We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert space in order to formulate the analytical solutions in a rapidly convergent series form in terms of their -cut representation. The approximation solution is expressed by -term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

1. Introduction

Building mathematical model of a specific phenomenon under uncertainty is essentially important for a large number of applications in economics, medicine, mathematics, physics, and engineering [15]. The concepts of uncertain integral equations have developed in recent years as a new branch of fuzzy mathematics. The uncertain integral equations are powerful tools to introduce uncertain parameters and to deal with their dynamical systems in natural fuzzy environments. They are actually of great importance in the fuzzy analysis theory and its application in fuzzy control models, atmosphere, artificial intelligence, measure theory, quantum optics, and so forth [610]. The experts in such areas extensively use these equations to make the uncertain problems, which are usually too complex to be defined in precise terms, more understandable. In many situations, information for real scientific and technological processes is provided under uncertainty, which may arise in the experiment part, data collection, and measurement process as well as when determining the initial values. In classical mathematics, however, crisp equations cannot cope with these situations. Therefore, it is necessary to have some mathematical apparatus to understand this uncertainty. Thus, it is immensely important to develop appropriate and applicable strategy to accomplish the mathematical construction that would appropriately treat uncertain problems and solve them.

During the last decades, many authors have devoted their attention to study solutions to uncertain integral equations using various numerical and analytical methods. Among these attempts are the homotopy analysis method [11], Adomian decomposition method [12], homotopy perturbation method [13], Lagrange interpolation method [14], differential transform method [15], and other methods [1618]. The purpose of this paper is to extend the application of the reproducing kernel Hilbert space (RKHS) method to provide analytic-numeric solutions for a class of uncertain Volterra integral equations of the second kind in the following form:where is continuous arbitrary crisp kernel function, is linear or nonlinear continuous increasing function, is a continuous fuzzy-valued function, and is an unknown function to be determined. If is a crisp function, then the solution to (1) is crisp. However, if is a fuzzy function, then this equation may only process fuzzy solutions. Sufficient conditions for the existence of a unique solution to (1) have been given in [18].

Generally, there exists no method that yields an explicit solution for nonlinear fuzzy Volterra integral equation due to the complexities of uncertain parameters involving these equations. Thus, we need an efficient reliable numerical technique for the solutions to such equations. Anyhow, by using the parametric form of fuzzy numbers, we convert the fuzzy Volterra integral equation into a crisp system of integral equations, which are solved numerically using the RKHS approach. The present method has the following characteristics: first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate and the results can be obtained easily; third, in the utilized method, the global approximation and its derivatives can be established on the whole solution domain; fourth, the method does not require discretization of the variables, and it is not affected by the computation round-off errors and one is not faced with necessity of large computer memory and time. For more details and descriptions about the methodology of the RKHS method including its history and theory, its modification, and its characteristics, we refer to [1926].

This paper is organized as follows. In the next section, we present some necessary definitions and preliminary results from the fuzzy calculus theory including fuzzy Riemann integrability concept. The procedure for solving fuzzy Volterra integral equation (1) is presented in Section 3. In Section 4, reproducing kernel algorithm is built and introduced in the Hilbert space . The numerical results are reported to demonstrate the superiority and capability of the proposed method by considering some numerical examples in Section 5. The last section is a brief conclusion.

2. Preliminaries

The material in this section is basic in certain sense. For the reader’s convenience, we present some necessary definitions and notations from fuzzy calculus theory which will be used throughout the paper. The reader is kindly requested to go through [2730] in order to know more details about fuzzy calculus and fuzzy differential and integral equations.

Let be a nonempty set, a fuzzy set in is characterized by its membership function , while is interpreted as the degree of membership of an element in the fuzzy set for each A fuzzy set on is called convex, if, for each and , ; is called normal, if there is such that ; and is called upper semicontinuous, if the set is closed for each The support of a fuzzy set is defined as

Definition 1 (see [28]). A fuzzy number is a fuzzy subset of with normal, convex, and upper semicontinuous membership function of bounded support.

