A Proposed Analytical Method for Solving Fuzzy Linear Initial Value Problems

In this article, we aim to define a universal set consisting of the subscripts of the fuzzy differential equation (5) except the two elements 0 and n , subsets of that universal set are defined according to certain conditions. Then, we use the constructed universal set with its subsets for suggesting an analytical method which facilitates solving fuzzy initial value problems of any order by using the strongly generalized Hdifferentiability. Also, valid sets with graphs for solutions of fuzzy initial value problems of higher orders are found.


Introduction:
The concept of the fuzzy derivative was first introduced by Chang and Zadeh (1), it was followed up by Dubios and Prade (2), and Puri and Ralescu (3). The fuzzy Laplace transform (FLT) is proposed to solve first order fuzzy differential equations (FDEs) by using the strongly generalized differentiability concept (4), and then some of wellknown properties of the fuzzy Laplace transform were investigated. In addition, an existence theorem was given for fuzzy-valued function which possesses the fuzzy Laplace transform (5). A formula of the fuzzy Laplace transform of the nthorder derivative was initially introduced in terms of the number of derivatives in form (ii) by Mohammad Ali ( 6), and Haydar and Mohammad Ali (7), it was followed by introducing another formula for the fuzzy Laplace transform on fuzzy nth-order derivative by concept of the strongly generalized differentiability (8). In the direction of solving n-th order FDEs numerically, many efforts have been introduced by a number of authors (9)(10)(11). So far, a few number of works have been introduced in the subject of finding the analytical solutions of FDEs for example (12)(13)(14). Also, some analytical methods for solving fuzzy differential equations are introduced in (15). Recently, approximated solutions of fuzzy initial value problems have been studied such as (16) and (17). In this paper, we extend the proposed method by Mohammed (18), for solving nth-order classical differential equations by the classical Laplace transform to solve nth-order FDEs by FLT under the strongly generalized H-differentiability. Also, we introduce theorem and some corollaries that help us in solving nth-order FDEs.
This paper is organized as follows: Basic concepts are given in Section 2. In Section 3, a new method for solving FIVPs of nth-order is introduced with some results. In Section 4, examples of several FIVPs are solved to show the activity of the method. In Section 5, discussion and conclusions are given.

Basic concepts
In this section, we are going to recall some basic concepts that we need in this paper.
ii. For all sufficiently small, Ө Ө and the limits (in ; .    where q is the number of (ii)-differentiable functions f ( i ) when ik  .
(2) If m is an odd number, we have: where q is defined as in (1) above.

A suggested method for solving nth-order fuzzy linear initial value problems
In this section, we are going to introduce a theorem and some corollaries which can be used for solving FIVPs of the nth-order.
where q is defined as in equation (1) above and "" o in the above subscript refers to the word ordinary for 1, 2,..., . in  Therefore, we note that: Suppose that we have the following FDE: satisfies the following relation: if and , nf are defined in equations (1) and (3) respectively and () up is a polynomial of p whose degree is less than or equal to 1 n  .
Proof. The FDE (5) can be written as: We suppose that 0 n is the number of derivatives in form (ii) which is less than the ith derivative for 1 1, ii  k n is the number of derivatives in form (ii) which is less than the ith derivative for is an even number. Therefore equation (7) can be written as follows: q is an even number, q is an even number, ( We take FLT to both sides of equation (8): [ Then, we can get: up is a polynomial of p its degree is less than or equal to 1 n  . It is clear that equation (12) can be written as: if is defined as in equation (1).
Similarly, if m is an odd number, (i.e., 1 m  is an even number and 2 m  is an odd number). Therefore, equation (7) can be written as follows: By taking FLT to both sides of the above equation and making some simplifications, we get: where , if is defined as in equation (1). It is clear that equations (13) and (14) can be written as in equation (6). By equation (6) io is defined as in equation (2) , where q is defined as in equation (1).
Also, we note that if m is an even number then yt are given as follows:  (4) and (2)

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, where q is defined as in equation (1).
Also, we note that if m is an even number then  yt are given as follows:    (22) and (23) can be written as in (18). We note that if we have 0    in Corollary 5, then we must put 0 a  instead of 0 a  in (18).

Illustrative examples
In this section, we shall introduce several examples by using the proposed method. Example 1. Consider the following FIVP:

Discussion
In this paper we suggest a method for finding analytical solutions for FIVPs of higher orders by using fuzzy Laplace transform by the concept of strongly generalized H-differentiability. This method depends on introducing the subscripts which appear in the fuzzy differential equation as a universal set , and defining other subsets which form a partition of that universal set. Several rules have been given for solving FIVPs directly by obtaining easy algebraic systems. For a FDE, multiple exact solutions can be found by the concept of strongly generalized H-differentiability, but the level sets are not necessarily identical for all the exact solutions even for the same FIVP as follows:

Conclusion:
An analytical method for solving FIVPs is introduced by using fuzzy Laplace transform and the concept of strongly generalized Hdifferentiability. The suggested method depends on defining a universal set and some certain subsets of that universal set for each FIVP. Also, multiple exact solutions can be found such that the level sets of solutions may be various for the same FIVP.