umerical algorithm for solving second order nonlinear fuzzy initial value problems

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 6, December 2020, pp. 6497~6506
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i6.pp6497-6506  6497
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
Numerical algorithm for solving second order nonlinear fuzzy
initial value problems
A. F. Jameel1
, N.R. Anakira2
, A. H. Shather3
, Azizan Saaban4
, A. K. Alomari5
1,4
School of Quantitative Sciences, Universiti Utara Malaysia (UUM), Malaysia
2
Department of Mathematics Faculty of Science and Technology Irbid National University, Jordan
3
Department of Computer Engineering Techniques, Al Kitab University College, Iraq
5
Department of Mathematics, Faculty of Science, Yarmouk University, Jordan
Article Info ABSTRACT
Article history:
Received Aug 23, 2019
Revised May 25, 2020
Accepted Jun 5, 2020
The purpose of this analysis would be to provide a computational technique
for the numerical solution of second-order nonlinear fuzzy initial value
(FIVPs). The idea is based on the reformulation of the fifth order Runge
Kutta with six stages (RK56) from crisp domain to the fuzzy domain by
using the definitions and properties of fuzzy set theory to be suitable to solve
second order nonlinear FIVP numerically. It is shown that the second order
nonlinear FIVP can be solved by RK56 by reducing the original nonlinear
equation intoa system of couple first order nonlinear FIVP. The findings
indicate that the technique is very efficient and simple to implement and
satisfy the Fuzzy solution properties. The method’s potential is demonstrated
by solving nonlinear second-order FIVP.
Keywords:
FIVP
Fuzzy differential equations
Fuzzy set theory
RK56
Second-order nonlinear
Six stages runge kutta
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
A. F. Jameel,
School of Quantitative Sciences,
Universiti Utara Malaysia (UUM),
Kedah, Sintok, 06010 Malaysia.
Email: homotopy33@gmail.com
1. INTRODUCTION
As a conceptual model, several complex real-life problems can be articulated. Such models’
equations may either be constructed as an ordinary or partial differential equation [1, 2]. These equations can
be solved via several method numerical or approximated method [3-6]. The FDEs are helpful in modelling
a complex structure when knowledge is insufficient on its behavior. The FIVPs arises as such models have not
been developed and are unpredictable in nature. These types of models can be defined by mathematical
dynamic systems models under vagueness of FDEs. Real words, phenomena such as population physics and
medicine take into account of fuzzy problems [5-9]. Ordinary differential. The efficacy of the methods
proposed can be tested on linear FIVPs as simplest consideration. The dynamics of the Range Kutta methods
depend on increasing number calculations in the methods function per each step that we can gain the method’s
ability to solve such problems [10].
The explicit fourth order Runge-Kutta methods have been widely used in the solution for first and
high order FIVPs [11, 12]. Various types of RungeKutte methods have been used in the last ducat to solve
linear second order FIVPs, including the second order RungeKutta method [10], the third order Runge Kutta
methods [13], and the fifth order Runge Kutta method of five stages in [14]. The focus is to develop and
evaluate RK56's use to provide the non-linear second order FIVP numerical solution. This is the first attempt,
to the best of our knowledge, to solve the nonlinear form of second order FIVP using RK56. The present
article shall be arranged accordingly. We start with certain standard concepts for fuzzy numbers in section 2.

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Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6497 - 6506
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The second order FIVP fuzzy analysis is described in section 3. We analyze and develop RK56 for
the solution of the second order of FIVP in section 4. For the nonlinear second order FIVP solution, section 5
uses RK56 for a test example and display results in the form of table and figures. Finally, in section 6,
we give the conclusion of this study.
2. PRELIMINARIES
Some definitions and most basic notes are standard and popular (see [15-19] for details), in this
section, which will be used in this paper, are introduced
Definition 2.1([20]): “The triangular fuzzy are linked to degrees of membership which state how true it is to
say if something belongs or not to a determined set. As shown if Figure 1 this number is defined by three
parameters  <  <  where the graph of μ(x) is a triangle with the base on the interval [, ] and vertex at
x =  and its membership function has the following form:
μ(x; α, β, γ) =
{
0 , if x < α
x − α
β − α
, if α ≤ x ≤ β
γ − x
γ − β
, if β ≤ x ≤ y
0 , if x > γ
Figure 1. Triangular fuzzy number
and its r-level is:[μ]r = [α + r (β − α), γ − r(γ − β)], r ∈ [0, 1].
Definition 2.2([4]): “A fuzzy number is a function ℝ: R → [0, 1] satisfying the following properties:
1.μ(x) is normal, i.e ∃t0 ∈ ℝ with μ(t0) = 1,
2.μ(x)Is convex fuzzy set, i. e. μ(λx + (1 − λ)t) ≥ min{μ(x), μ(t)} ∀x, t ∈ ℝ, λ[0,1],
3.μ upper semi-continuous on ℝ,
4.{t ∈ ℝ: μ(x) > 0} is compact.
Ẽ is called the space of fuzzy numbers and ℝ is a proper subset of Ẽ.