For , put and , where is the closure of . Then, it is easy to establish that is a fuzzy number if and only if is a compact convex subset of for each and [27]. Thus, if is a fuzzy number, then , where and for each Hence, the -level set is a nonempty compact interval for each and any , where is the set of fuzzy numbers on . The notation is called the -cut representation or parametric form of a fuzzy number . The last description leads to the following characterization theorem to define the parametric form of a fuzzy number in terms of the endpoint functions and

Theorem 2 (see [27]). Suppose that the functions satisfy the following conditions; first, is a bounded increasing function and is a bounded decreasing function with ; second, for each and are left-hand continuous functions at ; third, and are right-hand continuous functions at . Then, defined by is a fuzzy number with parameterization Furthermore, if is a fuzzy number with parameterization , then the functions and satisfy the aforementioned conditions.

In general, we can represent an arbitrary fuzzy number by an order pair of functions based upon the requirements mentioned in Theorem 2. Frequently, we will write simply and instead of and , respectively, for each .

The metric structure on is given by the Hausdorff distance mapping such that for arbitrary fuzzy numbers and . In [31], it has been proved that is a complete metric space.

Definition 3 (see [32]). Let , if there exists such that , then is called the -difference (Hukuhara difference) of and , and it is denoted by .

It is worth mentioning that the sign stands always for the -difference and If the -difference exists, then

For arbitrary fuzzy numbers and , we define the addition and scalar multiplication by as and . Moreover, if and only if ; that is, and .

Definition 4 (see [27]). Let be a fuzzy-valued function on a compact convex subset of . Then, we say that is continuous at if, for every , there exists such that , for all with .

Theorem 5 (see [27]). Let be a fuzzy-valued function, where , and then is continuous on if and only if both and are continuous on .

For the concept of fuzzy integral, we will define the integral of a fuzzy-valued function using the Riemann integral concept [27, 30], which has the advantage of dealing properly with fuzzy IEs, as follows.

Definition 6 (see [30]). Let be continuous fuzzy-valued function. For each partition of and for arbitrary points, , let and . Then, the definite integral of over is defined by , provided the limit exists in the metric space

Definition 7 (see [30]). Let be continuous fuzzy-valued function, where , and then exists, defined by , whereas and are integrable functions over

It should be noted that the fuzzy integral can be also defined using the Lebesgue-type approach [28] or the Henstock-type approach [31]. However, if is continuous function, then all approaches yield the same value and results. Moreover, the representation of the fuzzy integral using Definition 6 is more convenient for numerical calculations and computational mathematics.

3. Formulation of Fuzzy Volterra Integral Equation

Formulation of the fuzzy IEs is normally the most important part of the process. It consists of the determination of -cut representation form of nonlinear term , the selection of the integrability type, and the separation of the kernel function . In this section, we study the fuzzy IEs using the concept of Riemann integrability in which the FIEs are converted into equivalent system of crisp integral equations (CIEs). These can be done if the solution is fuzzy function, and consequently the integral must be considered as fuzzy integral.

In order to design RKHS algorithm for solving (1), we set and we write the fuzzy function in terms of its -cut representation form to get that By considering the parametric form for both sides of the FIE (1), one can getwhere and in which

Therefore, according to the previous results, the FIE (1) can be translated into the following equivalent form:

Let be continuous fuzzy-valued function. If satisfies (1), then we say that is a fuzzy solution to FIE (1). On the other aspect as well, the formulation of (1) together with the characterization Theorem 2 shows us how to deal with numerical solutions to FIEs. We can translate the original fuzzy IE equivalently into system of crisp IEs. In conclusion, one does not need to rewrite the numerical methods for crisp IEs in fuzzy setting, but, instead, we can use the numerical methods directly on the obtained crisp integral system.

4. Construction of the RKHS Method

In this section, the formulation of exact and approximate solutions to (1) and the implementation method are given. Initially, we utilize the reproducing kernel concept to construct the Hilbert space After that, we construct an orthogonal function system of the space using Gram-Schmidt orthogonalization process. Here,

Definition 8 (see [33]). Let be a nonempty abstract set. A function is a reproducing kernel of the Hilbert space if the following conditions are satisfied: firstly, for each Secondly, for each and

The last condition is called “the reproducing property” which means that the value of the function at the point is reproducing by the inner product of with . Indeed, a Hilbert space of functions on a set is called a reproducing kernel Hilbert space (RKHS) if there exists a reproducing kernel of . In functional analysis, RKHS is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels.

