World Applied Sciences Journal 17 (12): 1667-1674, 2012
ISSN 1818-4952
© IDOSI Publications, 2012
Corresponding Author: Ali F. Jameel, School of Mathematical Sciences, 11800 USM, University Science Malaysia, Penang,
Malaysia
1667
Numerical Solution of Fuzzy IVP with Trapezoidal and
Triangular Fuzzy Numbers by Using Fifth Order Runge-Kutta Method
Ali F. Jameel, Ahmad Izani Md. Ismail and Amir Sadeghi
School of Mathematical Sciences, 11800 USM, University Science Malaysia, Penang, Malaysia
Abstract: In this paper we formulate a fifth order Runge-Kutta method for solution of fuzzy linear initial
value problem involving ordinary differential equations with trapezoidal and triangular fuzzy numbers. We
conduct an error analysis and perform numerical experiments. It is shown that method yielded very
accurate results
Key words: Fifth order runge-kutta method • fuzzy differential equations • fuzzy initial value problem •
fuzzy trapezoidal and triangular numbers
INTRODUCTION
Fuzzy differential (FDEs) equations are used in
modeling problems in many applications including
engineering and sciences, population models [13, 19],
quantum optics, gravity [12] and medicine [5, 7]. An
Initial Value Problem (IVP) involving first order linear
FDE can be considered to be the simplest case to test
the effectiveness of proposed methods for solving FDE.
Taylor methods have been used to solve this equation in
[2], Euler method in [6, 17, 18, 23], Modified Euler in
[15], second order Runge Kutta in [3], third order
Runge-Kutta in [4.10] and fourth order Runge-Kutta in
[1, 21]. In this paper, we consider the use of the fifth
order Runge-Kutta method to solve an IVP involving a
fuzzy linear first order differential equation. We will
start with some preliminary concepts about fuzzy
numbers and fuzzy differential equations.
PRELIMINARIES
Fuzzy numbers are a subset from the real numbers
set and represent uncertain values. Fuzzy numbers are
linked to degrees of membership which state how
true it is to say if something belongs or not to a
determined set.
A trapezoidal fuzzy number is defined by four real
numbers a<ß< ? <?? [10]. The base of the trapezoid is
the interval [a, d] with vertices at x = ß, x = ?. A
trapezoidal fuzzy number will be denoted by µ= (a, ß,
?, ?? ), the membership function is defined as the
follows:
α
β x
≥ (x)
1
0
0.5
γ
δ
Fig. 1: Trapezoidal fuzzy number
(x;a,ß,?,d) =
0, if x
x
, if x
1, if x
x
, if x
0, if x
<α
−α
α≤ ≤β
β−α
β≤ ≤γ
δ−
γ≤ ≤δ
δ−γ
>δ
(2.1)
Note that:
(1) µ > 0 if a > 0;
(2) µ > 0 if ß > 0;
(3) µ > 0 if ? > 0; and
(4) µ > 0 if ?? > 0.
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 [24]. The r-level set
of trapezoidal fuzzy number can be defined as
follows: