# Convergence Analysis Hilbert Space Approach for Fuzzy Integro-Differential Models

Authors: Jingwen Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we present and demonstrate an innovative numerical method, which makes use of fuzzy numbers and fuzzy parameters that is effective in the solution of fuzzy type Volterra integro-differential equations, which was previously thought to be impossible using conventional methods. The first application of a technique for solving Volterra integro-differential equations of the fuzzy type, which was devised and tested in this paper, is shown here. This is the first time that this approach has been used. This system’s overall quality may be improved as a consequence of the use of the Hilbert space replicating kernel idea, which is a possibility. Separate evaluations are made of the algorithms’ correctness and sloppiness, as well as their foundations in the computationally effective kernel Hilbert space, which has been extensively researched in the past. Numerical examples are provided of the article to demonstrate how the technique outlined before may achieve convergence and accuracy. Here are a few illustrations to help understand that it is possible to deal with physical issues that require complicated geometric calculations with the assistance of the method explained in this article.


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## Body

## 1. Introduction

It is possible to achieve better results using differential and probabilistic techniques rather than conventional processes. It is possible to uncover system features with more precision and less work using differential and probabilistic techniques rather than traditional processes, as opposed to traditional processes. In contrast to this, traditional procedures require the placement of more resources and time to be successful. In an attempt to attain this goal, one strategy that may be used is the application of fuzzy integro-differential equations which are more accurate than typical approaches [1]. A significant increase in the number of theoretical and computational computations using fuzzy Volterra integro-differential equations, as well as the number of publications referring to these equations, both in terms of quantity and quality, has been seen in recent years [2–5]. Both the number of products available and the quality of those products have increased as a result of this expansion (also known as fuzzy Volterra integro-differential equations, or FVIDEs). Only a few academics (including [6–9] and other published works) have looked at the consequences of fuzzy modelling in a quantum gravity context to the best of our knowledge at the time of writing. With the exception of these issues, other areas such as fragile biopolymers, quantum gravity, and quantum optics, among others, have gotten only a sliver of attention in population dynamics research despite the fact that they are important. This, on the other hand, has just recently become the case. Biswas and Roy developed a second-order fuzzy differential equations (FVDE) technique that is based on fuzzy differential equations in order to deal with fuzzy differential equations in practice [10, 11]. It may be used to solve fuzzy differential equations as well as other issues because it is based on the differentiability extension concept established by Seikkala. The concept of differentiability extension developed by Seikkala, which provided as inspiration for the method, laid the groundwork for its development. In fact, upon closer examination, it becomes clear that fuzzy integro-differential equation (as well as the theory that underpins them) is addressed more thoroughly in references [2, 3], [10], and [12] than fuzzy differential equations (2), (3), and (10) Using fuzzy integro-differential equations as a reference point, it is clear that fuzzy integro-differential equations (and the theory that underpins them) are treated more leniently in the first two papers. Recently, it is observed that the Xue et al. presented the comprehensive study of decline approximations for fuzzy viscoelastic integral model [12, 13] and compound learning control of ambiguous nonlinear fractional-order models with actuator liabilities grounded on command sifting and fuzzy estimation [14]. One can find comprehensive related literature in Refs. [12–16].It is observed that the reproducing kernel philosophy has significant scientific applications in various fields like ordinary differential, numerical analysis, fractional differential, statistics, and probability models [17]. Ahmadian et al. recently developed some kind of reproducing kernel Hilbert space (RKHS) approaches to hand both ordinary and fractional-order fuzzy differential models [18, 19]. The author’s show many advantages of the proposed scheme like to start the procedure and choose any point lies in the limits of integration, and it requires less effort to investigate the results. Later, various researchers used this strategy to explore the two-point fuzz BVPs model [20, 21], fuzzy differential model [22], periodic first-order BVPs of integro-differential model Fredholm type [23], systems of periodic second-order BVPs [24], Lane–Emden equation, and fractional-order model of Lane–Emden [25, 26].On the basis of comprehensive literature review and author’s best knowledge, it is observed that no one introduced the algorithm with less computational cost toproduce more accurate solutions, which motivates us to fill this gap and provide an efficient scheme for the problem given in Equation (1).An in-depth explanation of the essay’s organisational structure is provided in the next section, which also includes samples of the essay throughout the rest of the section. The next part will continue the topic that started in Part I of this chapter by addressing fuzzy integrals and fuzzy number theories. These two subjects were addressed in more detail in the part that came before it in the chapter’s previous section. The following is a breakdown of how Part II of this chapter is organised. The use of methods such as erroneous integrals and fuzzy number theories, among other things, is required in order to create functions that take fuzzy numbers as both input and output values. In Section2, we will discuss how to use failure integrals and fuzzy number theories in the construction of fuzzy-valued functions. Failure integrals and fuzzy number theories are two concepts that are used in the construction of fuzzy-valued functions. Consider the ideas of fuzzy integrals and fuzzy number theories, which were introduced in Section 3 of this chapter. It becomes clear that these concepts may be understood differently depending on the context in which they are used. In the third part of this chapter, we will go over in depth a kind of differential equation known as second-order integro-differential equation, also known as Volterra integro-differential equation, which is also known as Volterra integro-differential equation.


