J. Nig. Soc. Phys. Sci. 2 (2020) 120–127

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

An Examination of a Second Order Numerical Method for
Solving Initial Value Problems

S. E. Fadugbaa,∗, S. N. Ogunyebia, B. O. Falodunb

aDepartment of Mathematics, Ekiti State University, Ado Ekiti, Ekiti State
bDepartment of Mathematics, University of Ilorin, Kwara State

Abstract

This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in
Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential
forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the
performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM),
Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at
the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the
method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than
IMEM and FOS and also compared favourably with the ES. By varying the step length, there are two-order decrease in the values of the final
absolute relative errors generated via SOCNM. Hence, SOCNM is found to be accurate, stable and a good tool for the numerical solutions of IVPs
of different characteristics in ODEs.

Keywords: Absolute relative error, accuracy, comparative study, stability.

Article History :
Received: 20 April 2020
Received in revised form: 31 May 2020
Accepted for publication: 04 June 2020
Published: 01 August 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: B. J. Falaye

1. Introduction

Most of the real world problems in science and engineering
are modelled and formulated by means of differential equations.
Some of these differential equations cannot be solved analyti-
cally or difficult to obtain their exact solutions. One way is to
use numerical methods for approximating the solution of the
differential equations using prescribed initial or boundary con-
ditions. In the recent years, many numerical methods of various

∗Corresponding author tel. no: +2348067032044
Email address: sunday.fadugba@eksu.edu.ng (S. E. Fadugba )

nature have been developed for the solution of IVPs in ODEs
of the form

dy
d x

= f (x, y), y(a) = y0, −∞ < y < ∞, a ≤ x ≤ b (1)

such as the Euler’s method, improved Euler’s method, Runge-
Kutta methods and so on. In [1], the authors developed a new
one-step scheme for the solution of IVPs in ODEs. Ahmad et al.
[2] studied the numerical accuracy of the Runge-Kutta method
of second, third and fourth order for the numerical solution of
differential equations. Fadugba and Idowu [3] derived a new
numerical method of third order accuracy via the transcendental
function of exponential type for the solution of IVPs in ODEs.

120



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 121

They also analysed the properties of the derived method. In [4],
accurate solutions of IVPs of ODEs with fourth order Runge
Kutta method were presented. Islam [5] compared the numer-
ical solutions of IVPs for ODEs with Euler’s and Runge-Kutta
methods. Refs. [6, 7, 8, 9, 10] also studied the numerical solu-
tions of IVPs in ODEs via several developed methods. In this
paper, the performance of SOCNM in terms of accuracy and
stability, that provides numerical solutions to IVPs in ODEs
is examined. The SOCNM is implemented on some selected
IVPS of different characteristics in ODEs. The rest of the paper
is structured as follows; Section Two presents the analysis and
the properties of SOCNM. In Section Three, three numerical
examples were solved via SOCNM in the context of the IMEM
[13], FOS [14] and ES. Section Four concludes the paper with
the discussion of results.

2. Analysis and the Properties of a Second Order Conver-
gence Numerical Method

This section presents the analysis and properties of SOCNM
as follows.

2.1. Analysis of SOCNM

Consider the IVP given by (1) whose exact solution is ob-
tained as y(x). In [1], a new numerical method of one step of
the form

yn+1 = yn + h
(

fn + f
(1)
n

)
+

(
e−h − 1

)
f (1)n (2)

was proposed and developed by means of the interpolating func-
tion of exponential and polynomial types

F(x) =
1∑

i=0

ai x
i
+ a2e

−x (3)

for the numerical solution of (1). Simplifying further, (2) be-
comes

yn+1 = yn + h fn +
(
h + e−h − 1

)
f (1)n (4)

The mesh point is defined as xn+1 = a + (n + 1)h, n = 0, 1, 2, ...
and the step length is defined as h = b−aN , where N is the number
of integration steps.

