Explicit Runge-Kutta Method for Evaluating Ordinary Differential

Equations of type vvi =f(u, v, v′)

MANPREET KAUR1, JASDEV BHATTI1,

SANGEET KUMAR2, SRINIVASARAO THOTA3∗

1Chitkara University Institute of Engineering and Technology

Chitkara University, Punjab, I1',$

2SGTB Khalsa College, Sri Anandpur Sahib, Punjab, I1',$

3Department of Mathematics, Amrita School of Physical Sciences

Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh–522503, I1',$

∗Corresponding Author

Abstract: - The initiative of this paper is to present the Runge Kutta Type technique for the development of

mathematical solutions to the problems concerning to ordinary differential equation of order six of structure vvi =

f(u, v, v′)denoted as RKSD with initial conditions. The three and four stage Runge-Kutta methods with order

conditions up to order seven (RKSD7) have been designed to evaluate global and local truncated errors for the

ordinary differential equation of order six. The framework and evaluation of equations with their results are

well established to obtain the effectiveness of RK method towards implicit function satisfying the required initial

conditions and for obtaining zero-stability of RKSD7 in terms of their accuracy with maximum precision under

minimal processing.

Key-Words: - Ordinary Differential Equations (ODE), Explicit Runge-Kutta Type methods, Local and Global

Truncation Error, Zero Stability.

Received: April 5, 2023. Revised: December 4, 2023. Accepted: February 5, 2024. Published: March 22, 2024.

1 Introduction

In the field of science and engineering there are many

real-life situations that can be modelized mathemat-

ically, [1], into the linear and higher order differen-

tial equations, [2]. In past, researchers analyzed many

numerical problems involving linear or ordinary dif-

ferential equations up to fourth order with different

initial conditions using Runge Kutta method, Laplace

transformations or many others. In applied sciences

and engineering, we observe large number of physical

problems involving initial value problems concerned

with higher order ordinary differential equations till

fourth order. For instance, free vibration analysis of

ring structures has been studied by [3]. Moreover

some researchers have developed methods likewise

schemes of Coupled compact for accuracy of sixth or-

der in space was developed to obtain numerical solu-

tions, [4], Langrages Polynomial with fictional points,

[5]. The study, [6], has used neural network for solv-

ing differential equations, while [7], contributed by

provided solution with B-series and coloring method-

ology in evolution of differential equations and fluid

dynamics which are non-linear in nature and fails to

discuss about the most important point, i.e., stabil-

ity of the solution for enhancing the efficiency of any

numerical method. Reducing higher order equations

to lower one for solving them is also an approach

used by many researchers, [8], [9], [10], [11]. Sit-

uations concerned to oscillatory problem. The au-

thors in [12], have solved such problems using ap-

proach of finite differences. Also, efficiency of the

methods designed by them, like, [13], solved many

applied physics problems using multi step methods.

Subsequently, two derivative RK method was derived

by [14], [15], for solving special first order DE, [15].

In 2021, [16], [17], contributed in analysis of er-

ror for differential equations of non-linear and lin-

ear type.The authors in [18], solved oscillating sys-

tems by optimizing sixth-order RKN method which

is explicit in nature.The authors in [19], solved fifth

order ODE using generalized Runge- Kutta integra-

tors. The authors in [20], had applied RKN methods

which are implicit in nature. Therefore, there are lots

of studies and analysis been done for providing the so-

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lutions to ordinary differential equation up to higher

order. There are several symbolic algorithms for reg-

ular initial and boundary value problems for differen-

tial equations as well as differential-algebraic equa-

tions, see for example, [21], [22], [23], [24]. But, in

case of ordinary differential equation of sixth order

the initiative had been done but not been proved ben-

eficial or in other word the solution had not been ini-

tiated towards minimizing error in shorter time, num-

ber of operation and using less memory space. Thus,

the current paper focus on evaluating the solutions

for sixth order ODE using a single-step method by

Runge-Kutta method denoted by RKSD. Moreover, it

proves accuracy and stability of the discussed method

in minimizing the error with its derivations.