Define the r-level set x ∈ ℝ,[μ]r = {x ∶ μ(x) ≥ r}, 0 ≤ r ≤ 1 where [μ]0 = {x ∶ μ(x) > 0} compact,
which is a closed bounded interval and denoted by is [μ]r = [μ(x), μ(x)]. In the single parametric form [2],
a fuzzy number is represented by an ordered pair of functions [μ(x), μ(x)], r ∈ [0,1] which satisfies:
1. μ(x) is a bounded left continuous non-decreasing function over [0,1].
2. μ(x)is a bounded left continuous non-increasing function over [0,1] .
3. μ(x) ≤ μ(x), r ∈ [0,1]. A crisp number r is simply represented by μ(r) = μ(r) = r, r ∈ [0,1].”
Definition 2.2([4]) “If Ẽ be the set of all fuzzy numbers, we say that f(x) is a fuzzy function if f: ℝ → Ẽ”
Definition 2.3([4]) “A mapping f: T → Ẽfor some interval T ⊆ Ẽis called a fuzzy function process and we
denote r-level set by:
[f̃(x)]r = [f(x; r), f(x; r)], x ∈ K, r ∈ [0,1]
  x
(x)
1
0
0.5


Int J Elec & Comp Eng ISSN: 2088-8708 
Numerical algorithm for solving second order nonlinear fuzzy… (A. F. Jameel)
6499
The r-level sets of a fuzzy number are much more effective as representation forms of fuzzy set than
the above. Fuzzy sets can be defined by the families of their r-level sets based on the resolution identity
theorem.”
Definition 2.5([18]): Each function f: X → Y induces another function f̃: F(X) → F(Y) defined for each fuzzy
interval U in X by:
f̃(U)(y) = {
Supx∈f−1(y)U(x), if y ∈ range(f)
0 , if y ∉ range(f)
This is called the Zadeh extension principle.
Definition 2.6 ([4]): “Considerx̃, ỹ ∈ Ẽ. If there exists z̃ ∈ Ẽsuch thatx̃ = ỹ + z̃, then z̃is called
the H-difference (Hukuhara difference) of x and y and is denoted byz̃ = x̃ ⊝ ỹ.”
Definition 2.7([4]) “If f̃: I → Ẽ and y0 ∈ I, where I ∈ [x0, K]. We say that f̃ Hukuhara differentiable at y0,
if there exists an element [f′̃]r
∈ Ẽsuch that for all h > 0 sufficiently small (near to 0), exists f̃(y0 + h; r) ⊝
f̃(y0; r),f̃(y0; r) ⊝ f̃(y0 − h; r) and the limits are taken in the metric (Ẽ , D):
lim
h→0+
f̃(y0 + h; r) ⊝ f̃(y0; r)
h
= lim
h→0+
f̃(y0; r) ⊝ f̃(y0 − h; r)
h
The fuzzy set [f′̃(y0)]r
is called the Hukuhara derivative of [f′̃]r
at y0.
These limits are taken in the space (Ẽ , D) if x0 or K, then we consider the corresponding one-side
derivation. Recall that x̃ ⊝ ỹ = z̃ ∈ Ẽ are defined on r-level set, where [x̃]r ⊝ [ỹ]r = [z̃]r, ∀ r ∈ [0,1]
By consideration of definition of the metric D all the r-level set [f̃(0)]r
are Hukuhara differentiable at y0,
with Hukuhara derivatives [f′̃(y0)]r
, when f̃: I → Ẽis Hukuhara differentiable at y0 with Hukuhara derivative
[f′̃(y0)]r
it’ lead to that f̃ is Hukuhara differentiable for all r ∈ [0,1] which satisfies the above limits i.e. if f
is differentiable at x0 ∈ [x0 + α, K] then all its r-levels [f′̃(x)]r
are Hukuhara differentiable at x0.”
Definition 2.8([4]) “Define the mapping f′̃: I → Ẽ and y0 ∈ I , where I ∈ [x0, K] . We say that f′̃Hukuhara
differentiable x ∈ Ẽ, if there exists an element [f̃(n)
]r
∈ Ẽ such that for allh > 0 sufficiently small (near to
0), exists f̃ (n−1)
(y0 + h; r) ⊝ f̃(n−1)
(y0; r),f̃(n−1)(y0; r) ⊝ f̃(n−1)
(y0 − h; r) and the limits are taken in
the metric (Ẽ , D)
lim
h→0+
f̃(n−1)
(y0 + h; r) ⊝ f̃(n−1)
(y0; r)
h
= lim
h→0+
f̃(n−1)(y0; r) ⊝ f̃(n−1)
(y0 − h; r)
h
exists and equal to f̃(n)
and for n = 2 we have second order Hukuharaderivative.”
Theorem 2.2([4]) “ Let f̃: [x0 + α, K] → Ẽ be Hukuhara differentiable and denote
[f′̃(x)]r
= [f′(x), f′(x)]r
= [f′(x; r), f′(x; r)].
Then the boundary functionsf′(x; r), f′(x; r) are differentiable we can write for second order fuzzy derivative
[f̃′′
(x)]r
= [(f′′(x; r))
′
, (f
′′
(x; r))
′
], ∀ r ∈ [0,1] .”
