Efficient Algorithm for Solving Fuzzy Singularly Perturbed Volterra Integro-Differential Equation

: In this paper, we design a fuzzy neural network to solve fuzzy singularly perturbed Volterra integro-differential equation by using a High Performance Training Algorithm such as the Levenberge-Marqaurdt (TrianLM) and the sigmoid function of the hidden units which is the hyperbolic tangent activation function. A fuzzy trial solution to fuzzy singularly perturbed Volterra integro-differential equation is written as a sum of two components. The first component meets the fuzzy requirements, however, it does not have any fuzzy adjustable parameters. The second component is a feed-forward fuzzy neural network with fuzzy adjustable parameters. The proposed method is compared with the analytical solutions. We find that the proposed method has excellent accuracy in findings, a lower error rate

Werren McCulloach and Walter Pitts in 1943 [1].In 1949, Donald Hebb developed a neuropsychological theory and its rule [2] that is known as Hebbian learning [3].The first neurocomputer was created in 1950 by Marvin Minsky and Dean Edmonds [4].We can consider 1956 as the year of artificial intelligence's birth, when McCarthy, Minsky, Rochester, and Shavron, AI pioneers conducted a summer symposium on AI at Dartmouth [5], [6].
Many physical and biological problems include the perturbed Volterra integro-differential and integral equation (see, e.g., [7], [8], [9]).In [10] , the authors provide a survey of singularly perturbed Volterra integral and integro-differential equations.When the perturbed parameter approaches zero for singly perturbed situations, the diameter of the boundary layer narrows.
Nonlinear FSPVIDE plays an important research material in science and engineering.In a fuzzy framework, these equations provide a usual system to simulate the doubt of energetic structures in several logical domains such as physics, geography, medicine, biology, applied mathematics, biophysics, quantum physics, medicine, bio-informatics, and gravity.It is frequently difficult to discover analytic solutions to these problems.Several mathematicians have investigated the numerical solutions to fuzzy equations in recent years [11], [12], [13], [14], [15], [16].
Scientists and engineers have recently conducted extensive research on the Adomian decomposition method (ADM) that is used to solve nonlinear differential and integral problems [11].Adomian [17] invented the ADM for solving several types of functional equations which is the theme of considerable approximation and analytical study.
In this paper, we consider the numerical discretization of fuzzy singularly perturbed Volterra integro-differential equations (FSPVIDE) in one and two dimensions, respectively as follows: With a fuzzy boundary condition, is called the perturbation parameter such that , F and K are given smooth functions on [0, X], .
The purpose of this paper is to design a neural system to solve a type of perturbation problem in the integral equations numerically (FSPVIDE) and to compare it with the exact solutions.
The paper is structured as follows.In section 2, basic concepts of fuzzy sets are given and we explain some important definitions.In section 3, a fuzzy integro-differential equation is discussed, we also explain the general form of the first order fuzzy Volterra integrodifferential equation.Fuzzy Neural Network is given in section 4. In section 5, we describe the fuzzy neural network architecture that is used in this paper through the structure of FNN.In section 6, we solve the Fuzzy Singularly Perturbed Volterra integro-differential equation.
An illustration of the method is done in section 8.In section 2, Numerical Results are presented to illustrate the numerical results of the method by taking two examples.Finally, we show the most important results of the paper through conclusions in section 9.

Basic concepts of fuzzy sets:
In this section, we provide several fundamental concepts related to fuzzy theory.
Definition 1 [16]: If U is a collection of substances, then a fuzzy set in U is a set of ordered pairs: {( ( )) ( ) }. ( ) is known as the membership function or grade of membership of in , it is also known as the degree of compatibility or degree of truth, which maps U to the membership space ( ).A is nonfuzzy when ( ) contains only the two points 0 and 1 and ( ) is the same as a nonfuzzy set's characteristic function.
Definition 2 [17] : The support of a fuzzy set , S( ) is the crisp set of all as well as ( ) > 0 and denoted S( ).
Definition 3 [18]: The crisp set of elements from the fuzzy set at least to the degree is called the -level set: { ( ) }. { ( ) } is called the strong -level set or the strong -cut.
Definition 4 [18] : . Alternatively, a fuzzy set is convex if all its -level sets are convex.
Definition 5 [19]: A fuzzy number is entirely determined by an ordered pair of functions ( ( ), ( )), 0 ≤ ≤ 1, which satisfies the following: The crispy number a is just signified by: ( ) = ( ) = a, 0 ≤ ≤ 1.The set of all the fuzzy numbers is denoted by .
Remark (), [19]: For subjective ( , ), ( ) and R, the addition and multiplication by C can be defined as follows: [20]: The distance function between arbitrary two fuzzy numbers ( , ) and ( ) is given as follows: Definition 6, [19] : The function : R ⟶ is called a fuzzy function.Also, we call every function that is defined from ⊆ into ⊆ by a fuzzy function.Definition 7, [19]: The fuzzy function : R ⟶ is said to be continuous if: For an arbitrary R and > 0 there exists an > 0 such that: |u -| < ⇒ ( (u), ( )) < ϵ, where is the distance function between two fuzzy numbers.Definition 8, [19]: Let I be the real interval.The -level set of the fuzzy function : I → can be denoted by: ( ) ( ) ( ) , x I , [0,1] The Seikkala derivative ˊ(u) of the fuzzy function (x) is defined by:

3.Fuzzy integro-differential equations
The first order fuzzy Volterra integro-differential equation is given by ,where and are fuzzy number in , is the kernel function over the rectangular region and is a specified function of x .If is a fuzzy function, ( ) is a given fuzzy function and is the fuzzy derivative of , this equation may only possess a fuzzy solution.Let ( ) ( ) ( ) be a fuzzy solution of the first order of FSPVIDE , therefore, by Definition 5 and Definition8 we have the equivalent system ( ) . Suppose ( ) is continuous in and changes its sign in finite points for fix t. [21] An FNN is a learning machine which is also known as a neuro-fuzzy system that describes the parameters of a fuzzy system by using neural network approximation techniques.There are some things in common between the neural networks and neuro-fuzzy systems.They are used to solve a problem, for example, pattern recognition, regression, or density estimation, if there is no mathematical model of the problem.They only have specific disadvantages and advantages that are almost entirely eliminated by accomplishing both notions.

4.Fuzzy Neural Network(FNN)
Neural networks can only be used if the problem is stated by a large enough number of observed examples.These observations are used to train the black box if there is no required prior knowledge of the issue.On the other hand, it is difficult to extract comprehensible rules from the structure of the neural network.
A fuzzy system, on the other hand, requires language rules rather than acquiring examples as prior knowledge.In addition, the input and output variables must be linguistically characterized.If the knowledge is partial, incorrect, or contradicting, the fuzzy system must be tweaked.Because there is no formal approach, tweaking is done heuristically.This is frequently time-consuming and error-prone.
If all of the variable parameters (weights and biases) are fuzzy numbers, the FNN is said to be completely FNN; otherwise, it is said to be partially FNN. Figure 1 depicts FNN with an input layer, a single hidden layer, and an output layer as a basic structural architecture for the sake of this study.

Structure of FNN
To explain the structure of FNNs for a design consisting of three layers with n of input units, m of hidden units, and s of output boards.Thus, the dimension of the fuzzy neural network is (n x m x s).Target vector, connection weights and biases are fuzzy numbers and the input vector is a real number.

Input unit
Hidden unit: Where where 6.Solving Fuzzy Singularly Perturbed Volterra integro-differential equation.

Solving FSPVIDE(One dimension).
To solve any FSPVIDE for one dimension with boundary condition (BC) by FNN, we reflect a three-layered neural networks with one unit entry x, one hidden layer with m hidden units (neurons), and one linear output unit, resulting in an FNN dimension of( 11).As it is shown in Figure 2. In this paper, we take the number of hidden layer neurons as five, in other words, m=5.The input -output each unit of our neural network can be rewritten for level set as follows : Input unit : x=x Hidden unit: Where Output unit : Where where

Solution of two-dimensional FSVIDE
To solve two dimensions of FSPVIDE using FNN, we reflect the 3 layers of the neural network with two unit entries x and y, one hidden layer consisting of m unit and one unit output ( ).Here, the dimension of FFNN (2×m×1) is shown in Figure 3.In this paper, we take the number of hidden layer neurons as seven, in other words, m=7.Also, the inputoutput each unit of our neural network can be rewritten for level set as follows: Input unit : x=x ,y=y Hidden unit : Output unit :

Conclusions
In this paper, we used a type of AI, which is within deep learning, by designing an artificial neural network to find the approximate solution for certain kinds of fuzzy integrative equations, which are the fuzzy singularly perturbed Volterra integro-differential equations with one and two variables, this type of equation is of great importance in many applications.It is often used in the transmission of radiant energy and ripple or oscillation issues.
In this design, three layers, the input and output layer and the hidden layer are used, in which the number of nodes is either five or seven according to the number of variables with an activation function of type sigmoid which is the hyperbolic tangent activation function.The results of this paper are clear through the application of this technique to some examples with one or two variables through the speed of convergence and the reduction of the error rate after comparing them to the exact solutions.It is possible that there are some future works for this paper by changing the structure of the neural network or changing the activation function, or it is possible to apply this technique to the type of equations in three dimensions.

Figure 1 :
Figure 1: The FNN The model's architecture demonstrates how FNN changes the n inputs ( outputs are fuzzy such that [ ] and are the fuzzy biases for the [ ] [ ] respectively, [ ] are the fuzzy weight connecting crisp neuron to fuzzy neuron [ ] , and [ ] are the fuzzy weight connecting [ ] to [ ] .And the hyperbolic tangent transfer function: (x) = tanh(x), and ′(x) = 1 -2 (x).