2 Runge-Kutta Type Sixth-order

ODE

The initial value problem of sixth order ODE exam-

ined in the present paper are:

vvi(u) = f(u, v, v′)(1)

with initial conditions as

v(u0) = α0, v′(u0) = α′

0, v′′(u0) = α′′

v′′′(u0) = α′′′

0, viv(u0) = αiv

0,(2)

vv(u0) = αv

v(n+1) =vn+hv′

n+h2

2v′′

n+h3

3! v′′′

n+h4

4! viv

+h5

5! vv

n+h6

i=1

biki(3)

v′

(n+1) =v′

n+hv′′

n+h2

2v′′′

n+h3

3! viv

+h4

4! vv

n+h5

i=1

b′

iki(4)

v′′

(n+1) =v′′

n+hv′′′

n+h2

2viv

n+h3

3! vv

n+h4

i=1

b′′

iki(5)

v′′′

(n+1) =v′′′

n+hviv

n+h2

2vv

n+h3

i=1

b′′′

iki(6)

viv

(n+1) =viv

n+hvv

n+h2

i=1

biv

iki(7)

(n+1) =vv

n+h

i=1

iki(8)

where

k1=f(un, vn, v′

n),(9)

ki=f(un+cih, vn+hciv′

n+(h2c2

2v′′

+(h3c3

3! v′′′

n+(h4c4

4! viv

n+(h5c5

5! vv

+h6

j=1

aij ki, v′

n+hciv′′

n+(h2c2

2v′′′

+(h3c3

3! viv

n+(h4c4

4! vv

n+h5

j=1

¯aij kj);

for i = 1,2,3, ...s. (10)

For numerical and algebraic calculations requiring

computation efforts, Mathematica software is used to

evaluate values of weights, nodes and coefficients and

arranged them in Butcher tableau (Table 1) form:

Table 1: The Butcher tableau RKSD Method

cA¯

bTb′Tb′′Tb′′′TbivT bvT

The principal motive in the construction of RKSD

explicit method is for finding the least value of trun-

cation local errors, [25], [26], [27], [28]. The method

computes the value to the vp

n+1 where p is the deriva-

tive i.e. p= 0,1,2, . . . , vn+1, parameters for ob-

taining the approximate value to v(un+1),v′(un+1),

v′′ (un+1),v′′′ (un+1),viv (un+1),vv(un+1)where

vn+1 is the calculated solution and v(un+1)is taken

as the analytic solution. Equation (3)-(8) be presented

vn+1 =vn+hψ, v′

n+1 =v′

n+hψ′,

v′′

n+1 =v′′

n+hψ′′ , v′′′

n+1 =v′′′

n+hψ′′′ ,

viv

n+1 =viv

n+hψiv, vv

n+1 =vv

n+hψv.

where

ψ(un, vn, v′

n) =v′

n+h

2v′′

n+h2

3! v′′′

n+h3

4! viv

+h4

5! vv

n+h5

i=1

biki(11)

ψ′(un, vn, v′

n) = v′′

n+h

2v′′′

n+h2

3! viv

+h3

4! vv

n+h4

i=1

b′

iki(12)

(13)

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ψ′′ (un, vn, v′

n) = v′′′

n+h

2viv

n+h2

3! vv

n+h3

i=1

b′′

iki

(14)

ψ′′′ (un, vn, v′

n) = viv

n+h

2vv

n+h2

i=1

b′′′

iki(15)

ψiv(un, vn, v′

n) = vv

n+h

i=1

biv

iki(16)

ψv(un, vn, v′

n) =

i=1

iki(17)

Elementary differentials of the scalar equations are

as follows:

F(6)

1=v(vi)=f(u, v, v′

n),(18)

F(7)

1== fu+fvv′+fv′vuu,(19)

F(8)

1=fuu +vufuv +fuv′vuu +v2

ufvv

+fvv′vuvuu +fvvuu

+fv′v′v2

uu +fv′vuuu.(20)

The local truncation error is obtained by having

τp

n+1 =hψp(un, vn, v′

n)−∆p(un, vn, v′

n),

where p= (0), ..., (v).(21)

Using (18)-(20), Taylor series functions of vp(u)

can be represented as:

∆ =v′

n+1

2hv′′

n+1

3!h2v′′′

n+1

4!h3viv

n+1

5!h4vv

6!h5F(6)

1+O(h6)(22)

∆′=v′′

n+1

2hv′′′

n+1

3!h2viv

n+1

4!h3vv

n+1

5!h4F(6)

6!h5F(7)

1+O(h6)(23)