3. ANALYSIS OF SECOND ORDER FIVP
Consider the second order nonlinear FIVP of the following form:
𝑦̃′′(𝑥) = 𝒩̃(𝑦̃(𝑥), 𝑦̃′(𝑥)) + 𝑛̃(𝑥), 𝑥 ∈ [𝑥0, 𝐾] (1)
Subject to the fuzzy initial conditions

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𝑦̃(𝑥0) = 𝛼̃0, 𝑦̃′(𝑥0) = 𝛽̃0 (2)
where the defuzzification details can be found in [4]:
“𝑦̃(𝑥): is a fuzzy function in terms of 𝑥 such that [𝑦̃(𝑥)] 𝑟 = [𝑦(𝑥)(𝑟)
, 𝑦(𝑥)(𝑟)
],
𝑦̃′(𝑥): is the first order fuzzy Hukuhara-derivative such that[𝑦̃′(𝑥)] 𝑟 = [𝑦′(𝑥)(𝑟)
, 𝑦
′
(𝑥)(𝑟)
],
𝑦̃′′(𝑥): is the second order fuzzy Hukuhara-derivative such that [𝑦̃′′(𝑥)] 𝑟 = [𝑦′′(𝑥)(𝑟)
, 𝑦
′′
(𝑥)(𝑟)
],
𝒩̃ : is fuzzy nonlinear function in terms of 𝑥, the fuzzy variable𝑦̃ and the fuzzy derivative 𝑦̃′
”
From the above analysis we define the following relations:
[𝒩̃(𝑦̃(𝑥), 𝑦̃′(𝑥))]
𝑟
= [𝒩(𝑥, 𝑦, 𝑦′
)(𝑟)
, 𝒩(𝑥, 𝑦, 𝑦
′
)(𝑟)
], such that
𝑛̃(𝑥) = 𝑝̃𝑡(𝑥) + 𝑞̃ (3)
where the nonhomogeneous term is [𝑛̃(𝑥)] 𝑟 = [𝑛(𝑥)(𝑟)
, 𝑛(𝑥)(𝑟)
] such that𝑝̃𝑡(𝑥) + 𝑞̃. The fuzzy coefficients
in (2) are triangular fuzzy numbers denoted by [𝑝̃] 𝑟 = [𝑝, 𝑝] 𝑟, [𝑞̃] 𝑟 = [𝑞, 𝑞] 𝑟 for all the fuzzy level
set 𝑟 ∈ [0,1].
𝑦̃0 : is the first fuzzy initial condition triangular numbers denoted by [𝑦̃(𝑥0)] 𝑟 = [𝑦(𝑥0)(𝑟)
, 𝑦(𝑥0)(𝑟)
],
such that[𝛼̃0] 𝑟 = [𝛼0, 𝛼0] 𝑟, finally:
𝑦̃′
: is the second fuzzy initial condition triangular numbers denoted by [𝑦̃′(𝑥0)] 𝑟 = [𝑦′
(𝑥0)(𝑟)
, 𝑦
′
(𝑥0)(𝑟)
].
such that[𝛽̃0] 𝑟
= [𝛽0, 𝛽0
] 𝑟.
Also, we can write
{
𝒩(𝑦̃(𝑥), 𝑦̃′(𝑥))(𝑟)
= 𝐿 [𝑦, 𝑦, 𝑦′
, 𝑦
′
]
𝑟
𝒩(𝑦̃(𝑥), 𝑦̃′(𝑥))(𝑟)
= 𝑈 [𝑦, 𝑦, 𝑦′
, 𝑦
′
]
𝑟
(4)
Let
𝒩̃(𝑦̃(𝑥), 𝑦̃′(𝑥)) + 𝑛̃(𝑡) = 𝐺̃(𝑥, 𝑦̃(𝑥), 𝑦̃′(𝑥)), (5)
𝑆̃(𝑥) = 𝑦̃(𝑥), 𝑦̃′(𝑥), (6)
where [𝑆̃(𝑥)] 𝑟
= [𝑆(𝑥)(𝑟)
, 𝑆(𝑥)(𝑟)
]such that 𝑦̃′′(𝑡) = 𝐺̃(𝑥, 𝑆̃(𝑥)) and from definition 2.5, we can define
the following relation:
{
𝐺(𝑥, 𝑆̃(𝑥))(𝑟)
= min{𝐺̃(𝑥, 𝜇̃(𝑟))|𝜇̃(𝑟) ∈ 𝑦̃(𝑥)}
𝐺(𝑥, 𝑆̃(𝑥))(𝑟)
= max{𝐺̃(𝑥, 𝜇̃(𝑟))|𝜇̃(𝑟) ∈ 𝑦̃(𝑥)}
(7)
where
{
𝐺(𝑥, 𝑆̃(𝑥))(𝑟)
= 𝐿∗
[𝑥, 𝑆̃(𝑥)] 𝑟
= 𝐿∗
(𝑥, 𝑆̃(𝑥)(𝑟)
)
𝐺(𝑥, 𝑆̃(𝑥))(𝑟)
= 𝑈∗
[𝑥, 𝑆̃(𝑥)] 𝑟
= 𝑈∗
(𝑥, 𝑆̃(𝑥)(𝑟)
)
(8)
From (7) and (8) we rewrite (1) as follows
{
𝑦′′(𝑥)(𝑟)
= 𝐿∗
(𝑡, 𝑆̃(𝑥)(𝑟)
)
𝑦(𝑥0)(𝑟)
= 𝛼0, 𝑦′
(𝑥0)(𝑟)
= 𝛽0
(9)
{
𝑦
′′
(𝑥)(𝑟)
= 𝑈∗
(𝑡, 𝑆̃(𝑥)(𝑟)
)
𝑦(𝑥0)(𝑟)
= 𝛼0, 𝑦
′
(𝑥0)(𝑟)
= 𝛽0
(10)

