Investigation of A Fuzzy Problem by the Fuzzy Laplace Transform

This paper is on the solutions of a fuzzy problem with triangular fuzzy number initial values by fuzzy Laplace transform. In this paper, the properties of fuzzy Laplace transform, generalized differentiability and fuzzy arithmetic are used. The example is solved in relation to the studied problem. Conclusions are given.

In this paper, the solutions of a fuzzy problem with triangular fuzzy number initial values are investigated by fuzzy Laplace transform. Generalized differentiability, fuzzy arithmetic are used. Purpose of this study is to investigate solutions using fuzzy Laplace transform for the studied problem.
It is given in section 2 preliminaries, in section 3 findings and main results, in section 4 conclusions. Let R F show the set of all fuzzy numbers.
The parametric form [u α , u α ] of a fuzzy number satisfying the following requirements is a valid α-level set. u α is left-continuous monotonic increasing (nondecreasing) bounded on (0, 1], u α is left-continuous monotonic decreasing (nonincreasing) bounded on (0, 1], u α and u α are right-continuous for α = 0, Definition 4. [8,24,25] Let be u, v ∈ R F . If u = v + w such that there exists w ∈ R F , w is the Hukuhara difference of u and v, w = u v. (i) If the function f is (1)-differentiable, the lower function f α and the upper function f α are differentiable, If the function f is (2)-differentiable, the lower function f α and the upper function f α are differentiable, Theorem 3. [18,19]Suppose that f is continuous fuzzy-valued function on [0, ∞) and exponential order α and that f is piecewise continuous fuzzy-valued function on [0, ∞) .
If the function f is (1) Theorem 4. [18,19]Suppose that f and f are continuous fuzzy-valued functions on [0, ∞) and exponential order α and that f is piecewise continuous fuzzy-valued function on [0, ∞) .
If the functions f and f are (1) differentiable, if the function f is (1) differentiable and the function f is (2) differentiable, if the function f is (2) differentiable and the function f is (1) if the functions f and f are (2) differentiable, Theorem 5. [17,19]Let be f (t), g (t) continuous fuzzy-valued functions and c 1 and c 2 constants, then L (c 1 f (t) + c 2 g (t)) = (c 1 L ( f (t))) + (c 2 L (g (t))) .

Findings and Main Results
We study the problem In this paper, (i,j) solution means that u is (i) differentiable, u is (j) differentiable. Case 1) If u and u are (1) differentiable, since and using the fuzzy arithmetic and Hukuhara difference, yields the equations Using the initial values, we get From this, taking the inverse Laplace transform of the above equations, the lower solution and the upper solution are obtained as u α (t) = A α (1 − cos (t)) + B α cos (t) +C α sin (t) , u α (t) = A α (1 − cos (t)) + B α cos (t) +C α sin (t) .

Conclusions
In this paper, solutions of a fuzzy problem with symmetric triangular fuzzy number inital values are investigated by fuzzy Laplace transform. Generalized differantiability, fuzzy arithmetic are used. Example is solved. It is shown whether the solutions are valid α−level sets or not. If inital values are symmetric triangular fuzzy numbers, then the solutions are symmetric triangular fuzzy numbers for any time. .
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