Analytical solutions forfuzzysystem using power series approach

: The aim of the present paper is present a relatively new analytical method, called residual power series (RPS) method, for solving system of fuzzy initial value problems under strongly generalized differentiability. The technique methodology provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. Several computational experiments are given to show the good performance and potentiality of the proposed procedure. The results reveal that the present simulated method is very effective, straightforward and powerful methodology to solve such fuzzy equations.


Introduction
Fuzzy differential equations (FDEs) are extensively used in modeling of complex phenomena arising in applied mathematics, physics, and engineering, including fuzzy controltheory, quantum optics, atmosphere, measure theory and dynamical systems [1][2][3][4]. In many cases, data about these physical phenomena is pervaded under uncertainty, which may arise in the experiment part, data collection, measurement process as well as when determining the initial values. Most of the uncertain practical problems under the differential sense require the solutions of the corresponding FDEs which satisfy the given fuzzy initial conditions; therefore, fuzzy problems should be solved. In general, there exists no method that yields an explicit solution for FDEs due to the complexities of uncertain parameters involving these equations. Therefore, an efficient reliable computer stimulation is required. To deal with this in more realistic situations, FDEs are commonly solved approximately using numerical techniques [5][6][7][8][9].On the other hand, applications for different problems using numerical methods can be found in [10][11][12][13][14][15][16].
Series expansions are very important aids in numerical calculations, especially for quick estimates made in hand calculation. Solutions of the FDEs can often be expressed in terms of series expansions. However, the RPS technique is an analytical method for solving different types of ordinary and partial differential equations [17][18][19]. The methodology is effectiveand easy to construct power series solution for strongly linear and nonlinear systems of FIVPs without linearization, perturbation, or discretization.Different from the classical power series method, the RPS technique does not need to compare the coefficients of the corresponding terms and recursion relations are not required, which computes the coefficients of its power series by a chain of linear equations of -variable, where is number of equations in the given system. Thetraditional higher order Taylor series method is computationally expensive for large orders and suited for the linear problems, while on the other hand, the proposed method is an alternative procedure for obtaining analytic Taylor series solution of systems of FIVPs. By using residual error concept, we get a series solution, in practice a truncated series solution [20][21][22]. More specifically, we discuss and provide numerical approximate solutions for system of FIVP of the form subject to the initial conditions where ∈ 0 , 0 + , 0 , ∈ ℝ, : 0 , 0 + × ℝ → ℝ are nonlinear continuous functions in term of .The unknown functions , = 1,2, … , , of independent variable to be determined.Throughout thispaper, we assume that , = 1,2, … , ,are analytic functionsand satisfies all the necessary requirements for the existence of a unique solution.
This article is organized as follows. In the next section, we revisit brieflysome necessary definitions and preliminary results from the fuzzy calculus theory including the strongly generalized differentiability. Formulation for solving the system of FIVPs is presented in Section 3.Numerical experiments and simulation results are presented inSection 4. This article ends in Section 5 with some concluding remarks.

2.Fuzzy Analysis Theory
In this section, we present some necessary definitions and notations from fuzzy calculus theory which be used throughout the paper.  The metric structure on ℝ is given by the Hausdorff distance : ℝ × ℝ → ℝ + ∪ 0 such that , = sup 0≤ ≤1 max ( ) − ( ) , − ( ) for arbitrary fuzzy numbers and .
Definition 2.2. [24]Let , ∈ ℝ . If there exists ∈ ℝ such that = + , then is called the H-difference (Hukuhara difference) of and , and is denoted by ⊖ . Definition 2.3. [24] Let : , → ℝ and 0 ∈ , . We say that is strongly generalized differentiable at 0 , if there exists an element ′ 0 ∈ ℝ such that either i) for eachℎ > 0 sufficiently close to 0, the H-differences 0 + ℎ ⊖ 0 , 0 ⊖ 0 − ℎ exist and Here, the limit is taken in the metric space ℝ , and at the endpoints of , , we consider only one-sided derivatives. If is differentiable at any point ∈ , , then we say that is differentiable on , .Furthermore, we say that is(1)-differentiableon , , if is differentiable in the sense of( ) and its derivative is denoted 1 , while is (2)differentiableon , , if is differentiable in the sense of( ) and its derivative is denoted 2 .Frequently, we will write simply 1 and 2 instead of and , respectively, for each ∈ 0,1 .
Theorem 2.2. [23] Let : , → ℝ and put Here, we say that is continuous on , , if is continuous at each ₀ ∈ , such that the continuity is one-sided at endpoints of , , that is, is continuous on , if and only if 1 and 2 arecontinuous on , .

Fuzzy System Initial Value Problems
In this section, we study the system ofFIVPs under the concept of strongly generalized differentiability in which the fuzzy differential equation is converted into equivalent system of crisp system of IVPs for each type of differentiability. These can be done if the initial value is fuzzy number, the solution is fuzzy function, and consequently the derivative must be considered as fuzzy derivative.
Let be a ( )-solution, then by utilizing Theorems 2.2, we can thus translate the fuzzy system (1) and (2) into system of crisp DEs, hereafter, called the corresponding ( )-system. In some cases, we can't decompose the membership where , = , − 0 .
On the other hand, Res ∞ 0 = Res 0 = 0, for each = 1,2, … , .To obtain the values of the coefficients , , we need to solve the following algebraic system Forsecond unknown coefficient, ,2 ,differentiate both sides of formula (7) with respect to ,put = 2,and then substitute = 0 to get that This procedure can be repeated till the arbitrary order coefficients of RPS solutionsare obtained.
Here, we will let Rem denote the difference between and its th Taylor polynomial; that is, where the functionsRem are called the th-remainder for the Taylor series of .

Numerical Experiments
To show the simplicity and effectiveness of the proposed algorithm, semi-analytical solutions for a class of both linear and nonlinear inhomogeneous fuzzu IVPs is constructed using the RPS approach. The method is applied in a direct way without using linearization, transformation, or restrictive assumptions. The results reveal that the method is highly accurate, rapidly converge, and convenient to handle a various engineering problems in fuzzy calculus.

Example 4.1Consider the linear fuzzy IVPs in the form
subject to the fuzzy initial conditions subject to the fuzzy initial conditions subject to the fuzzy initial conditions

Conclusion
In this paper, a new analytic-numeric method, so-called RPSM, is proposed and applied to handle the both linear and nonlinear fuzzy system of IVPs. Numerical results reveal the complete reliability and efficiency of the proposed method with a great potential in scientific applications. It may be concluded that the RPSM is very powerful, straightforward, and promising technique in finding analytic approximate solution for wide classes of fuzzy IVPs.