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4. ANALYSIS OF FUZZY RK56
Here we take into account the six steps Runge Kutta of order five process as a framework for
obtaining a numerical solution of second-order FIVP. In [21-23] the structure of this method to solve
the crisp initial values problems of was introduced. Also in [24] RK56 to solve first order linear FIVPs with
fuzzy intial conditions. In addition, we must defuzzify RKM56 in order to solve nonlinear second order
FIVPs. We consider (1) in section 3 and since RK56 is numerical method we need to reduce (1) in to system
of first order nonlinear FIVPs for all ∈ [0,1] :
{
𝑦̃′
1
(𝑥)(𝑟)
= 𝐺̃1(𝑡, 𝑆̃(𝑥)(𝑟)
)
𝑦̃′
2
(𝑥)(𝑟)
= 𝐺̃2(𝑡, 𝑆̃(𝑥)(𝑟)
)
𝑦̃1(𝑥0)(𝑟)
= 𝛼̃0(𝑟) = 𝑎̃1, 𝑦̃2(𝑥0)(𝑟)
= 𝛽̃0(𝑟)
(11)
where 𝑥 ∈ [𝑥0, 𝐾], and 𝛼̃0, 𝛽̃0are fuzzy numbers for all 𝑟 ∈ [0,1] such that
𝑦̃1,0(𝑟) = [𝑦1,0(𝑟), 𝑦1,0
(𝑟)] =[𝛼0, 𝛼0] 𝑟
,𝑦̃2,0(𝑟) = [𝑦2,0(𝑟), 𝑦2,0
(𝑟)] =[𝛽0, 𝛽0
] 𝑟.
Assume 𝑦̃(𝑥)(𝑟)
= [𝑦̃1(𝑥)(𝑟)
, 𝑦̃2(𝑥)(𝑟)
]
𝑡
be the fuzzy solution of system (11) on [𝑥0, 𝐾]then according to
section (3) for 𝑖 = 1,2 we have:
𝑦𝑖
′(𝑥)(𝑟)
= min {𝐺̃𝑖(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
); 𝑦̃𝑖(𝑥)(𝑟)
∈ [𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
] = 𝐺𝑖(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)} (12)
𝑦𝑖
′
(𝑥)(𝑟)
= max {𝐺̃𝑖(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
); 𝑦̃𝑖(𝑥)(𝑟)
∈ [𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
] = 𝐺𝑖(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)} (13)
and
{
[𝐺𝑖(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)]
𝑡
= 𝐿∗
(𝑥, 𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
) = 𝐿∗
(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)
[𝐺𝑖(𝑥𝑦̃𝑖(𝑥)(𝑟)
)]
𝑡
= 𝑈∗
(𝑥, 𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
) = 𝑈∗
(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)
(14)
From (11-13), system (10) can be written with following assumptions for 𝑖 = 1,2
{
𝑦′
(𝑥)(𝑟)
= 𝐿∗
(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)
𝑦(𝑥0)(𝑟)
= 𝑦0 ∈ [𝑥0, 𝐾]
(15)
where 𝑦(𝑥0)(𝑟)
= [𝑦𝑖,0(𝑟)]
𝑡
and𝑦′
(𝑥)(𝑟)
= [𝑦𝑖
′(𝑥)(𝑟)
]
𝑡
and
{
𝑦
′
(𝑥0)(𝑟)
= 𝑈∗
(𝑥, 𝑦̃𝑖(𝑥)(𝑟)
)
𝑦(𝑥0)(𝑟)
= 𝑦0
∈ [𝑥0, 𝐾]
(16)
where 𝑦(𝑥0)(𝑟)
= [𝑦𝑖,0
(𝑟)]
𝑡
and 𝑦
′
(𝑥0)(𝑟)
= [𝑦𝑖
′
(𝑥)(𝑟)
]
𝑡
.
