials of suitable degree 

satisfying bi(0) = 0 and bi(1) = bi, i = 1,...v

(8)

so as to satisfy the continuity conditions η(tn)=yn and __________________________

 Department of Mathematics, Periyar E.V.R.College, Tiruchirappalli-6200231 1

 Department of Mathematics, Mookambigai College of Engineering, Keeranur, Pudukkottai-622502 2

410 tn 1  yn 1

 

(9)

3

CONTINUOUS

EXTENSION

OF

RK

METHODS

FOR

DDE

The first order delay differential equation has the form

 



 





 







 

 

          0 0 , , , , , t t t t y t t t t y t t y t y t f t y _f   (10)

The standard method for solving the DDE, consists in solving step by step the local problems

 



 





 







 

              n n n n n n n n y t t t t t t t x t t f t 1 1 1 1

1 , , , ,

     (11)

 

 

 

           

1 1

0 0 ) ( n n n n t s t for s t s t for s t s for s s x    (12)

and  s is the continuous approximate solution computed

up to tn.

The overall method for DDE is presented as











t Y ,ηt τt ,Y



,i 1,... ..v

f a h y

Y _nj _{1, n}j_{1 n1}j _nj _{1 n}j₁

v 1 j ij 1 n n i 1

n        

 





(13)

t  y h  f



t Y ,η



t τ



t ,Yni 1







,0 1

i 1 n i 1 n i 1 n i 1, n v 1 i 1 n n

n          

   



 h bi

(14)

The Eq. (13) and Eq. (14) are called the RK method for DDE. The coefficients (A, b) are the underlying discrete RK method, whereas (A,b(θ)) are the interpolants. The pair formed by the underlying discrete RK method and interpolants is called the underlying continuous RK method.

In the mesh interval [tn, tn+1], the Eq. (13) and Eq. (14) takes the form

t θh y h b  θ f t Y ,y ,0 θ 1

η i i_{n 1}

1 n i 1, n v 1 i i 1 n n

n _  

          

  (15)



t Y Y



i v

f a h y

Y nj

j n j n v j ij n n i

n 1, 1, 1 , 1,...

1 1

1      

 





(16)

where the spurious stages i

n

y 1are given by



θ

 

f t Y ,Y



,

b h y

Y nj1

j 1 n j 1, n i 1 n v 1 j 1 n n i 1

n    

 

  



j

(17)

where  

i i

n 1 n 1

c_{i t , Y} i

n 1

h n 1

 

     _ 

 _.

If ti_n1  τ



t_ni_1,Y_ni_1



, t_n and

by





ni 1





i 1 n i 1 n i 1

n η t τ t ,Y

Y _  _  _{ }

(18) Otherwise, the system Eq. (15), Eq. (16) and Eq. (18) has to

be solved only for the stage values, j n

y 1, j = 1…v, the system

enlarged by Eq. (17) for some it has to be solved also for the relevant spurious stage value i

n 1

y _ .

4

CONTINUOUS

EXTENSION

FOR

IMPLICIT

RUNGE

KUTTA

METHOD

The seminal idea of continuous-stage methods was introduced by Butcher (1972)[3] ―continuous‖ extension of Runge Kutta (RK) methods allowed the number of stages to be infinite so that the discrete index set {1, 2, ….. , v} becomes the interval [0, 1]. The continuous-stage approaches provided numerical

solution of differential equations, the Butcher coefficients (as functions) are assumed to be ―continuous‖ or ―smooth‖ which potentially allows analytical techniques such as Taylor expansion, inner product, limit operation, orthogonal expansion, differentiation, integration, etc [11,12,13].Zennaro [15] developed the NCE of first class interpolants of RK method. The interpolant  t in equation (5) of order p is a

NCE of the RK method (4) of degree q if the polynomials, bi(θ), i =1, 2...v are such that  t satisfies the additional

asymptotic orthogonality condition as

       p1

t t h O dt t η t y t G 1 n n        (19)

For every sufficient smooth matrix valued function G and with the statements η(t0)=y0, η(t0+h)=y1 hold. We construct interpolants of the first class for Singly Diagonally Implicit Runge Methods. They are based on values of the solution and its derivative from integration steps. The RK m method is

accurate of order p (1) satisfies  





1 0      p h O y h t y . The approximate solution find iteratively on a mesh



t

t

t

n

t

N



t

f







_{0, 1}

,....