∆′′ =v′′′

n+1

2hviv

n+1

3!h2vv

n+1

4!h3F(6)

1+1

5!h4F(7)

6!h5F(8)

1+O(h6)(24)

∆′′′ =viv

n+1

2hvv

n+1

3!h2F(6)

1+1

4!h3F(7)

5!h4F(8)

1+1

6!h5F(9)

1+O(h6)(25)

∆iv =vv

n+1

2!hF (6)

1+1

3!h2F(7)

1+1

4!h3F(8)

5!h4F(9)

1+1

6!h5F(10)

1+O(h6)(26)

∆v=F(6)

1+1

2!hF (7)

1+1

3!h2F(8)

1+1

4!h3F(9)

5!h4F(10)

1+1

6!h5F(10)

1+O(h6)(27)

Further on, substituting the equations (18)-(20)

into equations (11)-(17), we get

i=1

biki=

i=1

biF(6)

i=1

bicihF (7)

1+1

i=1

bic2

ih2F(8)

i=1

bic3

ih3F(9)

1+ +O(h6).

Similarly,

i=1

iki=

i=1

iF(6)

i=1

icihF (7)

1+1

i=1

ic2

ih2F(8)

i=1

ic3

ih3F(9)

1+ +O(h6)(28)

where pis the derivative i.e. p= 0,1,2, .....v.

Using equations (11)-(17) and equations (22)-(27),

the local truncation errors equation (21) will be rep-

resented as

τn+1 =h6[∑biki−(1

6! F(6)

1+1

7! hF (7)

1+1

8! h2F(8)

1+1

9! h3F(9)

1+...)]

(29)

τ′

n+1 =h5[∑b′

iki−(1

5! F(6)

1+1

6! hF (7)

1+1

7! h2F(8)

1+1

8! h3F(9)

1+...)]

(30)

τ′′

n+1 =h4[∑b′′

iki−(1

4! F(6)

1+1

5! hF (7)

1+1

6! h2F(8)

1+1

7! h3F(9)

1+...)]

(31)

τ′′′

n+1 =h3[∑b′′′

iki−(1

3! F(6)

1+1

4! hF (7)

1+1

5! h2F(8)

1+1

6! h3F(9)

1+...)]

(32)

τiv

n+1 =h2[∑biv

iki−(1

2! F(6)

1+1

3! hF (7)

1+1

4! h2F(8)

1+1

5! h3F(9)

1+...)]

(33)

τv

n+1 =h[∑bv

iki−(F(6)

1+1

2! hF (7)

1+1

3! h2F(8)

1+1

4! h3F(9)

1+...)]

(34)

3 RKSD7 Method Order Conditions

The order conditions of RKSD7 are:

The order terms of v:

Sixth order :Xbi=1

720,(35)

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Seventh order :Xbici=1

5040.(36)

The order terms of v′:

F ifth order :Xb′

i=1

120,(37)

Sixth order :Xb′

ici=1

720,(38)

7th order :Xb′

ic2

i=1

2520,Xb′

i¯aij =1

5040.

(39)

The order terms of v′′ :

F ourth order :Xb′′

i=1

24,(40)

F ifth order :Xb′′

ici=1

120,(41)

Sixth order :Xb′′

ic2

i=1

360,Xb′′

i¯aij =1

720.

(42)

7th order :Pb′′

ic3

i=1

840 ,Pb′′

i¯aij cj=1

5040 ,

Pb′′

iaij =1

5040 ,Pb′′

ici¯aij =1

1680 .(43)

The order terms of v′′′ :

T hird order :Xb′′′

i=1

6,(44)

F ourth order :Xb′′′

ici=1

24,(45)

F ifth order :Xb′′′

ic2

i=1

60,(46)

Sixth order :Xb′′′

ic3

i=1

120,Xb′′′

i¯aij =1

720.

(47)

7th order :Pb′′′

ic4

i=1

210 ,Pb′′′

i¯aij cj=1

5040 ,

Pb′′′

iaij =1

5040 ,Pb′′′

ici¯aij =1

1260 .(48)

The order terms of viv :

Second order :Xbiv

i=1

2,(49)

T hird order :Xbiv

ici=1

6,(50)

F ourth order :Xbiv

ic2

i=1

12,(51)

F ifth order :Xbiv

ic3

i=1

20,(52)

Sixth order :Xbiv

ic4

i=1

30,Xbiv

i¯aij =1

720.