## 2. Important Concepts and Preliminaries

Many fundamental concepts and theorems will be reviewed in this part, and they will be used throughout the course. “Fuzzy numbers” (FN), “fuzzy functions” (FF), and derivative FF are only a few examples of what is available.Definition 1.
Reference [26]. A FN U⟶0,1 is a “fuzzy subset” (FS) of H with normal, convex, and superior “membership function” (MF) of “bounded support” (BS). Let Rℱ signify of FN. 0<β≤1, set Uβ=S∈H|US≥β and U0=S∈H|US>0. Formerly, the β-level Uβ is a compact interlude 0≤β≤1 and slightly U∈Rℱ. The representation Uβ=U1β,U2β signifies clearly the β-level set U. We remark to U1 and U2 inferior and superior divisions on U correspondingly.Theorem 1 (see [27]).
MappingU⟶0,1 is a FN with β-cut depiction U1β,U2β if and only if the succeeding circumstances are fulfilled:(i)
U1:0,1⟶H is restricted nondeclining(ii)
U2:0,1⟶H is restricted nongrowing(iii)
R∈0,1,limβ⟶R−U1β=U1Rlimβ⟶R−U2β=U2R(iv)
R∈0,1,limβ⟶R+U1β=U1Rlimβ⟶R+U2β=U2R(v)
U1β≤U2ββ∈0,1Definition 2 (see [26]).
SupposeY:a,c⟶Rℱ and x0∈a,c. We give or take Y is (1)-differentiable at x0 and some element Y′x0∈Rℱ and j>0 adequately near to 0, then there will be(1)Yx0+j−Yx0,Yx0−Yx0−j.
The limits(2)limj⟶0+Yx0+j−Yx0j=limj⟶0+Yx0−Yx0−jj=Y′x0.
In this circumstance, we signifyY′x0 by D11Yx0. Also, Y is (2)-differentiable, j<0 adequately near to 0, there be Yx0+j−Yx0,Yx0−Yx0−j, and the limits(3)limj⟶0−Yx0+j−Yx0j=limj⟶0−Yx0−Yx0−jj=Y′x0.
In circumstance, this imitative is signified byD21Yx0.Theorem 2 (see [28]).
LetY:a,c⟶Rℱ be a FF, then Yxβ=Y1,βx,Y2,βx,β∈0,1.(i)
IfY (1)-differentiable, then Y1,β and Y2,β are DF and D11Yx0β=Y1,β'x,Y2,β'x(ii)
IfY (2)-differentiable, then Y1,β and Y2,β are DF and D21Yx0β=Y2,β'x,Y1,β'xTheorem 3 (see [29]).
LetD11Y:a,c⟶Rℱ or D21Y:a,c⟶Rℱ be FF, where Yxβ=Y1,βx,Y2,βx, β∈0,1.(i)
IfD11Y is (1)-differentiable, then Y1,β′ and Y2,β′ are DF and Y″xβ=Y1,β″x,Y2,β″x(ii)
IfD11Y is (2)-differentiable, then Y1,β′ and Y2,β′ are DF and Y″xβ=Y2,β″x,Y1,β″x(iii)
IfD21Y is (1)-differentiable, then Y1,β′ and Y2,β′ are DF andY″xβ=Y2,β″x,Y1,β″x(iv)
IfD21Y is (2)-differentiable, then Y1,β′ and Y2,β' are DF and Y″xβ=Y1,β″x,Y2,β″xTheorem 4 (see [30]).
LetY:a,c⟶Rℱ be a continuous FF, where Yxβ=Y1,βx,Y2,βx. If Y1,βx and Y2,βx are integrable functions (IF) over a,c, then ∫acYxdx∈Rℱ, then we get(4)∫acYxdxβ=∫acY1,βxdx,∫acY2,βxdx.