2.2. Properties of SOCNM

The properties of SOCNM such as accuracy, consistency,
stability and convergence are discussed as follows:

2.2.1. Accuracy of SOCNM
Consider the Taylor’s series expansion for the exact solution

y(x) given by

y(xn + h) = y(xn) + h f (xn, y(xn))

+
h2

2!
f (1) (xn, y(xn))

+
h3

3!
f (2) (xn, y(xn)) + O(h

4) (5)

The local truncation error is given by

Tn+1 = y(xn + h) − yn+1 (6)

Substituting (4) and (5) into (6) yields

Tn+1 = y(xn) + h f (xn, y(xn)) +
h2

2!
f (1)(xn, y(xn))

+
h3

3!
f (2)(xn, y(xn)) + O(h

4)

−
[
yn + h fn + (h + e

−h
− 1) f (1)n

]
(7)

Simplifying further and by means of the localizing assumption
(that is, assume that no errors have been committed), (7) be-
comes

Tn+1 =
h3

3!

[
f (2)(xn, y(xn)) + f

(1)(xn, y(xn))
]

+ O(h4) (8)

which is the local truncation error for SOCNM. Hence, (8) con-
firms the second order accuracy of SOCNM.

2.2.2. Consistency of SOCNM
According to [15], a one step numerical method is said to

be consistent if it has at least order p = 1. It is clearly seen
from (8) that SOCNM is consistent since it has second order
accuracy and that

lim
h→0

Tn+1
h

= lim
h→0


h3
3!

[
f (2)(xn, y(xn)) + f (1)(xn, y(xn))

]
+ O(h4)

h

 = 0
2.2.3. Stability of SOCNM

According to [15], a one step numerical method is said to be
stable if it is capable of damping out small fluctuation carried
out in input data. To discuss the stability of SOCNM, consider
the IVP of the form

dy
d x

= −φy, y(0) = 1 (9)

The exact solution of (9) is obtained as

y(x) = e−φx, φ > 0 (10)

where φ is a complex constant. Expanding (10) at the points
x = xn and x = xn+1 and using the fact that h = xn+1 − xn, one
obtains

y(xn) = e
−φxn (11)

121



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 122

and

y(xn+1) = e
−φxn+1 = e−φxn e−φh = y(xn)e

−φh (12)

Using SOCNM and the series expansion of e−h, one obtains

yn+1 = yn

[
1 −φh +

φ2h2

2!

]
(13)

Setting

A =
[
1 − z +

z2

2!

]
, z = φh (14)

Equation (13) becomes

yn+1 = Ayn (15)

Comparing (12) and (15), it is clearly seen that (14) consists of
the first-three terms of the series expansion of e−φh. Hence, the
stability function of SOCNM requires that

‖A‖ < 1 (16)

Equation (16) shows that SOCNM is stable.

2.2.4. Convergence of SOCNM
According to [16], the necessary and sufficient conditions

for convergence are consistency and stability. In other words, a
one step numerical method is said to be convergent if it is con-
sistent and stable. It is clearly seen that SOCNM is convergent
since it is consistent and stable. The convergence of SOCNM
is of second order.

3. Numerical Examples

Here, the performance of SOCNM in terms of the accu-
racy and stability is examined. Also the comparative study of
SOCNM, IMEM given by yn+1 = yn + hk3, where k1 = f (xn, yn),
k2 = f (xn, yn + k1), k3 = f

(
xn +

h
2 , yn +

h
2 k2

)
, FOS given by

yn+1 = yn +
(
h + h2 + 23 h

3
)

fn and ES is presented. All the cal-
culations were carried out with the aid of computational soft-
ware ”MATLAB R2014a, 8.3.0.552, 32 bit (win 32)” in double
precision.

Problem 1
Consider the IVP of the form

dy
d x
− x sin(x) = 0, y(0) = 1, x ∈ [0, 1]

whose exact solution is given by

y(x) = 1 + x sin(x) − x cos(x)

Problem 2
Consider the IVP of the form

dy
d x
− 1 − (y − x)2 = 0, y(0) =

1
2
, x ∈ [0, 1]

whose exact solution is given by

y(x) = x +
1

2 − x

Problem 3
Consider the Riccati equation coupled with the initial value

given by

dy
d x

+ 4 − 4y + y2 = 0, y(0) = 1, x ∈
[
0,

2
5

]
whose exact solution is given by

y(x) =
2x − 1
x − 1

and has a pole in x = 1.
The results generated via the SOCNM, IMEM, FOS in the con-
text of ES for Problems 1-3 are displayed in the Figures 1-
18. The comparative results analyses of SOCNM, IMEM, FOS
and ES for Problems 1-3 with h = 0.1 are shown in Figures
19,21,23, respectively. The absolute relative errors (AREs) gen-
erated via SOCNM, IMEM and FOS with h = 0.1 for Problems
1-3 are displayed in Figures 20,22,24, respectively. The rel-
ative absolute errors (FAREs) generated via SOCNM, IMEM
and FOS for Problems 1-3 at the final nodal point of the asso-
ciated integration interval with different values of step length,
h = 0.1, 0.01, 0.001, 0.0001, 0.00001 are displayed in the Fig-
ures 25,26,27, respectively.

Figure 1. The results generated via SOCNM and ES for Problem 1

122



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 123

Figure 2. The ARE generated via SOCNM for Problem 1

Figure 3. The results generated via IMEM and ES for Problem 1

Figure 4. The ARE generated via IMEM for Problem 1

Figure 5. The results generated via FOS and ES for Problem 1

Figure 6. The ARE generated via FOS for Problem 1

Figure 7. The results generated via SOCNM and ES for Problem 2

123



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 124

Figure 8. The ARE generated via SOCNM for Problem 2

Figure 9. The results generated via IMEM and ES for Problem 2

Figure 10. The ARE generated via IMEM for Problem 2

Figure 11. The results generated via FOS and ES for Problem 2

Figure 12. The ARE generated via FOS for Problem 2

Figure 13. The results generated via SOCNM and ES for Problem 3

124



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 125

Figure 14. The ARE generated via SOCNM for Problem 3

Figure 15. The results generated via IMEM and ES for Problem 3

Figure 16. The ARE generated via IMEM for Problem 3

Figure 17. The results generated via FOS and ES for Problem 3

Figure 18. The ARE generated via FOS for Problem 3

Figure 19. The comparative results analyses of SOCNM, IMEM, FOS and ES
for Problem 1

125



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 126

Figure 20. The AREs generated via SOCNM, IMEM and FOS for problem 1

Figure 21. The comparative results analyses of SOCNM, IMEM, FOS and ES
for Problem 2

Figure 22. The AREs generated via SOCNM, IMEM and FOS for problem 2

Figure 23. The comparative results analyses of SOCNM, IMEM, FOS and ES
for Problem 3

Figure 24. The AREs generated via SOCNM, IMEM and FOS for problem 3

Figure 25. The FAREs generated via SOCNM, IMEM and FOS with varying
step length values for problem 1

126



Fadugba et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 120–127 127

Figure 26. The FAREs generated via SOCNM, IMEM and FOS with varying
step length values for problem 2

Figure 27. The FAREs generated via SOCNM, IMEM and FOS with varying
step length values for problem 3