(53)

7th order :Pbiv

ic5

i=1

42 ,Pbiv

i¯aij cj=1

5040 ,

Pbiv

iaij =1

5040 ,Pbiv

ici¯aij =1

1008 .(54)

The order terms of vv:

F irst order :Xbv

i= 1,(55)

Second order :Xbv

ici=1

2,(56)

T hird order :Xbv

ic2

i=1

3,(57)

F ourth order :Xbv

ic3

i=1

4,(58)

F ifth order :Xbv

ic4

i=1

5,Xbv

i¯aij =1

120.

(59)

6th order :Pbv

ic5

i=1

Pbiv

i¯aij =1

720 ,

Pbv

i¯aij cicj=1

144 .(60)

7th order :Pbv

ic6

i=1

7,Pbv

i¯aij cj=1

5040 ,

Pbv

iaij =1

5040 ,Pbv

ici¯aij =1

840 ,

Pbv

iaij cicj=1

840 .(61)

4 Zero-Stability of RKSD7 Method

The most important precondition for obtaining the

convergence of numerical problem is evaluating zero-

stability of the system, as explained by [1]. The

methodology used in current research paper be writ-

ten in an array representation as:







100000

010000

001000

000100

000010

000001













vn+1

hv′

n+1

h2v′′

n+1

h3v′′′

n+1

h4viv

n+1

h5vv

n+1







=





1 1 1

120

0 1 1 1

0 0 1 1 1

0 0 0 1 1 1

0 0 0 0 1 1

0 0 0 0 0 1













hv′

h2v′′

h3v′′′

h4viv

h5vv







The characteristic equation represented by ρ(ξ)

can be presented as:

ρ(ξ) = |I.ξ −A|

Hence, ρ(ξ) = (ξ−1)6we we get the roots to be

ξ= 1,1,1,1,1,1, which is the zero-stability of the

given proposed method.

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5 Construction of RKSD Methods

5.1 A Third Stage Seventh Order RKSD

The motive of current section is the derivation of

a third stage with 7th order RKSD method, where

we use the conditions of Equations (35)-(61) respec-

tively, as simultaneous equations for calculating the

values of ci,bp

ifor i= 1,2,3as follows:

c2=1

5(1−5c3

1−4c3

), b1=28c2c3−4(c2+c3)+1

20160c2c3

b2=4c3−1

20160c2(c3−c2), b3=1−4c2

20160c3(c3−c2),

b′

1=378c2c3−63(c2+c3) + 18

45360c2c3

b′

2=63c3−18

45360c2(c3−c2),

b′

3=18 −63c2

45360c3(c3−c2),

b′′

1=15c2c3−3(c2+c3)+1

360c2c3

b′′

3=1−3c2

360c3(c3−c2),

b′′′

1=20c2c3−5(c2+c3)+2

120c2c3

b′′′

2=5c3−2

120c2(c3−c2),

b′′′

3=2−5c2

120c3(c3−c2),

biv

1=6c2c3−2(c2+c3)+1

12c2c3

biv

2=2c3−1

12c2(c3−c2),

biv

3=1−2c2

12c3(c3−c2),

1=6c2c3−3(c2+c3)+2

6c2c3

biv

2=3c3−2

6c2(c3−c2),

3=2−3c2

6c3(c3−c2),

The errors norms of v(u),v′(u),v′′ (u),v′′′ (u),

viv(u),vv(u)are as

For evaluating the minimal value to the error

norms of 7th order Equations (5.1)-(5.6) we find the

value of parameters ci,bi,b′

i,b′′

i,b′′′

i,biv

i,bv

ifor

i= 1,2,3and arranged in mnemonic device known

as Butcher tableau. Hence, the result values of er-

ror norms are τ(7)2= 0,τ′(7)2=−3.70074 ∗

10−19,τ′′(7)2=−1.5873 ∗10−4,τ′′′(7)2=

−4.42177 ∗10−4,τiv(7)2=−4.198251 ∗10−3

and τv(7)2=−4.2635 ∗10−2.

