Homotopy Analysis Method for Solving Fuzzy Fractional Volterra ‐ Fredholmintegro ‐ differential Equations

: The fuzzy fractional Volterra-Fredholm integro-differential equation is introduced by using the fuzzy Caputo derivative under the generalized Hukuhara difference, and the existence and uniqueness of the solution of this equation are proved by using the fixed point theorem. The homotopy analysis method is used to study the numerical solutions of linear and nonlinear fuzzy fractional integro-differential equations. Several numerical examples are given to illustrate the effectiveness and applica-bility of the method.


Introduction
The motivation of fractional calculus research is its application in biology, electrochemistry, fractal theory, control theory, fluid flow and viscoelasticity [1][2][3].In 1823, the application of fractional derivatives was introduced by Abel, who applied fractional calculus to solve the integral equation in the Tautochrone problem [4].The fixed point theorem is often used to study the existence and uniqueness of solutions of fractional calculus equations [5][6][7][8][9], and its numerical methods are also widely studied.Because of its wide application of fractional calculus equations, in many models, such as the novel coronavirus infection system model in recent years, researchers often face parameter uncertainties.In order to solve this problem, fuzzy concepts are introduced.Zadeh [10,11] introduced the concept of fuzzy numbers and its arithmetic operations, Mizumoto and Tanaka [12] further enriched the related concepts of fuzzy numbers.Dubois and Prade [13] introduced the concept of fuzzy function sets.The Hukuhara derivative of fuzzy valued functions and fuzzy initial value problems was proposed in [14] and studied in [15].In 1992, Liao proposed a general analytical method for solving linear and nonlinear problems by using the basic idea of homotopy in topology and differential geometry [16], which is called homotopy analysis method (HAM).Hamoud and Ghadle applied HAM to solve fuzzy Volterra-Fredholm integro-differential equations [17].Hussain [18] used HAM to study fuzzy integral-differential equations.Hanan [19] uses HAM to solve MultiFractional order integro-differential equations.Adomian decomposition method (ADM) [20,21], Laplace transform method [22,23], implicit finite difference method [24], variational iteration method [25] and other numerical methods are also applied to solve fuzzy integral, differential and integro-differential equations.In this paper, we mainly study HAM processing fuzzy fractional Volterra-Fredholm integro-differential equation.
In addition, the existence of solutions of fuzzy equations is also the focus of re-searchers.In 2015, Arshad et al [26] studied the existence and uniqueness of solutions for Riemann-Liouville fuzzy fractional differential equations.In [27], the existence and uniqueness of solutions for a class of fractional differential equations with fuzzy initial values defined in the sense of fuzzy Caputo were discussed.Ahmad and Ullah [28] also studied the existence and uniqueness of solutions of fuzzy fractional Volterra-Fredholm integrodifferential equation in the sense of Caputo.Allahviranloo et al. [29] studied the fuzzy fractional Volterra-Fredholm integro-differential equation in the sense of fuzzy Caputo derivative under the generalized Hukuhara difference(gHdifference), and proved the existence and uniqueness of the solution of the equation.Armand and Gouyandeh [30] introduced the existence and uniqueness of solutions of nonlinear fuzzy fractional Fredholm integral-differential equations under generalized fuzzy Caputo derivative.In this paper, We will discuss the existence and uniqueness of fuzzy fractional Volterra-Fredholm integro-differential equation solution in the sense of fuzzy Caputo under gH-difference.
and E denotes the fuzzy number space.Inspired by the above literature, the main contributions of this paper are as follows.We know that HAM is often used to solve linear and nonlinear differential-integral problems.In this paper, HAM is used to solve the approximate solutions of linear and nonlinear fuzzy fractional Volterra-Fredholm integro-differential equation, and the effectiveness and applicability of the method are illustrated by numerical examples.Another contribution is to prove the existence and uniqueness of solution by using fixed point theorem.Based on the parametric representation of fuzzy numbers, the set of one-dimensional fuzzy numbers can be regarded as a closed convex set in Banach space.
The structure of this paper is as follows: Section 2 mainly introduces the fuzzy definition and fixed point theorem.
Section 3 introduces the iterative scheme of HAM and its application in fuzzy fractional Volterra-Fredholm integrodifferential equation.In Section 4, the existence and uniqueness of solutions for fuzzy fractional Volterra-Fredholm integro-differential equation are studied.We provide several numerical examples and analysis of numerical results in Section 5. Finally, Section 6 gives a brief conclusion.

Preliminaries
In this section, we will give the related concepts of fuzzy numbers, and some important theorems and symbols used in the article.
Definition2.1.[31,32] R denotes the set of all fuzzy sets on R. Let h ∈ R , if h satisfies (i) h is a normal fuzzy set, i.e., there exists s ∈ R such that h s 1, (ii) h is a convex fuzzy set, i.e., h δs 1 δ s min h s , h s for all s , s ∈ R and δ ∈ 0,1 , (iii) h is an upper semi-continuous function, (iv) The closure of the support of h is compact, i.e., h is compact; then h is called as a fuzzy number.The set of all fuzzy numbers is known as the fuzzy number space, denoted by E.
Definition2.2.[33] Given 0 r 1 , a fuzzy number h in parametric form is represented by an ordered function pairs h r , h r satisfying (i) h r is a bounded left continuous non decreasing function, (ii) h r is a bounded left continuous non increasing function, (iii) h r h r .
For h h, h , v v, v ∈ E and δ ∈ R, the sum of v h and the scalar multiplication δh can be defined by v h r v r h r , v h r v r h r , ∀r ∈ 0,1 , and δh δh, δh , δ 0, δh, δh , δ 0.

Definition2.3. [29]
For any two fuzzy numbers w and h, defin where v v r , v r , h h r , h r .It has the following useful propertie.
(vi) D w s ds, h s ds D w s , h s ds, (vii) D w * h, 0 D w, 0 D h, 0 with the fuzzy multiplication * is based on the extension principle that can be proved by α -cuts of fuzzy numbers w, h ∈ E.Here 0 ∈ E is defined by (see [25]) 1, 0, 0( ) 0, .

Homotopy Analysis Method
To illustrate the basic idea of HAM, consider the following differential equation where N is a nonlinear operator and ( ) w x is an unknown function.
The following homotopy can be constructed when [ ( ; ), ( ), ( ), , ], q L x q w x qhH x N x q H x q w x H x h q where 0 ( ) w x is the initial conjecture of the exact solution ( ) , h H are auxiliary parameters and auxiliary functions, respectively.L is an auxiliary linear operator.When ( ) 0, [ ( )] 0 q  is the embedding parameter.Let homotopy Eq. ( 4) be zero, that is 0 1 [ ( ; ), ( ), ( ), , ] 0.
The Zero-order deformation equation from Eq.( 4)and Eq. ( 5) when 0 q  , from Eq.( 6) and when , the Eq.( 6) is equivalent to It can be obtained from Eq.( 7) and Eq.( 8) that as the embedding parameter q increases from 0 to 1, ( ; ) x q  changes continuously from the initial conjecture 0 ( ) w x to the exact solution, which is called homotopy deformation.
Using Taylor 's theorem, ( ; ) x q  is expanded to the following power series with respect to q 0 1 ( ; ) ( ) ( ) , where By choosing appropriate , etc.to make ( ; ) x q  converge at 1 q  , then under these assumptions, we have the following series solution Substituting Eq.( 9) into Eq.( 6) yields Simplifying Eq.( 12) yields 0 1 , , The n-th differential of Eq (13) with respect to q is calculated and assigned at 0 q  . 1 1 Therefore where Application of HAM to fuzzy fractional Volterra-Fredholm integro-differential equations.
x N x r q D x r q g x r a x x r q where 0 0 ( , ; ) [ ( , ; ), ( , ; )], ( , ) [ ( , ), x r q x r q x r q w x r w x r x r q w x r qhN x r q And meet the following initial conditions 0 ( , ) (0, ; ) (0, ) w x r r q w r    The next steps are similar to (3)- (19), and now by selecting the appropriate parameters, we can make converge ( , ; ) x r q  to 1 q  , and obtain the following series solution where .
Acting on operator J  on both sides of Eq. ( 20), there is

Existence and Uniqueness of the Solution
In this section, we will give the existence and uniqueness results of the solutions of Eq. ( 1), and prove it.The following assumptions are given before proof.

D F w x r F w x r L D w x r w x r D F w x r F w x r L D w x r w x r
  H(3) There exist two continuous functions 1 0, Next, the uniqueness of the solution of the equation will be proved using Balach's fixed point theorem.

Tw x r w r x s g s r ds
x s a s w s r ds ( 1,2).
( 1) The above equation shows that T is a compressive map, and the equation has a unique solution by Banach's fixed point theorem.

Numerical Results
Example1 Consider the application of HAM to solve the following linear fuzzy fractional Volterra-Fredholm integrodifferential equations The exact solution is x n dt x t w t r dt The left and right boundary errors between the approximate solution and the exact solution are obtained in Table 1 and Table 2.    31 and Table 4. From the error results of Table 1-Table 4, the error between the exact solution and the approximate solution is small, so HAM is effective and practical for dealing with linear and nonlinear fuzzy fractional integral and differential problems.

Conclusion
In this paper, a class of fuzzy fractional order mixed integro-differential equations is studied.The fixed point theorem is used to prove the existence and uniqueness of the solution.The appropriate auxiliary parameters and initial conditions are selected to solve the fuzzy fractional Volterra-Fredholm integro-differential equation by using the method of homotopy analysis.The validity and applicability of the method can be obtained by several numerical examples and their results.In the future, the fuzzy nonlinear fractional Volterra-Fredholm integral-differential will be studied.Based on the HAM, the nonlinear term will be approximated and its stability will be studied.
Let the definition of operator : [0,1] sr w sr d s left and right boundary errors between the approximate solution and the exact solution are obtained in Table

Table 1 .
left bound of error

Table 2 .
right left bound of error

Table 3 .
left bound of error

Table 4 .
right left bound of error