## 3. Modelling of Fuzzy Integral Equation

This section contains the modelling of the fuzzy integro-differential equation of Volterra type, where the fuzzy integro-differential equations of Volterra type are transformed into corresponding system of integro-differential equations [31]. It comprises the discovery of α-cut representation form of g. With the intention of develop the reproducing kernel Hilbert space algorithm to examine the accurate solutions of fuzzy integro-differential equations of Volterra type, first we assume(5)x″η=fη+∫aηKη,ξGxξdξ,a≤η,xa=b1,x′a=b2.According to section2, the above second-order fuzzy Volterra equations is converted to the following system of equations as(6)x1,α″η=f1,αη+∫anV¯η,αds,x2,α″η=f2,αη+∫ηηV¯η,αdξ,x2,α″η=f2,αη+∫ηηV¯η,αds,x1,α″η=f1,αη+∫ηηV¯η,αdξ.By means of the well-known Zadeh expansion principle given in reference [28] and if the function Gxξ present in Equation (1) is a function of strictly increasing, then(7)V¯η,α=Kη,ξGx2,αξ,Kη,ξ≥0,Kη,ξGx1,αξ,Kη,ξ<0,V¯η,α=Kη,ξGx1,αξ,Kη,ξ≥0,Kη,ξGx2,αξ,Kη,ξ<0.Similarly, if the functionGxξ is strictly decreasing, then we obtain(8)V¯η,α=Kη,ξGx2,αξ,Kη,ξ≥0,Kη,ξGx1,αξ,Kη,ξ<0,V¯η,α=Kη,ξGx1,αξ,Kη,ξ≥0,Kη,ξGx2,αξ,Kη,ξ<0.It is important to mention that the sufficient conditions for the existence and uniqueness of solution for the problem subject to Equation (1) are presented in reference [15].