The global error of three stage seventh order is cal-

culate as follows:

τ(7) 2=1

5040s(2 −15c3−5040(b1+b3−5b1c3−10b3c3+ 20b3c2

3))2

(5 −20c3)2,(62)

τ′(7) 2=1

5040s(−3+5c3−5040b′

3c3(1 −10c3+ 20c2

3))2

(5 −20c3)2,(63)

τ′′(7) 2=1

2520s(−8 + 25c3−2520b′′

3c2

3(1 −10c3+ 20c2

3))2

(5 −20c3)2,(64)

τ′′′(7) 2=1

840s(−13 + 45c3−840b′′′

3c3

3(1 −10c3+ 20c2

3))2

(5 −20c3)2,(65)

τiv(7) 2=1

210s(−18 + 65c3−210biv

3c4

3(1 −10c3+ 20c2

3))2

(5 −20c3)2,(66)

τv(7) 2=1

42s(−23 + 85c3−42bv

3c5

3(1 −10c3+ 20c2

3))2

(5 −20c3)2,(67)

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τ(7)

g2= 4.284379 ∗10−2(68)

Figure 1: Butcher Table of 3 Stage 7th Order RKSD

Method

5.2 Construction of 4-stage Seventh Order

RKSD Methods

Similar to section 5.1, the current section is designed

for the derivation of 4-stage 7th order RKSD method,

where we have used conditions of Equations (35)-(61)

respectively as simultaneous equations for calculating

the values of ci,bi,b′

i,b′′

i,b′′′

i,biv

i,bv

ifor i= 1,2,3,4

as follows

c2=5−24c3

24 −90c3

, b2=−12c3c4−3(c3+c4) + 1

60480c2(c2−c3)(c4−c2),

b3=−12c2c4−3(c2+c4)+1

60480c3(c2−c3)(c3−c4),

b4=−12c2c3−3(c2+c3)+1

60480c4(c3−c4)(c4−c2),

b′

2=−28c3c4−8(c3+c4)+3

20160c2(c2−c3)(c4−c2),

b′

3=−28c2c4−8(c2+c4)+3

20160c3(c2−c3)(c3−c4),

b′

4=−28c2c3−8(c2+c3)+3

20160c4(c3−c4)(c4−c2),

b′′

2=−21c3c4−7(c3+c4)+3

2520c2(c2−c3)(c4−c2),

b′′

3=−21c2c4−7(c2+c4)+3

2520c3(c2−c3)(c3−c4),

b′′

4=−21c2c3−7(c2+c3)+3

2520c4(c3−c4)(c4−c2),

b′′′

2=−5c3c4−2(c3+c4)+1

120c2(c2−c3)(c4−c2),

b′′′

3=−5c2c4−2(c2+c4)+1

120c3(c2−c3)(c3−c4)

b′′′

4=−5c2c3−2(c2+c3)+1

120c4(c3−c4)(c4−c2),

biv

2=−10c3c4−5(c3+c4)+3

60c2(c2−c3)(c4−c2),

biv

3=−10c2c4−5(c2+c4)+3

60c3(c2−c3)(c3−c4),

biv

4=−10c2c3−5(c2+c3)+3

60c4(c3−c4)(c4−c2),

2=−6c3c4−4(c3+c4)+3

12c2(c2−c3)(c4−c2),

3=−6c2c4−4(c2+c4)+3

12c3(c2−c3)(c3−c4),

4=−6c2c3−4(c2+c3)+3

12c4(c3−c4)(c4−c2),

c2=1

6(1−6(c2+ 6c3) + 30c2c3

−5(c2+ 6c3) + 20c2c3

The errors norms of vp(u)with p= 0, i, ...., v as an

derivative.

τ(7) 2=1

5040s(11 −73c3−5040(55b1+ 55b3+ 31b4−24b1c3−48b3c3+ 66b4c3+ 90b3c2

3))2

(24 −90c3)2,(69)

τ′(7) 2=1

5040s(−13 + 12c3−5040[b′

3c3(5 −48c3+ 90c2

3)−b′

4(19 −66c3)])2

(24 −90c3)2,(70)

τ′′(7) 2=1

2520s(−37 + 102c3−2520[b′′

3c2

3(5 −48c3+ 90c2

3)−b′′

4(19 −66c3)])2

(24 −90c3)2,(71)

τ′′′(7) 2=1

840s(−61 + 192c3−840[b′′′

3c3

3(5 −48c3+ 90c2

3)−b′′′

4(19 −66c3)])2

(24 −90c3)2,(72)