Now we show that under the assumption for the fuzzy functions 𝐺̃𝑖, for 𝑖 = 1,2 solution for system
(11) for each r-level set can obtain a unique fuzzy [11]. Now, for the analysis of fuzzy RK56 with a view to
finding a numerical solution of system (11), we first the followings:
𝑦̃(𝑥)(𝑟)
= [𝑦̃1(𝑥)(𝑟)
, 𝑦̃2(𝑥)(𝑟)
]
𝑡
such that:
𝑧̃1 = 𝑦̃1(𝑥)(𝑟)
, 𝑧̃2 = 𝑦̃2(𝑥)(𝑟)
where 𝑧̃𝑖 ∈ [𝑧𝑖, 𝑧𝑖] for 𝑖 = 1,2.
𝑉𝑖1 (𝑥; 𝑦(𝑥)(𝑟)
) = min {𝐺̃𝑖(𝑥, 𝑧̃1, 𝑧̃2); 𝑧̃𝑖 ∈ [𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
]}
𝑉𝑖1(𝑥; 𝑦(𝑥)(𝑟)
) = max {𝐺̃𝑖(𝑥, 𝑧̃1, 𝑧̃2); 𝑧̃𝑖 ∈ [𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
]}
) = min {𝐺̃𝑖 (𝑥 +
ℎ
2
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖1 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖1(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = max {𝐺̃𝑖 (𝑥 +
ℎ
2
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖1 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖1(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = min {𝐺̃𝑖 (𝑥 +
ℎ
4
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖2 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖2(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}

 ISSN: 2088-8708
6502
) = max {𝐺̃𝑖 (𝑥 +
ℎ
4
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖2 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖2(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = min {𝐺̃𝑖 (𝑥 +
ℎ
2
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖3 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖3(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = max {𝐺̃𝑖 (𝑥 +
ℎ
2
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖3 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖3(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = min {𝐺̃𝑖 (𝑥 +
3ℎ
4
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖4 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖4(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = max {𝐺̃𝑖 (𝑥 +
3ℎ
4
, 𝑧̃1, 𝑧̃2) ; 𝑧̃𝑖 ∈ [𝑠𝑖4 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖4(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = min {𝐺̃𝑖(𝑥 + ℎ, 𝑧̃1, 𝑧̃2); 𝑧̃𝑖 ∈ [𝑠𝑖5 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖5(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
) = max {𝐺̃𝑖(𝑥 + ℎ, 𝑧̃1, 𝑧̃2); 𝑧̃𝑖 ∈ [𝑠𝑖5 (𝑥, 𝑦(𝑥)(𝑟)
, ℎ) , 𝑠𝑖5(𝑥, 𝑦(𝑥)(𝑟)
, ℎ)]}
where
𝑠𝑖1 (𝑥; 𝑦(𝑥)(𝑟)
, ℎ) = 𝑦𝑖(𝑥)(𝑟)
+
ℎ
2
𝑉𝑖1(𝑟) , 𝑠𝑖1(𝑥, 𝑦(𝑥)(𝑟)
, ℎ) = 𝑦𝑖
(𝑥)(𝑟)
+
ℎ
2
𝑉𝑖1(𝑟)
, ℎ) = 𝑦𝑖(𝑥)(𝑟)
+
ℎ
16
(3𝑉𝑖1(𝑟) + 𝑉𝑖2(𝑟))
𝑠𝑖2(𝑥, 𝑦(𝑥)(𝑟)
, ℎ) = 𝑦𝑖
(𝑥)(𝑟)
+
ℎ
16
(3𝑉𝑖2(𝑟) + 𝑉𝑖2(𝑟))
, ℎ) = 𝑦𝑖(𝑥)(𝑟)
+
ℎ
2
𝑉𝑖3(𝑟), 𝑠𝑖3(𝑥, 𝑦(𝑥)(𝑟)
, ℎ) = 𝑦𝑖
(𝑥)(𝑟)
+
ℎ
2
𝑉𝑖3(𝑟)
, ℎ) = 𝑦𝑖(𝑥)(𝑟)
+
ℎ