4. Conclusion

In this paper, an examination of SOCNM for solving IVPs
in ODEs is presented. The SOCNM has been derived via the
interpolating function comprises of polynomial and exponen-
tial forms. The analysis and the properties of SOCNM were
discussed extensively. Three numerical examples were solved
successfully to examine the performance of SOCNM in terms
of accuracy and stability. The comparative study of SOCNM,
IMEM, FOS and ES is presented. By varying the step length,
the absolute relative errors at the final nodal point of the as-
sociated integration interval are computed. Furthermore, the
analysis of the properties of SOCNM shows that the method is
consistent, stable, convergent and has second order accuracy.
The comparative study of the results generated via SOCNM,
IMEM, FOS and ES is displayed in Figures 1-18. Moreover,
the numerical results show that SOCNM is more accurate than
its counterparts, compared favourably and agreed with ES as

seen in the Figures 19,21,23. It is also observed from Figures
20,22,24 that the error curves for SOCNM show that SOCNM
follows the ES curves elegantly. Also, by varying the step
length (h = 0.1, 0.01, 0.001, 0.0001, 0.00001), it is observed
from Figures 25-27 that there are two-order decrease in the val-
ues of the final absolute relative errors generated via SOCNM.
Hence, SOCNM is found to be accurate, stable and a good ex-
plicit method for the numerical solutions of IVPs of different
characteristics in ODEs.

References

[1] S. Fadugba & B. Falodun, “Development of a new one-step scheme for
the solution of initial value problem (IVP) in ordinary differential equa-
tions”, International Journal of Theoretical and Applied Mathematics, 3
(2017) 58.

[2] N. Ahmad, S. Charan & V. P. Singh, “Study of numerical accuracy of
Runge-Kutta second, third and fourth order method”, International Jour-
nal of Computer and Mathematical Sciences, 4 (2015) 111.

[3] S. E. Fadugba & J. O. Idowu, “Analysis of the properties of a third or-
der convergence numerical method derived via transcendental function
of exponential form”, International Journal of Applied Mathematics and
Theoretical Physics, 5 (2019) 97.

[4] Md. A. Islam, “Accurate solutions of initial value problems for ordinary
differential equations with fourth order Runge Kutta method”, Journal of
Mathematics Research, 7 (2015) 41.

[5] Md. A. Islam, “A comparative study on numerical solutions of initial
value problems (IVP) for ordinary differential equations (ODE) with
Euler and Runge Kutta methods”, American Journal of Computational
Mathematics, 5 (2015) 393.

[6] Y. Ansari, A. Shaikh & S. Qureshi, “Error bounds for a numerical scheme
with reduced slope evaluations”, J. Appl. Environ. Biol. Sci., 8 (2018) 67.

[7] J. C. Butcher, Numerical methods for ordinary differential equations, 3rd
Edition, John Wiley & Sons, Chichester, West Sussex, 2016.

[8] M. E. Davis, Numerical methods and modeling for chemical engineers,
Courier Corporation, New York, 2013.

[9] S. E. Fadugba & J. T. Okunlola, “Performance measure of a new one-step
numerical technique via interpolating function for the solution of initial
value problem of first order differential equation”, World Scientific News,
90 (2017) 77.

[10] S. E. Fadugba & A. O. Ajayi, “Comparative study of a new scheme and
some existing methods for the solution of initial value problems in or-
dinary differential equations”, International Journal of Engineering and
Future Technology, 14 (2017) 47.

[11] J. D. Lambert, Numerical methods for ordinary differential systems: the
initial value problem, John Wiley & Sons, Inc., New York, 1991.

[12] S. Fadugba & S. Qureshi, “Convergent numerical method using transcen-
dental function of exponential type to solve continuous dynamical sys-
tems”, Punjab University Journal of Mathematics, 51 (2019) 45.

[13] O. Abraham, “Improving the modified Euler method”, Leonardo J. Sci.,
10 (2007) 1.

[14] S. E. Fadugba & T. E. Olaosebikan, “Comparative study of a class of one-
step methods for the numerical solution of some initial value problems
in ordinary differential equations”, Research Journal of Mathematics and
Computer Science, 2 (2018) 1.

[15] J. D. Lambert, Computational methods in ordinary differential equations,
John Wiley & Sons Inc., 1973.

[16] J. D. Lambert, Numerical methods for ordinary differential systems: the
initial value problem, John Wiley & Sons, Inc., New York, 1991.

127