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τiv(7) 2=1

210s(−85 + 282c3−210[biv

3c4

3(5 −48c3+ 90c2

3)−biv

4(19 −66c3)])2

(24 −90c3)2,(73)

τv(7) 2=1

42s(−109 + 372c3−42[bv

3c5

3(5 −48c3+ 90c2

3)−bv

4(19 −66c3)])2

(24 −90c3)2,(74)

For the least value of error norms of 7th or-

der Equations above we find value of the parame-

ters ci,bi,b′

i,b′′

i,b′′′

i,biv

i,bv

ifor i= 1,2,3,4as

shown in the Butcher Figure 1. Hence the result

values of error norms are τ(7)2= 3.250539 ∗

10−2,τ′(7)2=−2.97306 ∗10−19,τ′′(7)2=

−1.33788 ∗10−18,τ′′′(7)2= 2.65617 ∗10−4,

τiv(7)2=−2.23367 ∗10−3and τv(7)2=

−1.166532 ∗10−2.

The global error of four stage seventh order is cal-

culated in Figure 2 as:

τ(7)

g2= 3.460838 ∗10−2(75)

Figure 2: Butcher Table of Four Stage Seventh Order

RKSD Method

6 Numerical Example

The results of the given methods discussed in the Sec-

tion 5.1 and Section 5.2 are tested with the help of

example of sixth order. The result were also tested

shown in Figure 3 to compare it with existing implicit

RK methods of the same order and with the direct

method of solving the sixth order differential equa-

tion with constant coefficient.

Example 1. Consider the homogeneous linear

equation given as:

vvi(u) + v′(u) = 0

with initial conditions as

v(u0) =0, v′(u0) = 1, v′′(u0) = 1, v′′′ (u0) = 0,

viv(u0) =1, vv(u0) = 2.

The exact solution is

v(u) =c1+c2e−u+e(1+√5

4)u[c3cos(p10 −2√5

4)u

+c4sin(p10 −2√5

4)u] + e(1−√5

4)u

[c5cos(p10 + 2√5

4)u+c6sin(p10 + 2√5

4)u]

where

c1=5242880 + 5242880√5

16(163840 + 163840√5),

c2=1048576√5

(40 −8√5)(163840 + 163840√5)

c3=256(−256 −512√5)

163840 + 163840√5,

c4=4√10(204800 + 40960√5)

5(163840 + 163840√5)(p5−√5)

c5=32√5(−11796480 −1310720√5)

5(163840 + 163840√5)(640 −128√5)

c6=(83886080√10)

(163840 + 163840√5)(640 −128√5)(p5−√5)

7 Conclusion

This paper gives Runge-Kutta technique for solving

and assessing the local truncated error for sixth or-

der ODE of form vvi =f(u, v, v′)possessing ini-

tial conditions. The initiative of introducing the three

and four stage seventh order RKSD7 to the sixth or-

der ODE is proved to be beneficial for evaluation of

values of norms of zero stability and error of RKSD7

with efficient values of the weights biand the nodes

ciand arranged them in the form of Butcher tableau.

The objective of providing accurate solution by min-

imizing error in shorter time, number of operations,

use less memory space for ODE of sixth order has

been attained in numerical form. Hence, the cur-

rent paper findings proved be beneficial for analyzing

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.20

Manpreet Kaur, Jasdev Bhatti,

Sangeet Kumar, Srinivasarao Thota

E-ISSN: 2224-2880

173

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Figure 3: Efficiency curves for RK, three

and four stage RKTF7 with step size h=

0.1,0.2,0.25,0.4,0.5

many problems related to engineering, science, med-

ical areas with more accuracy by minimizing the er-

rors. From numerical results, the best outcome been

received is that the number of function evaluations of

both RKSD7 methods are less than number of func-

tion evaluations for other existing RK methods.

Acknowledgment:

The authors thank the reviewers, editor and editing

department for giving valuable inputs and

suggestions to get the current form of the manuscript.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Dr. M. Kaur and J. Bhatti are involved in the forma-

tion and derivation of the mathematical calculations.

Dr. S. Kumar and Dr. S. Thota are involved in sug-

gestions, revisions and verification of the mathemat-

ical calculations. All authors read and approved the

final manuscript.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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