## 4. Important Results and Convergence Analysis

This segment encloses the preliminaries, notation, development, and application of reproducing kernel Hilbert space scheme to seek the exact and numerical solutions of second kind Volterra integral equation. By means of Gram–Schmidt orthogonalization procedure, we build system of orthogonal function ofH230,1⊕H230,1.Definition 3.
Reference [32]. Let H and A be denoted by Hilbert space and abstract set, respectively, then a function f1:A×A⟶ℝ is known as reproducing kernel of the Hilbert space H if it holds the following conditions:(9)f1⋅,t∈H,∀t∈A,ϕ,f1⋅,t=ϕt,∀ϕ∈H,t∈A.
The second condition is also known as the reproducing property.Definition 4 (see [32]).
Suppose thatH2ma,b is an inner product space and defined as absolutely continuing, then H2ma,b=xη|xη,x′η,…,xmηare absolutely continuous,xna=0forn=0,1,2,…,m−1andxmηbelongs toL2a,b.
In the time being, the norm and inner product inH2ma,b are defined as(10)xη,xηH2m=xηH2m,xη,zηH2m=∑n=0m−1xnazna+∫abxmvzmvdv.
In the above equation, the functionsxη,zη are belongs to H2ma,b.Definition 5 (see [32, 33]).
Hilbert spaceH2ma,b⊕H2ma,b for m=1,2,…,n, can be given as(11)H2ma,b⊕H2ma,b=x=x1,x2T|x1,x2∈H2ma,b.
The inner product and norms of spaceH2ma,b⊕H2ma,b are given as(12)∑i=12xiηH2m2=xηH2m⊕H2m,xη,zηH2m⊕H2m=∑i=12xiη,ziηH2m.Definition 6 (see [32, 33]).
Hilbert spaceH2ma,b is called the reproducing kernel under the condition that ∀η∈a,b,∃Rη,ξ∈H2m so that xη=xξ,Rη,ξH21,∀xη∈H21a,b,ξ∈a,b.Theorem 5 (see [32, 33]).
LetH2ma,b be the Hilbert space which is also complete reproducing kernel space. Then, Rηξ reproducing kernel function is given as(13)Rηξ=∑i=12m−1piηξi,ξ≤η,∑i=12m−1qiηξi,ξ>η.The reproducing kernel function illustrationRηξ in the Hilbert space H230,1, by means of Maple 2015, is delivered by(14)Rηξ=1+12!⋅3!η2ξ2η+3+ηξ1−14!η4+15!η5,ξ≤η,1+12!⋅3!η2ξ2η+3+ηξ1−14!η4+15!η5,ξ>η.In order to apply the proposed scheme on the Hilbert spaceH23a,b⊕H23a,b, first, we introduce a linear and invertible operator as(15)L:H23a,b⊕H23a,b⟶H21a,b⊕H21a,b,as Lxη=x″η so that(16)L=L100L2,and xη=x1,αη,x2,αηT. Therefore, the problem under study can be transformed as given below:(17)Lxη=Fη,fη,hxη,xa=b1,x′a=b2.In the above equations,hxη=∫aηKη,ξGxξdξ,fη∈H21a,b⊕H21a,b, and xη∈H23a,b⊕H23a,b. Also, Fη,fη,hxη is Fη,f1,αη,f2,αη,h1,αxη,h2,αxη.Theorem 6 (see [32, 33]).
The invertible and linear operatorL defined as L:H23a,b⊕H23a,b⟶H21a,b⊕H21a,b is bounded.Proof.
It is very easy to prove that the operatorL is linear from the space H23a,b⊕H23a,b to the space H21a,b⊕H21a,b. Now, we have to prove that the operator L is bounded and bounded by some constant γ such that(18)LxH21⊕H21≤γxH23⊕H23,γ>0,∀ x belongs to H23a,b⊕H23a,b, then we have(19)LxηH21⊕H212=L1x1,αηH212+L2x2,αηH212=L1x1,αη,L1x1,αηH21+L2x2,αη,L2x2,αηH21=L1x1,αη2+L2x2,αη2+∫abL1x1,αηη2dη+∫abL2x2,αηη2dη.