,....

of the interval (t0,tf),such that

     

0 0h

q t t t

y t t O h

m a x



 

   

(20)

TABLE 1.ORDER CONDITION FOR CONTINUOUS RK METHOD.

Order Conditions

1. b  θ θ

v 1 i i 





2.   2

i v 1 i i θ 2 1 c θ b   

It is to be noticed that the collocation polynomial for any one-step collocation method is as NCE of degree q = v.

Theorem: 4.1

Every RK process Eq.(5) and Eq.(7) of order p has a NCE

 t

 of minimal degree, then

        2 1 p q . Theorem: 4.2

If the interpolant Eq. (6) of order (and degree) q is an NCE of

the RK method Eq.(5) and Eq. (6) of order p, then

        2 1 p q . Theorem: 4.3

Every Runge-Kutta method of Eq. (3) of order p1 has a

continuous extensionη t of order q=1 ...        2 1

p _{. The}

polynomial b_i

 

θ satisfies the condition _{ } i

b 0  0 a n d

  r

i c b d θ θ b

θ _{i i}

1

0

r _{ }



r = 0,...q . (21)

5

CONTINUOUS

EXTENSION

SINGLY

DIANGONAL

IMPLICITY

RUNGE

KUTTA

METHOD

411

TABLE 2.IRKCOEFFICIENT TABLEAU

c1 γ

c2 a21 γ

c3 a31 a32 γ

c4 a41 a42 a43 γ

b1 b2 b3 b4

From theorem 4.1 it follows that NCE of minimal degree

        2 1 p

q = 2 for p=4,

The second degree polynomial stated as

 θ ξθ ηθ

b i 2 i

i   where ηibiξi, i = 1,2,3,4 (22)

satisfies the order condition from Table 1 up to order 2, Then the equations used are given as,

 θ θ b v 1 i i 



 ₍₂₃₎

  2

i v 1 i i θ 2 1 c θ b 



 (24)

Putting i = 1,2,3,4 in equation (22) we get

  2  _{1 1}θ

1

1 θ ξ θ b ξ

b    ₍₂₅₎

 θ ξ θ b ξ θ

b₂  ₂ 2 ₂ ₂ (26)

 θ ξ θ b ξ θ

b 3 3

2 3

3    (27)

  2  

4 4 4 4

b θ ξ θ  b ξ θ ₍₂₈₎ From Eq. (23) and Eq. (24), we arrive at

       

1 2 3 4

b θ b θ b θ b θ 

        1 2

b θ c b θ c b θ c b θ c θ

1 1 2 2 3 3 4 4 2

   

To find b₄ θ :

  2    

4 4 2 2 3 3

1

b c b c b c 2

       (29)

Substituting Eq. (26) and Eq. (27) in Eq. (29) we get,

  2 2   2  

4 4 2 2 2 2 3 3 3 3

1

b θ c θ ξ θ b ξ θ c ξ θ b ξ θ c 2

   

 _   _ _   _

  θ ξ c θ b c θ ξ c θ ξ c θ b c θ ξ c θ

2 1 c θ

b_{4 4}  2 _{2 2} 2  _{2 2}  _{2 2}  _{3 3} 2 _{3 3}  _{3 3}

  θ 4 c 3 c 3 ξ θ 4 c 3 c 3 b 2 θ 4 c 3 c 3 ξ θ 4 c 2 c 2 ξ θ 4 c 2 c 2 b 2 θ 4 c 2 c 2 ξ 2 θ 4 2c 1 θ 4

b       

  θ c c ξ θ c c b θ c c ξ θ c c b θ c c ξ c c ξ 2c 1 θ b 4 3 3 4 3 3 4 2 2 4 2 2 2 4 3 3 4 2 2 4

