International Journal of Advanced Mathematics V.1, N.1, 2017, pp.18 to 25.
SOLUTION OF FUZZY INTEGRODIFFERENTIAL EQUATION BY FUZZY LAPLACE TRANSFORM METHOD S. Priyadharsini1, P. Gowri2 and T. Aparna3 1 Department of Mathematics Sri Krishna Arts and Science College Coimbatore, INDIA e-mail:
[email protected] 2
Department of Mathematics Sri Krishna Arts and Science College Coimbatore, INDIA e-mail:
[email protected] 3
Department of Mathematics Sri Krishna Arts and Science College Coimbatore, INDIA e-mail:
[email protected] Abstract This work establishes method of finding solution of fuzzy linear integrodifferential system. In this article fuzzy integrodifferential equation under fuzzy initial condition is solved by using fuzzy Laplace transformation. The concept of generalized Hdifferentiability is used. Finally, some numerical illustrations are provided to explain the proposed theory. Keywords: Fuzzy numbers, integrodifferential equation, Fuzzy Laplace transformation AMS 2010 subject classification: 94D05, 34A07. 1
Introduction The subject of fuzzy integrodifferential equations is one of the most powerful mathematical tools for modeling uncertainty and for processing vague or subjective information in mathematical models. It has enormous applications in many physical problems. These types of fuzzy differential equations arise in the mathematical modeling of various physical phenomena. The concept of fuzzy sets and set operations is a field of mathematical study which was first introduced by Zadeh [8].
1∗
Corresponding Author - Priyadharsini S © Tyrex Publishing; all rights reserved.
The generalized derivative is defined for a larger class of fuzzy number valued functions than Hukuhara derivative. Hence, this differentiability concept is used in the present paper. Under suitable conditions, the fuzzy initial value problem considered has locally two solutions. The existence and uniqueness theorems for fuzzy differential equations can be found in [5,6]. Fuzzy integrodifferential equations (FIDEs) are used to model the system with uncertainty in dynamical environments. When a physical system is modeled under the differential sense; it finally gives a fuzzy differential equation or a fuzzy integrodifferential equation and hence, the solution of integrodifferential equations have a major role in the fields of science and engineering. We cannot give a perfect model, while transforming a real world problem into a deterministic ordinary differential equation. For example, the initial values may contain uncertain parameters. For the initiation of this aspect of fuzzy theory, has been introduced. Consequently the study of the theory of FIDEs has recently been growing rapidly. In this paper, fuzzy Laplace transform technique is used to compute the solution of fuzzy integrodifferential equation. This work is organized as follows. In section 2, some basic definitions and results are given. In section 3, Fuzzy Laplace transform method for solving fuzzy integrodifferential equation is proposed. In section 4, some examples are implemented to illustrate the proposed theory. Finally, conclusion is drawn in section 5. 2 Preliminaries In this section, we shall present some basic definitions of Fuzzy sets including the definition of fuzzy numbers and fuzzy functions. Definition: Fuzzy number An arbitrary fuzzy number is represented by an ordered pair of functions (
); 0≤r≤1 which
satisfy the following conditions. 1. (r) is a bounded monotonic increasing left continuous function; 2.
(r) is a bounded monotonic decreasing left continuous function;
3.
(r)≤ (r), 0≤r≤1.
Definition: Membership function Let X be any set of elements. A fuzzy set : X [0,1] written as the set of points. ={(x,
X))|x∈X, 0≤
(X)≤1}
is characterized by a membership function
Definition: Fuzzy Laplace Transformation The fuzzy Laplace transform of a fuzzy real valued function f(t) is defined as follows: F(s)=L{f(t)}=∫
dt=
∫
dt
whenever the limit exist.
The symbol L denotes fuzzy Laplace transformation, which acts on fuzzy real valued function f=f(t) and generates F(s)=L{f(t)}. The lower and upper Laplace transform of a fuzzy real valued function f(t) are given as follows: F(s,α)=L{f(t,α)}=[l{ (t,α)}, l{ (t,α)}]
Where,
l{ (t,α)}= ∫
(t,α)dt
∫
(t,α)dt , 0≤α≤1
t,α)}= ∫
(t,α)dt
∫
(t,α)dt , 0≤α≤1
l{
Definition (8): Fuzzy convolution theorem [5] The convolution of two fuzzy real valued functions f,g defined for t≥0 by (f*g)(t)- ∫ Theorem 1: If f and g are piecewise continuous fuzzy real valued function on [0, ) with exponential order p, then L{(f*g)(t)}=L{f(t)}L{g(t)}=F(s)G(s), s>p. 3. Fuzzy Integro-Differential Equation: In this section, solution of fuzzy integro-differential equations with separable kernels is investigated. Let y(t) be a fuzzy-valued function to be solved for, f(t) is given known function and k(t,s) is a known real-valued integral kernel. The fuzzy integrodifferential equation explain in the form y(n)(t)=f(t)+∫
------------------------------------------------------
where f(z):R to E and x [a,b], b< MAIN METHOD Consider the following fuzzy integro-differential equation: y(n)(t)=f(t) ∫
, with y(k)(0)= =(
Where y(n) denotes the nth derivative of y.
,
);
0≤ k ≤n-1.
(1)
Taking fuzzy Laplace transformation on both sides of equation (1), we get L[y(n)(t)]= L[f(t)]+L[∫ By definition of fuzzy Laplace transformation and using fuzzy convolution theorem, we have, snL[y(t)]-sn-1y(0)-sn-2y΄(0)-…........yn-1(0)
= L[f(t)]+L[∫
This implies that, snl{ (t;α)}-sn-1
-sn-2
……
=l{
, 0≤ α ≤1
-sn-2
…..