16
(−3𝑉𝑖2(𝑟) + 6𝑉𝑖3(𝑟) + 9𝑉𝑖4(𝑟))
, ℎ) = 𝑦𝑖
(𝑥)(𝑟)
+
ℎ
16
(−3𝑉𝑖2(𝑟) + 6𝑉𝑖3(𝑟) + 9𝑉𝑖4(𝑟))
, ℎ) = 𝑦𝑖(𝑥)(𝑟)
+
ℎ
7
(𝑉𝑖1(𝑟) + 4𝑉𝑖2(𝑟) + 6𝑉𝑖3(𝑟) − 12𝑉𝑖4(𝑟) + 8𝑉𝑖5(𝑟))
, ℎ) = 𝑦𝑖
(𝑥)(𝑟)
+
ℎ
7
(𝑉𝑖1(𝑟) + 4𝑉𝑖2(𝑟) + 6𝑉𝑖3(𝑟) − 12𝑉𝑖4(𝑟) + 8𝑉𝑖5(𝑟))
where 𝑉𝑖𝑗 (𝑥; 𝑦(𝑥)(𝑟)
, ℎ) = 𝑉𝑖𝑗(𝑟) and 𝑉𝑖𝑗(𝑥; 𝑦(𝑥)(𝑟)
, ℎ) = 𝑉𝑖𝑗(𝑟) for 𝑗 = 1,2, . . ,6. Now consider
the followings
𝐿∗
𝑖 (𝑥; 𝑦(𝑥)(𝑟)
, ℎ) = 7𝑉𝑖1(𝑟) + 32𝑉𝑖3(𝑟) + 12𝑉𝑖4(𝑟) + 32𝑉𝑖5(𝑟) + 7𝑉𝑖6(𝑟)
𝑈∗
𝑖(𝑥; 𝑦(𝑥)(𝑟)
, ℎ) = 7𝑉𝑖1(𝑟) + 32𝑉𝑖3(𝑟) + 12𝑉𝑖4(𝑟) + 32𝑉𝑖5(𝑟) + 7𝑉𝑖6(𝑟)
where 𝑥 ∈ [𝑥0, 𝐾]and 𝑟 ∈ [0,1]. If the exact and numerical solution in the i-th𝑟 -level at 𝑥 𝑚for 0 ≤ 𝑚 ≤ 𝑁
can be denoted by 𝑌̃𝑖(𝑥)(𝑟)
= [𝑌𝑖(𝑥)(𝑟)
, 𝑌𝑖(𝑥)(𝑟)
], 𝑦̃𝑖(𝑥)(𝑟)
= [𝑦𝑖(𝑥)(𝑟)
, 𝑦𝑖
(𝑥)(𝑟)
]respectively. According to
section (2) the i-th coordinate r-level set of RK56 is:
𝑦𝑖
[𝑚+1](𝑥)(𝑟)
= 𝑦𝑖
[𝑚](𝑥)(𝑟)
+
1
90
𝐿∗
𝑖(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝑦𝑖
[0](𝑟) = 𝑌𝑖
[0]
(𝑟) (17)
𝑦𝑖
[𝑚+1]
(𝑥)(𝑟)
= 𝑦𝑖
[𝑚]
(𝑥)(𝑟)
+
1
90
𝑈∗
𝑖(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝑦𝑖
[0]
(𝑟) = 𝑌𝑖
[0]
(𝑟) (18)
where
𝑦̃[𝑚](𝑥)(𝑟)
= [𝑦[𝑚](𝑥), 𝑦
[𝑚]
(𝑥)]
𝑟
, 𝑦[𝑚](𝑟) = [𝑦1
[𝑚](𝑟), 𝑦2
[𝑚](𝑟)]
𝑡
and 𝑦
[𝑚]
(𝑟) = [𝑦1
[𝑚]
(𝑟), 𝑦2
[𝑚]
(𝑟)]
𝑡
Now we let
𝐿∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ) =
1
90
(𝐿∗
1(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝐿∗
2(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ))
𝑡
(19)

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𝑈∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ) =
1
90
(𝑈∗
1(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝑈∗
2(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ))
𝑡
(20)
The RK5 numerical solution for all r-level sets of (17-18) is as follows
𝑦[𝑚+1](𝑥)(𝑟)
= 𝑦[𝑚+1](𝑥)(𝑟)
+ ℎ 𝐿∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝑦[0](𝑟) = 𝑌[0](𝑟) (21)
𝑦
[𝑚+1]
(𝑥)(𝑟)
= 𝑦
[𝑚]
(𝑥)(𝑟)
+ ℎ 𝑈∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ), 𝑦
[0]
(𝑟) = 𝑦
[0]
(𝑟) (22)
and
𝐿∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ) =
1
90
[7𝑉𝑖1(𝑟) + 32𝑉𝑖3(𝑟) + 12𝑉𝑖4(𝑟) + 32𝑉𝑖5(𝑟) + 7𝑉𝑖6(𝑟) (23)
𝑈∗
(𝑥 𝑚, 𝑦̃[𝑚](𝑥)(𝑟)
, ℎ) =
1
90
[7𝑉
𝑖1
(𝑟) + 32𝑉𝑖3(𝑟) + 12𝑉𝑖4(𝑟) + 32𝑉𝑖5(𝑟) + 7𝑉𝑖6(𝑟) (24)
5. NUMERICAL EXAMPLE
Consider the second-order fuzzy nonlinear differential equation [25]:
𝑦′′(𝑥) = −(𝑦′
(𝑥))2
, 0 ≤ 𝑥 ≤ 0.1
𝑦(0) = (𝑟, 2 − 𝑟), 𝑦′(0) = (1 + 𝑟, 3 − 𝑟), ∀𝑟 ∈ [0,1] (25)
From [25] the exact solution of (25) is:
{
𝑌(𝑥)(𝑟)
= ln[(𝑒 𝑟
+ 𝑒 𝑟
𝑟)𝑥 + 𝑒 𝑟].
𝑌(𝑥)(𝑟)
= ln[(3𝑒2−𝑟
− 𝑒2−𝑟
𝑟)𝑥 + 𝑒2−𝑟] .
(26)
Figure 2 is displayed the three-dimensional exact solution (26) corresponding with (25):
Figure 2. 𝑌(𝑥)(𝑟)
and 𝑌(𝑥)(𝑟)
Exact solutions of (25) for all 𝑟 ∈ [0,1] and 𝑥 ∈ [0,0.1].