4     

       

Putting 2

4

c 

  3

4

c 

  in the above equation

  θ c μc c θ c c b θ c λc c θ c c b θ c μc c c λc c 2c 1 θ b 4 3 4 4 3 3 4 2 4 4 2 2 2 4 3 4 4 2 4 4

4     

       

2 2 2 3 3

2 3 2 3

4 4 4

b c b c

1

λ c μ c θ θ λ c θ θ + μ c θ

2 c c c

 

_   _   

 

 

2

2 3 2 2 3 3 2 3

4 4

1 θ

λ c μ c θ b c b c λ c θ c θ

2 c c

 

_   _     

 

 

2

2 3 2 2 3 3 2 3

4 4

1 θ

λ c μ c θ b c b c λ c θ μ c θ

2 c c

 

_   _    

 

2

2 3 4 4 2 3

4 4

1 θ 1

λ c μ c θ b c λ c θ μ c θ

2 c c 2

   

_   _  _  _ 

 

 

2 4 4

2 3 2 3

4 4 4

b c θ

1 θ

λ c μ c θ λ c θ μ c θ

2 c 2 c c

 

_   _    

 

2

2 3 4 2 3

4 4

1 θ

λ c μ c θ b θ λ c θ μ c θ

2 c 2 c

 

_   _    

 

Hence,

  b θ λc θ μc θ

2c θ θ μc λc 2c 1 θ

b 4 2 3

4 2 3 2 4

4     

        

  2

4 2 3 4 2 3

4 4

1 1

b λ c μ c θ b θ θ λ c μ c

2 c 2 c

   

 _   _   _   _

    ₍₃₀₎ To find b₃

 

θ :

  2  2 4  4

2 3

3 θ b θ c b θ c

2 1 c θ

b   

(31)

Substituting Eq. (26) and Eq. (30) in Eq. (31) we get,

    b  θ

c c θ b c c θ 2c 1 θ b ₄ 3 4 2 3 2 2 3

3   

                                

 2 3

4 4 2 3 2 4 3 4 2 2 2 2 3 2 2 3

3 λc μc

2c 1 θ θ b θ μc λc 2c 1 c c θ ξ b θ ξ c c θ 2c 1 θ b

  1 2 c2 2





c4 1 2 1

b θ θ c λθ b c λ θ λc μ c θ b θ θ λc μ c

3 _{2 c c} 4 2 _{4 c 2 c} 2 3 4 _{2 c} 2 3

3 3 3 4 4

                 _{   }         _{   }   2

2 2 2 2

c c μ θ

c c λ θ c b θ c c λ θ c θ c c λ θ b c θ c θ

1 2 2 4 2 2 2 4 4 2 4 3 4 4 4 4

b θ θ

3

2 c c c c 2 c c c c c 2 c c

3 3 3 3 3 4 3 3 3 3 4

c c μ θ c c λ θ

3 4 2 4 c c 3 3           

2 2 2 4 4

4 4

3 3 3

c b θ b c θ θ

c μ θ c μ θ

c c 2 c

     



2 2 4 4



4 2 4

3 3

θ θ

c b b c c μ θ c μ

c 2 c

      

4 2 4 3 3

3 3

θ θ 1

c μ θ c μ θ b c

2 c c 2

 

    _  _

 

2 3 3

4 4

3 3 3

b c θ

θ θ

c μ θ c μ θ

2 c 2 c c

    

Hence,

  2

3 4 4 3

b   c    c    b  (32)

To find b₂  :

 

2

 

 

2 2 3 3 4 4

1

b c b c b c

2

       (33)

