Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

: In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.


Introduction
Fractional calculus has been concerned with integration and differentiation of fractional (noninteger) order of the function. Riemann and Liouville defined the concept of fractional order intgrodifferential equations [1]. Fractional calculus has developed an extensive attraction in current years in applied mathematics such as physics, medical, biology and engineering [2][3][4][5][6][7][8]. Whenever dealing with the fractional integro-differential equation many authors consider the terms Caputo fractional derivative, Riemann-Liouville and Grunwald-Letnikvo [9][10][11][12][13]. The subject fractional calculus has many applications in widespread and diverse field of science and engineering such as fractional dynamics in the trajectory control of redundant manipulators, viscoelasticity, electrochemistry, fluid mechanics, optics and signals processing etc.
Fractional integro-differential equations having some uncertainties in the form of boundary conditions, initial conditions and so on [14][15][16]. To resolve these type of uncertainties mathematicians introduced some concepts fuzzy set theory is one of them.
Aim of our study is to extend OHAM for solution of system of fuzzy Volterra integro differential equation of fractional order of the following form represents the fuzzy fractional derivative in Caputo sense for fractional order of  with respect to x , : [ , ] h a b → F is fuzzy valued function, ( , ) k x t is arbitrary kernel 0 () ux  F is an unknown solution.
F represent set of all fuzzy valued function on real line. The remaining paper is structured as follows: A brief overview on some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus are discussed in section 2. Analysis of the technique is presented in section 3. Proposed method is applied to solve fuzzy fractional order Volterra integro-differential equations in section 4. Result and discussion of the paper is given in section 5 and section 6 is the conclusion of the paper.

Preliminaries
In literature there exist various definitions of fuzzy calculus and fuzzy fractional calculus [50]. Some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus related to this study are provided in this section.
Definition 2.2. Caputo partial fractional Derivative operator x D  of order  with respect to x is defined as follow [50]: which clearly shows that  6. A fuzzy real valued function 12 , : [ , ] , ab  → then in [54]: with the given initial condition The homotopy of OHAM [41][42][43], constructed as follow:    Putting Eq (3.9) into Eq (3.1), we can found our residual given as follow: Optimum solution contains some auxiliary constants; the optimal values of these constants are obtained through various techniques. In the present work, we have used the least square method [56,57]. The method of least squares is a powerful technique for obtaining the values of auxiliary constants. By putting the optimal values of these constants in Eq (8), we obtain the OHAM solution.
.   Figure 5 shows the comparison of OHAM with the exact solution at different values of and r=0.5 while Figure 6 shows the comparison of OHAM with the exact solution at different values of r and =0.5 for problem 2.

Conclusions
In the research paper, a powerful technique known as Optimal Homotopy Asymptotic Method (OHAM) has been extended to the solution of system of fuzzy integro differential equations of fractional order. The obtained results are quite interesting and are in good agreement with the exact solution. Two numerical equations are taken as test examples which show the behavior and reliability of the proposed method. The extension of OHAM to system of fuzzy integro differential equations of fractional order is more accurate and as a result this technique will more appealing for the researchers for finding out optimum solutions of system of fuzzy integro differential equations of fractional order.