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.
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 Section 2, 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 is a “fuzzy subset” (FS) of with normal, convex, and superior “membership function” (MF) of “bounded support” (BS). Let signify of FN. set and . Formerly, the -level is a compact interlude and slightly . The representation signifies clearly the -level set . We remark to and inferior and superior divisions on correspondingly.
Theorem 1 (see [27]). Mapping is a FN with -cut depiction if and only if the succeeding circumstances are fulfilled:(i) is restricted nondeclining(ii) is restricted nongrowing(iii) (iv) (v)
Definition 2 (see [26]). Suppose and . We give or take is (1)-differentiable at and some element and adequately near to , then there will beThe limitsIn this circumstance, we signify by . Also, is (2)-differentiable, adequately near to , there be , and the limitsIn circumstance, this imitative is signified by .
Theorem 2 (see [28]). Let be a FF, then .(i)If (1)-differentiable, then and are DF and (ii)If (2)-differentiable, then and are DF and
Theorem 3 (see [29]). Let or be FF, where , .(i)If is (1)-differentiable, then and are DF and (ii)If is (2)-differentiable, then and are DF and (iii)If is (1)-differentiable, then and are DF and(iv)If is (2)-differentiable, then and are DF and
Theorem 4 (see [30]). Let be a continuous FF, where . If and are integrable functions (IF) over , then , then we get
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 . 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
According to section 2, the above second-order fuzzy Volterra equations is converted to the following system of equations as
By means of the well-known Zadeh expansion principle given in reference [28] and if the function present in Equation (1) is a function of strictly increasing, then
Similarly, if the function is strictly decreasing, then we obtain
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 of .
Definition 3. Reference [32]. Let and be denoted by Hilbert space and abstract set, respectively, then a function is known as reproducing kernel of the Hilbert space if it holds the following conditions:The second condition is also known as the reproducing property.
Definition 4 (see [32]). Suppose that is an inner product space and defined as absolutely continuing, then
In the time being, the norm and inner product in are defined asIn the above equation, the functions are belongs to
Definition 5 (see [32, 33]). Hilbert space for , can be given asThe inner product and norms of space are given as
Definition 6 (see [32, 33]). Hilbert space is called the reproducing kernel under the condition that so that
Theorem 5 (see [32, 33]). Let be the Hilbert space which is also complete reproducing kernel space. Then, reproducing kernel function is given as
The reproducing kernel function illustration in the Hilbert space , by means of Maple 2015, is delivered by
In order to apply the proposed scheme on the Hilbert space , first, we introduce a linear and invertible operator asas so thatand Therefore, the problem under study can be transformed as given below:
In the above equations, and . Also, is
Theorem 6 (see [32, 33]). The invertible and linear operator defined as is bounded.
Proof. It is very easy to prove that the operator is linear from the space to the space . Now, we have to prove that the operator is bounded and bounded by some constant such that∀ belongs to , then we haveBy means of the property of reproducing kernel of , thenThus, we haveBy means of the continuity of the function on the interval , we obtainAccordingly, we haveIn the above equation, is maximum of and
To apply the illustration form of exact and numerical solutions of second kind Volterra integral equation, we next formulate the system of orthogonal functions for of space, and thus we assumeIn the above equation, is known as the adjoint operator of the operator and . The system of orthogonal system for in space can be computed by means of the Gram–Schmidt orthogonalization procedure of for as given below:The coefficients can be obtained by means of the following relation
Theorem 7. Let for is dense in the interval , is the solution of fuzzy integro-differential model given in Equation (1) and , then is the convergent series in the logic of.
Proof. In order to prove the required result, first we need to show that for is a complete system and belongs to asIn contrast, for each assume alsoSince for is dense, then , and sin e in invariable which implies that We know that the sequence for is complete in the space andThen the system for is complete orthonormal in the space By means of the Fourier series expansion around for , we obtainThis implies that the above series is nothing but Fourier series in the space . As is the Hilbert space, the above series is convergent with the logic
For numerical procedure, we place the initial function and the nth-term of the numerical solution of the problem under study is given as
Theorem 8. Let the exact solution of (11) be , and denote its approximate solution, then(i)Suppose is the bounded sequence and the sequence dense on the interval then as , (ii)As then (iii)As then uniformly(iv)If as , is the bounded, as , then the function is continuous and the functions are continuous, then
Proof. (i)By means of orthonormality of and Equation (12), we have where We know that is bounded and , then the sequence is convergent and also for If as , then Since the space is Hilbert, then as , (ii)By means of the reproducing property of the function , then This implies that as (iii)We know that uniformly converges to , then This implies that as (iv)First of all, we show that as asWe know that this implies that as , NowAs the function is symmetric, as and this implies that Therefore,
We know that the functions and are continuous and , then as and as By means of the continuity of the function , we have .
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 asAllowing to the anticipated reproducing kernel Hilbert space algorithm, the problem given in Equation (13) transformed into following systemThe exact solutions of the problems (14)-(15) are given asNumerous simulation have been implemented for various values of parameter and . The comparison between the exact and approximate solutions when and numerous values of 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 , the analytic solution is obtained of the discussed problem using proposed scheme when .
Problem 2. Suppose the following fuzzy integro-differential model of Volterra type is given asAllowing the anticipated reproducing kernel Hilbert space algorithm, the problem given in Equation (13) transformed into following systemThe exact solutions of the problems (17)-(18) are given asFor reproducing the kernel Hilbert space algorithm applied for solving this problem when and numerous values of . 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 , which is the beauty of the suggested algorithm [34].
(a)
(b)
Problem 3. Suppose the following fuzzy integro-differential model of Volterra type is given asThe exact solutions are given asFor reproducing the kernel Hilbert space algorithm applied for solving this problem when and numerous values of . 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 , which is the beauty of the suggested algorithm.
(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.
Data Availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that this article is free of conflicts of interest.