Substituting Eq. (32) and Eq. (30) in Eq. (31) we get

                               

 2 3

4 4 2 3 2 4 4 3 4 2 4 3 2 2

2 λc μc

412                                    

 2 3

4 4 2 3 2 4 2 4 3 4 2 4 2 3 2 2

2 λc μc

2c 1 θ θ b θ μc λc 2c 1 c c θ b μθ c μθ c c c θ 2c 1 θ b   2 4 3 2 4 2 2 4 4 2 4 4 2 2 4 3 2 2 4 2 2 4 2 4 2 3 3 2 3 4 2 2 4 3 2 2 2 c θc μc c θc λc c 2c θc c θc b c θ c μc c θ c λc c 2c θ c c θc b c μθc c c μθ c c θ 2c 1 θ

b           

2

3 3 4 4

4 4

2 2 2

b θ c b θ c θ

λ c θ λ θ c

c c 2 c

     





2

3 3 4 4 4 4

2 2

θ θ

b c b c λ c θ λ θ c

c 2 c

     

2

2 2 4 4

2 2

θ 1 θ

b c λ c θ λ θ c

c 2 2 c

    _  _     2 2 2 4 4

2 2 2

b c θ

θ θ

λ c θ λ θ c

2 c c 2 c

     

Therefore,

  2

2 2 4 4

b   b   c    c (34)

To find b₁  :

       

1 2 3 4

b     b   b  b  (35)

Substituting b2

 

 , b3

 

 & b4

 

 in Eq. (35) we get

 



 



                                    

 2 3

4 4 2 3 2 4 3 4 2 4 4 2 4 2

1 λc μc

2c 1 θ θ b θ μc λc 2c 1 θ b μθ c μθ c λθc θ λc θ b θ θ b

  λc θ μcθ

2c θ θ b θ μc θ λc 2c θ θ b θ μc μθ c θ λc θ b θ λc θ θ

b 2 3

4 4 2 3 2 2 4 2 3 4 2 4 4 2 2 4

1              

    _                         

 _{2 3}

4 4 4 3 2 4 4 4 2 4 3 2

1 λc μc

2c 1 μc λc θ μc λc 2c 1 μ c λc θ b b b 1 θ θ b          _                       

 _{4 2 4 3}

4 3 4 2 4 4 2 1

1 λc c μ c c

2c 1 θ c c μ c c λ 2c 1 θ θ b θ b

For the finding the NCEs of order q=2, put r = 1 in

  i i

1

0

i θ d θ b c

b

θ  



 

i i i

ξ  3 b 2 c 1

We know that  _ 2_



_



i

i i

b   b  

  2   

i i i i i

3 b 2 c 1 θ b 3 b 2 c 1 θ

    

  2  

i i i i i i

3 b 2 c 1 θ b 6 b c 3 b θ

    

 θ 3b 2c 1θ 4b 6b c θ

b i i i

2 i i

i    

and therefore,

    2  

3 2 1 2 2 3

i i i i i

b   b c    b  c 

(36)

where  

3 4 2 b c 1 2c 3 λ   ,   3 4 3 b c 1 2c 3 μ  

Table 3. gives the Coefficients of Implicit RK fourth order method. [9]

TABLE 3.IMPLICIT RKFOURTH ORDER METHOD COEFFICIENTS

0.20 0.20

0.05 -0.15 0.20

0.40 -0.7819 0.98519 0.20

0.80 0.70672 -0.19819 0.09147 0.20 0.40741 0.01693 0.10174 0.46851

From Table 3, the continuous Extension singly diagonal implicit Runge -kutta polynomial as

b1(θ)= -0.73θ2+1.14θ b2(θ)= -0.053θ2+0.06θ b3 (θ)= -0.06θ2+0.17θ b4 (θ)= 0.84θ2-0.37θ