By means of the property of reproducing kernel ofRηξ, then(20)x1,αη=x1,αξ,RηξH23,x2,αη=x2,αξ,RηξH23.
Thus, we have(21)L1x1,αη=x1,αξ,L1RηξH23,L1Rηη=x2,αξ,L2RηξH23,L1x1,αηη=x1,αξ,L1RηξηH23,L1Rηη=x2,αξ,L2RηξηηH23.
By means of the continuity of the functionRηξ on the interval a,b, we obtain(22)L1x1,αη=x1,αξ,L1RηξH23≤x1,αH23L1RηξH23≤γ1x1,αH23,L2x2,αη=x2,αξ,L2RηξH23≤x2,αH23L2RηξH23≤γ1x2,αH23,L1x1,αηη=x1,αξ,L1RηξH23≤x1,αH23L1RηξηH23≤γ3x1,αH23,L2x2,αηη=x2,αξ,L2RηξH23≤x2,αH23L2RηξηH23≤γ4x2,αH23.
Accordingly, we have(23)LxηH21⊕H212≤γ12x1,αH232+γ22x2,αH232+b−aγ32x1,αH232+b−aγ42x2,αH232=γ12+b−aγ32x1,αH232+γ22+b−aγ42x2,αH232≤γ2x1,αH232+x2,αH232=γ2∑i=12xi,αH232=γ2xH23⊕H232.
In the above equation,γ is maximum of γ12+b−aγ32 and γ22+b−aγ42.
To apply the illustration form of exact and numerical solutions of second kind Volterra integral equation, we next formulate the system of orthogonal functionsχijη for i=1,2,…,∞,j=1,2 of H23a,b⊕H23a,b space, and thus we assume(24)χijη=L1∗00L2∗Θij=L∗Θij.
In the above equation,L∗ is known as the adjoint operator of the operator L and Θij=Θi1,Θi2T. The system of orthogonal system χ¯ijη for i=1,2,…,∞,j=1,2 in H23a,b⊕H23a,b space can be computed by means of the Gram–Schmidt orthogonalization procedure of χijη for i=1,2,…,∞,j=1,2 as given below:(25)χ¯lmη=∑i=1l∑j=1mβijlmχijη.
The coefficientsβijlm can be obtained by means of the following relation(26)β11lm=1χ11,βijlm=1χij2−∑q=1i−1χijη,χ¯iqη2,fori=j≠1,βijlm=−∑q=ji−1χijη,χ¯iqηβqjlmχij2−∑q=1i−1χijη,χ¯iqη2,fori>j.Theorem 7.
Letηi for i=1,2,…,∞ is dense in the interval a,b, xη is the solution of fuzzy integro-differential model given in Equation (1) and xη∈H23a,b⊕H23a,b, then(27)xη=∑l=1∞∑m2∑i=1l∑j=1mβijlmFjηi,fηi,hxηiχ¯lmη,xη is the convergent series in the logic of. ⋅H23⊕H23.Proof.
In order to prove the required result, first we need to show thatχijη for i=1,2,…,∞,j=1,2 is a complete system and belongs to H23a,b⊕H23a,b as(28)χijη=L∗Θijη=L∗Θijξ,RηξH23⊕H23=Θijξ,LξRηξH21⊕21=LξRηξ|ξ=ηi∈H23a,b⊕H23a,b.
In contrast, for eachxη∈H23a,b⊕H23a,b, assume xη,χijηH23⊕H23=0, also(29)xη,χijηH23⊕H23=x1,αη,χi1ηH23+x2,αη,χi2ηH23=x1,αη,L1∗Θi1ηH23+x2,αη,L2∗Θi2ηH23=L1x1,αη,Θi1ηH21+L2x2,αη,Θi2ηH21=L1x1,αηi+L2x2,αηi=Lxηi.
Sinceηi for i=1,2,…,∞ is dense, then Lxη=0, and sin e L in invariable which implies that xη=0. We know that the sequence χijη for i=1,2,…,∞,j=1,2 is complete in the space H23a,b⊕H23a,b and(30)χ¯lmη=∑i=1l∑j=1mβijlmχijη.