From section 3, (25) can be written into first order system as follows
{
𝑦̃′
1
(𝑥)(𝑟)
= 𝑦̃2(𝑥)(𝑟)
𝑦̃1(𝑥0)(𝑟)
= [𝑟, 2 − 𝑟]
𝑦̃′
2
(𝑥)(𝑟)
= −[𝑦̃2(𝑥)(𝑟)
]
2
𝑦̃2(𝑥0)(𝑟)
= [1 + 𝑟, 3 − 𝑟]
(26)
According to section 4, we apply RK5 to system (26) with selected ℎ = 0.1 which is enough to
obtain convergence results as follows. According to RK56 in section 4, we have:

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𝑃1(𝑥, 𝑦̃′
1
(𝑥)(𝑟)
, 𝑦̃2(𝑥)(𝑟)
) = 𝑃1(𝑥, 𝑦̃1, 𝑦̃2) = 𝑦̃2(𝑥)(𝑟)
𝑃2(𝑥, 𝑦̃′
1
(𝑥)(𝑟)
, 𝑦̃2(𝑥)(𝑟)
) = 𝑃2(𝑥, 𝑦̃1, 𝑦̃2) = [𝑦̃2(𝑥)(𝑟)
]
2
𝑉̃1(𝑟) = ℎ𝑃1(𝑥𝑖, 𝑦̃1, 𝑦̃2)
𝐴̃1(𝑟) = ℎ𝑃2(𝑥𝑖, 𝑦̃1, 𝑦̃2)
𝑉̃2(𝑟) = ℎ𝑃1 (𝑥𝑖 +
ℎ
2
, 𝑦̃1 +
𝑉̃1(𝑟)
2
, 𝑦̃2 +
𝐴̃1(𝑟)
2
)
𝐴̃2(𝑟) = ℎ𝑃2 (𝑥𝑖 +
ℎ
2
, 𝑦̃1 +
𝑉̃1(𝑟)
2
, 𝑦̃2 +
𝐴̃1(𝑟)
2
)
𝑉̃3(𝑟) = ℎ𝑃1 (𝑥𝑖 +
ℎ
4
, 𝑦̃1 +
3𝑉̃1(𝑟) + 𝑉̃2(𝑟)
2
, 𝑦̃2 +
3𝐴̃1(𝑟) + 𝐴̃2(𝑟)
2
)
𝐴̃3(𝑟) = ℎ𝑃2 (𝑥𝑖 +
ℎ
4
, 𝑦̃1 +
3𝑉̃1(𝑟) + 𝑉̃2(𝑟)
2
, 𝑦̃2 +
3𝐴̃1(𝑟) + 𝐴̃2(𝑟)
2
)
𝑉̃4(𝑟) = ℎ𝑃1 (𝑥𝑖 +
ℎ
2
, 𝑦̃1 +
𝑉̃3(𝑟)
2
, 𝑦̃2 +
𝐴̃3(𝑟)
2
)
𝐴̃4(𝑟) = ℎ𝑃2 (𝑥𝑖 +
ℎ
2
, 𝑦̃1 +
𝑉̃3(𝑟)
2
, 𝑦̃2 +
𝐴̃3(𝑟)
2
)
𝑉̃5(𝑟) = ℎ𝑃1 (𝑥𝑖 +
3ℎ
4
, 𝑦̃1 +
−3𝑉̃1(𝑟) + 6𝑉̃3(𝑟) + 9𝑉̃4(𝑟)
16
, 𝑦̃2 +
−3𝐴̃1(𝑟) + 6𝐴̃3(𝑟) + 9𝐴̃4(𝑟)
16
)
𝐷̃5(𝑟) = ℎ𝑃2 (𝑥𝑖 +
3ℎ
4
, 𝑦̃1 +
−3𝑉̃1(𝑟) + 6𝑉̃3(𝑟) + 9𝑉̃4(𝑟)
16
, 𝑦̃2 +
−3𝐴̃1(𝑟) + 6𝐴̃3(𝑟) + 9𝐴̃4(𝑟)
16
)
𝑉̃6(𝑟) = ℎ𝑃1 (𝑥𝑖 + ℎ, 𝑦̃1 +
𝑉̃1(𝑟) + 4𝑉̃2(𝑟) + 6𝑉̃3(𝑟) − 12𝑉̃4(𝑟) + 9𝑉̃5(𝑟)
7
, 𝑦̃2
+
𝐴̃1(𝑟) + 4𝐴̃2(𝑟) + 6𝐴̃3(𝑟) − 12𝐴̃4(𝑟) + 9𝐴̃5(𝑟)
7
)
𝐷̃6(𝑟) = ℎ𝑃2 (𝑥𝑖 + ℎ, 𝑦̃1 +
𝑉̃1(𝑟) + 4𝑉̃2(𝑟) + 6𝑉̃3(𝑟) − 12𝑉̃4(𝑟) + 9𝑉̃5(𝑟)
7
, 𝑦̃2
+
𝐴̃1(𝑟) + 4𝐴̃2(𝑟) + 6𝐴̃3(𝑟) − 12𝐴̃4(𝑟) + 9𝐴̃5(𝑟)
7
)
then
𝑦̃1(𝑥𝑖+1)(𝑟)
= 𝑦̃1(𝑥𝑖)(𝑟)
+
7𝑉̃1(𝑟)+32𝑉̃3(𝑟)+12𝑉̃4(𝑟)+32𝑉̃5(𝑟)+7𝑉̃6(𝑟)
90
(27)
𝑦̃2(𝑥𝑖+1)(𝑟)
= 𝑦̃2(𝑥𝑖)(𝑟)
+
7𝐴̃1(𝑟)+32𝐴̃3(𝑟)+12𝐴̃4(𝑟)+32𝐴̃5(𝑟)+7𝐴̃6(𝑟)
90
(28)
where
𝑦̃1(𝑡𝑖; 𝑟) = [𝑦1(𝑥𝑖; 𝑟), 𝑦1
(𝑥𝑖; 𝑟)] ,
𝑈1(𝑥𝑖, 𝑦̃1, 𝑦̃2) = [𝑈1 (𝑡𝑖, 𝑦1, 𝑦2) , 𝑈1(𝑥𝑖, 𝑦1
, 𝑦2
)] , 𝑈2(𝑥𝑖, 𝑦̃1, 𝑦̃2) = [𝑈2 (𝑥𝑖, 𝑦1, 𝑦2) , 𝑈2(𝑥𝑖, 𝑥1, 𝑥2)],
𝐾̃𝑗(𝑟) = [𝐾𝑗(𝑟), 𝐾𝑗(𝑟)],𝐷̃𝑗(𝑟) = [𝐷𝑗(𝑟), 𝐷𝑗(𝑟)]and𝑥𝑖+1 = 𝑥𝑖 + ℎ for 𝑗 = 1, … ,6, 𝑖 = 1, … , 𝑛
and 𝑟 ∈ [0,1].
Define the absolute error of the numerical solution of (25) for all 𝑟 ∈ [0,1]as follows
[𝐸] 𝑟
= |𝑌(𝑥)(𝑟)
− 𝑦(𝑥)(𝑟)
| , [𝐸] 𝑟
= |𝑌(𝑥)(𝑟)
− 𝑦(𝑥)(𝑟)
| (29)
From (27-28) we numerical solution of (25) as displayed in Table 1 and Figures 3 and 4:
From Table 1, we can note that the high accuracy of the suggested method in ten step size only and