CUBIC SPLINE INTERPOLATION METHOD

Given a set [(xi, yi ) i = 0,1….n] of n+1 point value pairs, where x0 < x1<...< xn, we wish to fit a piecewise–cubic spline f(x) to the point .That is, the curve f(x) is made up a cubic polynomial fi(x)=ai+bix+cix

2 +dix

3

for i=0,1….n-1, where if x falls in the range x ≤ x ≤ x then the value of called knot. The

continuity of f(x) ensures the condition that satisfy f(xi) = fi(0)=y0 for i=0,1,2,…n-1, f(xi+1)=fi(1)=yi+1 for i =0,1,2….n-1. The first derivative at each knot is continuous f′(x ) = f′(1) = f ′ (0) , Further, let k and n, k<n, be non- negative integers.

The resulting piecewise cubic polynomial S(x) will interpolate f(x) at x0,x1,…..xn based on the condition on each sub-interval [xi, xi+1].

1. S(x) coincides with an algebraic polynomial of degree at most n.

2. S(x) and its derivatives up to order k continuously differentiable on [a, b].

The second order derivative of S(x) exists but may not be continuous at the knots. On the sub-interval [xi, xi+1 ] the cubic spine interpolation represents the polynomial as S(x) = a(x-xi)

3

+b(x-xi) 2

+c(x-xi)+d, where a,b,c,d are its coefficients.

CUBIC HERMITE INTERPOLATION METHOD

The Hermite interpolation polynomial interpolates not only the function f(x) but also its derivatives at a given set of tabular point. Piecewise cubic Hermite interpolating polynomial H(x) have properties that met these request [7] . Given an interval [a,b], a function f :[a, b]  R derivative f  :[a, b]  R and a set of a partition point x⃗ = (x , x , x … . x ) with a =x0x1….

xnb , cubic Hermite spline is defined by a set of polynomial h0,h1,…….hn-1with hixi =f xi,

hixi+1 =f xi+1 , h′(x) = f′(x) h′(x ) = f′(x )and for i = 0,1,…n-1. The spline formed by this collection of polynomials can be defined as H(x) = ∑ f(x)h(x) + f′h′(x)The hermite

form has two control points and two control tangents for each polynomial.They are simple to calculate, in term of time required to determine the interpolant and to evaluate it, but at the same time they are two powerful. The slopes at xi are chosen in such a way that H(x) preserves the shape of the data and respects monotonicity [4].

6

RESULTS

AND

DISCUSSION

Numerical solutions to Equations (1) and (2) are computed using the CESDIRK method. The delay argumentis approximated using CESDIRK polynomial, Cubic Spline Interpolation polynomial(CSI) and Piecewise Cubic Hermite Interpolation. The temperature anomalies found over the range 0≤t≤1 using step size h=0.1 , a=1, b =0.75,

413

TABLE 4:NUMERICAL SOLUTION OF LINEAR DAO EQUATION

T DDE23 NCE CSI CHI

0.0 0.55000000 0.55000000 0.55000000 0.55000000 0.1 0.56446042 0.56446453 0.56603787 0.56603787 0.2 0.58044159 0.58045732 0.58464476 0.58422178 0.3 0.59810344 0.59813872 0.60575143 0.60465858 0.4 0.61762274 0.61768548 0.62931968 0.62747982 0.5 0.63919481 0.63929215 0.65557747 0.65293952 0.6 0.66303555 0.66317239 0.68483711 0.68134480 0.7 0.68938354 0.68956015 0.71745103 0.71304013 0.8 0.71850246 0.71871059 0.75381301 0.74841026 0.9 0.75068372 0.75090058 0.79436276 0.78788481 1.0 0.78625062 0.78642854 0.83959119 0.83194348

FIGURE1: ERROR GRAPH FOR LINEAR DAO EQUATION

TABLE 5:NUMERICAL SOLUTION OF NON-LINEAR DAO EQUATION

T DDE23 NCE CSI CHI

0.0 0.55000000 0.55000000 0.55000000 0.55000000 0.1 0.54709864 0.54709857 0.54707147 0.54707147 0.2 0.54416752 0.54416724 0.54399871 0.54406386 0.3 0.54120349 0.54120283 0.54083241 0.54099600 0.4 0.53820328 0.53820207 0.53762300 0.53788692 0.5 0.53516350 0.53516155 0.53437676 0.53473718 0.6 0.53208061 0.53207772 0.53108990 0.53154298 0.7 0.52895092 0.52894687 0.52775779 0.52830004 0.8 0.52577053 0.52576511 0.52437552 0.52500384 0.9 0.52253539 0.52252834 0.52093798 0.52164961 1.0 0.51924119 0.51923225 0.51743975 0.51823234

FIGURE2: ERROR GRAPH FOR NON-LINEAR DAO EQUATION

7

CONCLUSION

The theory of CESDIRK methods are examined. The continuous extension interpolants polynomial of first class derived for CESDIRK methods. The delay term is computed using CESDIRK, Cubic and Hermite polynomials. Particularly, we intensively discuss its applications ENSO DAO first order linear and non-linear delay differential equation. From the results obtained via the numerical experiment, we verified that the method is appropriate for first order DDE problems.

REFERENCES

[1] A. Bellen and M. Zennaro. ―Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method‖, Numerical. Mathematics, pp. 301-316, 1985.

[2] A. Bellen, ―One-step collocation for delay differential equations‖, J. computation. Applied Mathematics.Vol. 10, pp. 275-283, 1984.

[3] J.C Butcher, ‖An algebraic theory of integration methods‖, Mathematics of Computation., Vol. 26, pp. 79-106, 1972. [4] R.E Calson and F.N Fritsch, ―Monotone piecewise Cubic

interpolation‖, SIAM Journal on Numerical Analysis, Vol. 17, pp. 238-246. 1980.

[5] W.H Enright, K.R Jackson, S.P Nørsett and P.G,Thomson ,― Interpolants for Runge-Kutta formulas‖, ACM Trans Math Soft , Vol. 12, pp. 193–218, 1986.

[6] M.K Horn, ―Fourth and fifth-order scaled Runge-Kutta algorithms for treating dense output ―, SIAM Journal Numerical Analysis, Vol. 20, pp. 558–568, 1983.

[7] S.R.K Iyengar., M..KJain and R.K. Jain,‖ Numerical Methods for Scientific and Engineering Computation‖, New AGE International(P) Limited Publishers, New Delhi, Third Edition, 1992.

[8] Max J. Suarez and Paul S. Schopf, ―A delayed action oscillator for ENSO‖, Journal Of Atmospheric Sciences, Vol. 45, Issue No: 20, pp. 3283-3287, 1988.

[9] Nur Izzati Che Jawias, Fudziah Ismail , Mohamed Suleiman and Azmi Jaafar, ―Fourth Order Four-Stage Diagonally Implicit Runge-KuttaMethod for Linear Ordinary Differential Equations‖ , Malaysian Journal of Mathematical Sciences, Vol. 4(1), pp. 95-105, 2010.

414

[11]W. Tang, G. Lang and X. Luo, ― Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage‖, Applied Mathematics and Computation, Vol. 286, pp. 279–287, 2016.

[12]W.Tang and J. Zhang, ―Symplecticity-preserving continuous-stage Runge-Kutta-Nystr¨om methods‖, Applied Mathematics and Computation, Vol. 323, pp. 204–219, 2018.

[13]W .Tang and Y. Sun, ―Construction of Runge-Kutta type methods for solving ordinary differential equations‖, Applied Mathematics and Computation, Vol. 234 pp.179–191, 2014. [14]K.Wright, ―Some relationships between implicit Runge-Kutta

collocation and Lanczos T-mehod and their stability properties‖, Bit, Vol. 10, pp.217-227, 1970.