=
, 0 ≤α ≤1
And Snl{ (t;α)}-sn-1
Now we discuss the following cases: Case 1: if y(t;α) is positive, then =l{
}l{
}
=l{
}
Case 2: if y(t;α) is negative, then =l{
}l{
}
=l{
}l{
}
snl{ (t;α)}-sn-1
-sn-2
……
=l{
snl{ (t;α)}-sn-1
-sn-2
…..
=
l{ }+ l{
}l{
},0≤α≤1 and },0≤α≤1
In compact form: l{ (t;α)}= l{ (t;α)}=
, 0≤ α ≤1 and
(
{
})
, 0≤ α ≤1
By taking the inverse of fuzzy Laplace transformation on both sides of above relations we can easily obtain the value of (t;α) and (t;α) where 0≤α≤1. 4. Examples In this section, some examples are solved by using fuzzy Laplace transform method, which is explained in section 3. Solutions are plotted by using MATLAB.
Example1: Consider the first order FIDE y΄+2y+∫
dt=0 with y(0)=(r-1, 1-r).
By taking Laplace transform on both sides, we get, sL[y]-y(0)+2L[y]+ L[y]=0 Here, L[1*y(t)]=L[1].L[y] On simplifying we get, l( )=
, l( )=
By taking inverse Laplace transform, we have = (r-1)sin(t) and = (1-r)sin(t) Actual solution and fuzzy solution for r=0.5 are plotted in fig 1. Comparison between fuzzy and actual solution 1 y1 y2 y
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6 Actual solution -0.8
-1
1
2
3
4
5
6
7
8
9
10
Fig 1 Example 2: Consider the second order FIDE y-y+∫
ds, with initial condition y(0)=(0,0) & y΄(0)= (r-1, 1-r)
By taking fuzzy Laplace transform on both sides, we have s2L[y]-sy(0)-y΄(0)-L[y]+L[sin(t)].L[y]=0 Here, L[sin(t)*y(t)]=L[sin(t)].L(y) On simplifying we get, s2l( )-l(
+
l( )=(r-1)
and
s2l( )-l( )+
l(
l( )= (1-r)
and l( )= (r-1)
=(1-r) where, 0≤ r ≤1
By taking inverse Laplace transform we have (
)
Solution of this problem plotted in fig 2 for different values of r. Here r=0.3, 0.5 and 0.8 is considered. Solution when r=0.3, 0.5 and 0.8 150 r=0.3 r=0.5 r=0.8 r=0.8 r=0.5 r=0.3
100
50
0
-50
-100
-150
0
1
2
3
4
5
6
7
8
9
10
Fig 2 Example 3: Consider the second order FIDE with damping y-y΄+y=aʘt+∫
ds, where a=(α+2,4-α) with the initial condition
y(0)=(α+1,3-α), y΄(0)= (α,2-α). By taking Laplace transform, we have [s2L(y)-sy(0)-y΄(0)]-[sL[y]-y(0)]+L[y] = aL[t]+L[t].L[y] Since, L[aʘt]= a L[t] & L[t*y]=L[t].L[y]. This implies, l( )=
+
+
l( )=
+
+
By taking inverse Laplace transform, we get = (α+2)sin(t)+(α+1) -cos(t) =-α sin(t)+(3-α)
-cos(t).
Here =0.4 is considered. Solutions
and y are plotted in fig 3.
4
6
x 10
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
10
Fig 3 5. Conclusion: In this work, fuzzy integrodifferential equation is basically solved by using fuzzy Laplace transform method. To show the applicability of the method, some examples are presented. Also, crisp solution and fuzzy solution of the system is compared by using MATLAB. Reference: [1] Abood, M.F., Linkens, D.A. and Mahfof, M., “survey of utilization of fuzzy Technology in Medicine and Healthcare”, Fuzzy sets and systems, 120 (2001), 331 - 349. [2] Data, D.P., “The Golden Mean, Scale Free Extension of Real Number System”, Fuzzy Sets System in Physics and Biology, Chaos, Solutions and Fractals, 17 (2003), 781 – 788. [3] El-Nachie, M.S., “On a Fuzzy Kahler Manifold which is Consistent with Slit Experiment”, International Journal of Nonlinear Science and Numerical Simulation, 6 (2005), 95-98. [4] Y. Feng, The solutions of linear fuzzy stochastic differential systems, Fuzzy Sets and Systems, 140 (2003), 541-554 [5] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301-317. [6] V. Lupulescu, On a class of fuzzy functional differential equations, Fuzzy Sets and Systems, 160 (2009), 1547-1562. [7] S. Priyadharsini, V. Parthiban and A. Manivannan, Solution of Fractional Integrodifferential systems with fuzzy initial conditions, International Journal of Pure and Applied Mathematics, 8 (2016), 107 – 112. [8] Zadeh L.A.Zadeh, Fuzzy sets, Inform and control 8 (1965), 338 – 353.
Priyadharsini S Prof. Priyadharsini S is working in Sri Krishna Arts and Science College, Affiliated to Bharathiar University, Coimbatore, Her areas of interests are mathematics, Differential equations, Fluid Mechanics, and Fuzzy Mathematics. She published more than 10 research papers. She obtained the Dr. Padmini Adiseshiah Award and Sadhanai Aarvalar Award. Gowri P Prof. Gowri P is working in Sri Krishna Arts and Science College, Affiliated to Bharathiar University, Coimbatore, Her areas of interests are mathematics, Differential equations, Fluid Mechanics, and Fuzzy Mathematics. She published more than 10 research papers. Aparna T Prof. Aparna T is working in Sri Krishna Arts and Science College, Affiliated to Bharathiar University, Coimbatore, Her areas of interests are mathematics, Differential equations, Graph Theory, and Fuzzy Mathematics. She published more than 10 research papers.