Then the systemχijη for i=1,2,…,∞,j=1,2 is complete orthonormal in the space H23a,b⊕H23a,b. By means of the Fourier series expansion around χijη for i=1,2,…,∞,j=1,2, we obtain(31)xη=∑l=1∞∑m=12xη,χ¯lmηH23⊕H23χ¯lmη=∑l=1∞∑m=12xη,∑l=1l∑m=1mβijlmχijηH23⊕H23χ¯lmη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmxη,χijηH23⊕H23χ¯lmη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmxη,L∗ΘijηH23⊕H23χ¯lmη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmLxη,ΘijηH21⊕H21χ¯lmη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmFjη,fη,hxη,ΘijηH21⊕H21χ¯lmη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmFjηi,fηi,hxηiχ¯lmη.
This implies that the above series is nothing but Fourier series in the spaceH23a,b⊕H23a,b. As H23a,b⊕H23a,b is the Hilbert space, the above series is convergent with the logic ⋅H23⊕H23.
For numerical procedure, we place the initial functionx0ηi=xηi and the nth-term of the numerical solution of the problem under study is given as(32)xnη=∑l=1∞∑m=12∑i=1l∑j=1mβijlmFjηi,fηi,hxi−1ηiχ¯lmη.Theorem 8.
Let the exact solution of (11) be xη, and xnη denote its approximate solution, then(i)
SupposexnH23⊕H23 is the bounded sequence and the sequence ηii=1∞ dense on the interval a,b, then as n⟶∞, xn−xH23⊕H23⟶0(ii)
Asn⟶∞ then xn−xc⟶0(iii)
Asn⟶∞ then xnηη⟶xηη uniformly(iv)
If asn⟶∞xn−1−xH23⊕H23⟶0, xn−1H23⊕H23 is the bounded, as n⟶∞ηn⟶τ, then the function Fη,fη,hxη is continuous ∀η∈a,b and the functions hη,fη are continuous, then(33)Fηn,fηn,hxn−1ηn⟶Fη,fη,hxη.Proof.
(i)
By means of orthonormality ofχijηj,i=1,12,∞ and Equation (12), we have(34)xn+1H23⊕H232=x0H23⊕H232+∑m=1n+1∑l=12Alm2,whereAij=∑i=1l∑j=1mβijlmFjηi,fηi,hxi−1ηiχ¯lmη. We know that xnH23⊕H23 is bounded and xnH23⊕H23≤xn+1H23⊕H23, then the sequence xnH23⊕H23 is convergent and also ∑m=12Alm2∈l2 for l=1,2,…,∞. If n<m as m,n⟶∞, then(35)xm−xnH23⊕H232=∑l=n+1m∑m=12Alm2⟶0.Since the spaceH23a,b⊕H23a,b is Hilbert, then as n⟶∞, xn−xH23⊕H23⟶0.(ii)
By means of the reproducing property of the functionRηξ, then(36)xnη−xη=xnξ−xξ,RηξH23⊕H23≤xnξ−xξH23⊕H23RηξH23⊕H23≤γ5xn−xH23⊕H23⟶0,asn⟶∞.This implies thatxn−xC⟶0 as n⟶∞.(iii)
We know thatxn uniformly converges to x, then(37)xn″η−x″η=d2dη2xnη−xη=d2dη2xnξ−xξ,RηξH23⊕H23=xnξ−xξ,d2dη2RηξH23⊕H23≤xnξ−xξH23⊕H23d2dη2RηξH23⊕H23≤γ6xn−xH23⊕H23⟶0,asn⟶∞.This implies thatxn″−xC″⟶0, as n⟶∞.(iv)
First of all, we show thatxn−1ηn⟶xτ as n⟶∞ as(38)xn−1ηn−xτ=xn−1ηn−xn−1τ+xn−1τ−xτxn−1ηn−xn−1τ+xn−1τ−xτ.
We know thatn⟶∞,xn−1−xH23⊕H23⟶0, this implies that as n⟶∞, xn−1τ−xτH23⊕H23⟶0. Now(39)xn−1ηn−xn−1τ=xn−1η,Rηnξ−xn−1η,Rτξ=xn−1η,Rηnξ−Rτξ≤xn−1ηH23⊕H23Rηnξ−RτξH23⊕H23.
As the functionRηξ is symmetric, as n⟶∞;Rηnξ−RτξH23⊕H23⟶0, and this implies that n⟶∞,xn−1ηn−xn−1τ⟶0. Therefore, xn−1ηn−xτ⟶0,asn⟶∞.
We know that the functionsh and f are continuous and ηn⟶τ, then as n⟶∞;fηn⟶fτ and hxn−1ηn⟶hxτ as n⟶∞. By means of the continuity of the function F, we have Fηn,fηn,hxn−1ηn⟶Fτ,fτ,hxτ.


