^{1}

^{2}

^{3,4,5}

^{1}

^{2}

^{3}

^{4}

^{5}

Fuzzy and fractional differential equations are used to model problems with uncertainty and memory. Using the fractional fuzzy Laplace transformation we have solved the fuzzy fractional eigenvalue differential equation. By illustrative examples we have shown the results.

Fractional calculus is the generalization of the standard calculus. That involves the derivative of functions to arbitrary orders. But the fractional derivatives are nonlocal so they provided the mathematical models to non-Markov processes and memory processes. Fractional calculus has found many applications in science, engineering, and so forth [

This paper is arranged in the following manner.

After an introduction to the present work, in Section

Fractional calculus deals with generalizations of integer order derivatives integrals to arbitrary order. In this section we present basic definitions and properties which will be used in the subsequent sections [

A fuzzy number is a fuzzy set

there are real numbers

The set of all the fuzzy numbers (as given in Definition

A fuzzy number

A popular fuzzy number is the triangular fuzzy number

Its parametric form is

Triangular fuzzy numbers are fuzzy numbers in

Initial value problems are considered in fractional differential equations and solved by analytical and numerical methods [

Let us consider the fractional equation with fuzzy condition; that is,

As a pursuit of fractional fuzzy differential in the following section we generalized fuzzy Laplace transformation method to fractional fuzzy Laplace method. Now, we solve illustrated examples in the subsequence sections.

Consider the following fuzzy fractional eigenvalue differential equations as

Now, we solve these equations according to the two following cases, using the generalized fractional fuzzy Laplace transform (FFLT). The equation with lower functions is

Now, we use the FFLT for solving (

After using Laplace transform on (

Thus, we have

Taking the fuzzy inverse Laplace transform we obtain

In a similar manner we are led to

Therefore, the general solution will be

In Figures

Let us consider the following fuzzy fractional differential equation as

Applying Laplace transform on (

In view of (

Also, by taking inverse Laplace transform of (

Likewise, by doing the same calculation (

Therefore, we have the final solution

In Figures

Consider the following fuzzy fractional differential equations with fuzzy Caputo initial condition as

So (

Applying Laplace transform and inverse Laplace transform on (

Therefore, the general solution will be as follows:

Figures

Suppose the following fuzzy fractional differential equation with fuzzy initial Riemann-Liouville condition:

So its lower and upper functions equations are

Using the same manner we get the solutions

Finally, we obtain general solution

Figures

Let us consider the fuzzy fractional differential equation involving fuzzy Riemann-Liouville initial condition:

Equation (

The solutions for (

And the general solution will be as

Figures

In this work, we have generalized the fractional Laplace transformation to the fuzzy fractional Laplace transformation. Then, we have solved the fractional fuzzy differential equation using suggested fuzzy fractional Laplace transformation. Riemann-Liouville and Caputo fractional derivatives were used in the fractional fuzzy differential equations. Moreover, Liouville and Caputo fractional initial condition is chosen in the example to show the difference. The illustrated graphs present the difference between fuzzy, fractional, and ordinary differential equations.

The authors declare that there is no conflict of interests regarding the publication of this paper.