the results or the lower and upper bounds of (25) flow the fuzzy solution as in definition 2.1. Additionally,
Figure 3 shows that the RK56 solution over the all (25) given interval and for all the fuzzy level sets from 0
to 1 flow the fuzzy solution as in definition 2.1. Overall, as displayed in Table 1 and Finger 3 we conclude
that RK56 numerical solution with 10 iterations solution at x= 0.1 and for all 0 ≤ r ≤1 satisfy the fuzzy

6505
differential equations solution in the form of triangular fuzzy numbers. Figure 4 displayed the compression of
the RK56 and exact solutions of (25).
Table 1. RK56 solution and accuracy of (25) at 𝑥 = 0.1 with step size ℎ = 0.1
r RK56𝑦(0.1)(𝑟)
𝐸 RK56 RK56𝑦(0.1)(𝑟)
RK56𝐸
0 0.095310179804423 9.84490267086357 × 10−14
2.26236426450975 4.22675228151092 × 10−11
0.25 0.367783035656732 3.49276163547074 × 10−13
1.992946178637013 2.66238142643260 × 10−11
0.5 0.639761942376130 9.71445146547012 × 10−13
1.723143551330192 1.59825486178988 × 10−11
0.75 0.911268147598408 2.28594920770319 × 10−12
1.452940844005739 9.04853969529995 × 10−12
1 1.182321556798717 4.76285677564192 × 10−12
1.182321556798717 4.76285677564192 × 10−12
Figure 3. RK 56 numerical solutions of (25) for all 𝑟 ∈ [0,1] and 𝑥 ∈ [0,0.1]
Figure 4. RK 56 and exact solutions of (25) for all 𝑟 ∈ [0,1] and 𝑥 = 0.1
6. CONCLUSION
In this paper a computational approach has been presented and developed for solving fuzzy ordinary
differential equations. An approach is based on RK6 for solving nonlinear second order FIVP. A full fuzzy
analysis is presented to analyze RK56 to be suitable for solving the proposed problem. Numerical illustration
of non-linear second-order FIVP under fuzzy initial conditions demonstrate that the RKM56 is a capable and
accurate process with few steps. The findings also by RK56 satisfy the characteristics of fuzzy numbers in
triangular shape.
RK56
RK56
Exact
0.2 0.4 0.6 0.8 1.0
r level
0.0
0.5
1.0
1.5
2.0
y
RK56 & Exact

 ISSN: 2088-8708
6506
ACKNOWLEDGEMENTS
The authors of this research are extremely grateful that they have given us the Fundamental
Research Grants Scheme (FRGS) S/O No. 14188 provided by the Higher Education Ministry of Malaysia.
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