## 5. Results and Discussion

A fuzzy integro-differential problem of the Volterra type is investigated in this part using the expected reproducing kernel Hilbert space method and the approximate solution of the integro-differential equation. It is important to note that the suggested technique is very simple to use, quick convergent, and trustworthy, and it may be used to find both approximate and precise solutions to the issue under consideration. The Maple code is being created for this mentioned issue utilising a replicating kernel Hilbert space method, and it is being used to produce fresh findings as well as to make comparisons between results [34–37].Problem 1.
Suppose the following fuzzy integro-differential model of Volterra type is given as(40)xηη=−1+α,1−α+∫0ηxξdξ,α,t∈0,1,x0=−1+α,1−α,xη0=0,0.
Allowing to the anticipated reproducing kernel Hilbert space algorithm, the problem given in Equation (13) transformed into following system(41)x1,α″=−1+α+∫0ηx1,αξdξ,x2,α″=1−α+∫0ηx2,αξdξ,0≤η≤1,x1,α0=−1+α,x1,α′0=0,x2,α0=1−α,x2,α′0=0,x1,α″=1−α+∫0ηx2,αξdξ,x2,α″=−1+α+∫0ηx1,αξdξ,0≤η≤1,x1,α0=1−α,x1,α′0=0,x2,α0=−1+α,x2,α′0=0.
The exact solutions of the problems (14)-(15) are given as(42)xη=cosη2−12sinη2α−1e1/2η,cosη2−12sinη21−αe1/2η,xη=cos32η−13sin32ηα−1e1/2η,cos32η−13sin32η1−αe1/2η.
Numerous simulation have been implemented for various values of parameterα and η. The comparison between the exact and approximate solutions when η=0.5 and numerous values of α for problem (13) are deliberated in Table 1. It can be observed that the proposed scheme demonstrates very accurate solution for solving the problem under study. Also, the accuracy does not depend on the selection of α, the analytic solution is obtained of the discussed problem using proposed scheme when η=1.Table 1
Comparison of exactx1,α,x2,α and approximate x˜1,α,x˜2,α solutions of problem (13) when η=0.5 for different values ofα.
↓αx1,αx˜1,αx2,αx2,αSolution of problem (14)0.0−0.89025−0.888490.890250.888490.2−0.71220−0.710790.712200.710790.4−0.5342−0.533090.534150.533090.6−0.35610−0.355390.356100.355400.8−0.17805−0.177700.178050.177691.000.00.0Solution of problem (15)0.0−0.85445−0.852170.854450.852170.2−0.68356−0.681740.683560.681740.4−0.51267−0.511300.512670.511300.6−0.34178−0.340870.341780.340870.8−0.17089−0.170430.170890.170431.000.00Problem 2.
Suppose the following fuzzy integro-differential model of Volterra type is given as(43)xηη=−10+5αe1−η−1+12αη,−3−2αe1−η−0.6+0.1α−0.2α2η+∫0η−0.2+0.1α,−0.1eξ−1xξdξ,α,t∈0,1,x0=−10+5αe,−3−2αe,xη0=3+2αe,10−5αe.
Allowing the anticipated reproducing kernel Hilbert space algorithm, the problem given in Equation (13) transformed into following system(44)x1,α″=−10+5αe1−η−1+12αη+∫0η−0.2+0.1αeξ−1x1,αξdξ,x2,α″=−3−2αe1−η−0.6+0.1α−0.2α2η−∫0η0.1eξ−1x2,αξdξ,x1,α0=−10+5αe,x1,α′0=3+2αe,x2,α0=−3−2αe,x2,α′0=10−5αe,x1,α″=−3−2αe1−η−0.6+0.1α−0.2α2η−∫0η0.1eξ−1x1,αξdξ,x2,α″=−10+5αe1−η−1+12αη+∫0η−0.2+0.1αeξ−1x2,αξdξ,x1,α0=−3−2αe,x1,α′0=10−5αe,x2,α0=−10+5αe,x2,α′0=3+2αe.
The exact solutions of the problems (17)-(18) are given as(45)xη=−10+5αe1−η−3−2αe1−η,xη=3+2αe1−η,10−5αe1−η.
Forη∈0,1, reproducing the kernel Hilbert space algorithm applied for solving this problem when α=0.5 and numerous values of η. Figure 1 is plotted to show the comparison of approximate and exact solutions. It is significant to indicate that the estimated solution attained using the proposed scheme is well-matched with the exact solutions for every values of α, which is the beauty of the suggested algorithm [34].Figure 1
Comparison of the approximateX1,α,X2,α and exact solutions x1,α,x2,α for (a) α=0.5 and (b) α=0.9.
(a)(b)Problem 3.
Suppose the following fuzzy integro-differential model of Volterra type is given as(46)xηη=3+3αsinη+320α2−6α+5ηcos2η9−9αsinη−920α2−6α+5ηcos2η+∫0η0.1+0.3α,0.5−0.1ηcosξxξdξ,α,t∈0,1,x0=−10+5αe,−3−2αe,xη0=3+2αe,10−5αe.
The exact solutions are given as(47)xη=3+3αsinη,9−9αsinη,xη=−9+9αsinη,3−3αsinη.
Forη∈0,1, reproducing the kernel Hilbert space algorithm applied for solving this problem when α=0.5 and numerous values of η. The comparison of the approximate and exact solutions is deliberated in Figure 2. It is important to mention that the approximate solution achieved by means of the proposed scheme is well-matched with the exact solutions for every values of α, which is the beauty of the suggested algorithm.Figure 2
Comparison of the approximateX1,α,X2,α and exact solutions x1,α,x2,α for (a) α=0.5 and (b) α=0.9.
(a)(b)


## 6. Conclusion

Using replicated kernel theory, we were able to create a whole new technique for solving complex second-order FVIDs, which we have just published. This method is explained in more depth farther down this page. The technique under consideration generates responses that are both broad in scope and particular in kind. Numerical simulations have been performed in order to demonstrate the robustness of the technique under discussion. In the future, we expect that our technique will be used to solve fuzzy ordinary differential algebraic equations, as well as fuzzy partial integro-differential equations and various types of fuzzy differential algebraic equations, among other things.



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