Definition 9 (see [34]). The inner product space defined as is absolutely continuous real-valued function, . Meanwhile, the inner product and norm in are defined, respectively, byand , where .

Definition 10 (see [34]). The Hilbert space is called a reproducing kernel if, for each fixed , there exists such that for any and .

It is important to mention here that the reproducing kernel function of a Hilbert space is unique and the existence of is due to the Riesz representation theorem, where completely determines the space .

Theorem 11. The Hilbert space is a complete reproducing kernel space and its reproducing kernel function is given by

Proof. Considerwhere
Through iterative integrations by parts for (8), we obtain that Thus, it becomes If and , then . For each , if also satisfiesthen .
For , the characteristic equation of (11) is given by , and then the characteristic values are with multiplicity 2. So, let the expression of the reproducing kernel function be as defined in (7). But, on the other aspect as well, if satisfies , then, by integrating (11) from to with respect to and letting , we have the jump degree of at such that . Hence, through the last descriptions, the unknown coefficients of (7) can be obtained.

The unique representation of the reproducing kernel function in the space is provided in [26] and is given by

The space is complete Hilbert space with some special properties. So, all the properties of the Hilbert space will be held. Further, this space possesses some special and better properties which could make some problems solved easier. However, these properties require no more integral computation for some functions, instead of computing some values of a function at some nodes. In fact, this simplification of integral computation not only improves the computational speed but also improves the computational accuracy.

In order to illustrate the analytical solution to the model problem, we consider that is an invertible bounded linear operator such that . Thus, system (5) can be converted into the equivalent form as follows:where , and , , , and .

Let , where , for each , is the conjugate operator of and is a countable dense subset on . Next, the subscript of means that the operator acts on the function of in which .

Theorem 12. If is dense on , then the orthogonal functions system is a complete system of and .

Proof. Clearly, . Now, for each fixed , let , , which means that . But since is dense on , we must have . It follows that from the existence of and the continuity of . So, the proof of the theorem is complete.

In order to utilize the representation form of analytic and approximate solutions to (1), we will use the Gram-Schmidt process that produces an orthonormal sequence of the space from such thatwhere are orthogonalization coefficients given as in which .

Theorem 13. If is dense on and , are the exact solutions to system (13), and then and satisfy the following form, respectively:and the approximate solutions can be obtained bywhere and (fixed) .

Proof. From Theorem 12 and the process of (14), it is obvious that the sequence forms an orthonormal system in . Since and are the exact solutions to system (13), then can be expanded to Fourier series in terms of normal orthogonal basis in as follows: and since the space is the Hilbert space, then the series is convergent in the norm . Thus, can be written as Similarly, one can get . The approximate solutions can be obtained directly by taking finitely many terms in the series representation for the exact solutions and to (16) and (17). So, the proof of the theorem is complete.

Lemma 14. If , then there exists , such that , where .

Theorem 15. Suppose that and are bounded in (18). If is dense on and and for any , then the approximate solutions and in iterative formulas (18) are convergent to the analytic solutions and to system (13) in and the exact solutions are expressed as , , where

Proof. The proof can be divided into two steps. Firstly, we will prove the convergence of and . From (18), we infer that and . The orthogonality of yields that In another formulation, it holds that and . Due to the condition that and are bounded, then and are convergent as It implies that , . This means that and (). On the one hand, since , , and it follows for that Considering the completeness of , there exist and such that and as in sense of .
Secondly, we will prove that and are the solutions to system (13). By Lemma 14 and since is dense on , we know that and converge uniformly to and , respectively. Taking limits in (18), one can get and . Since it follows that Now, if , then and . Again, if , then and . In the same way, it is easy to see by induction that and . Since is dense on , then, for any , there exists subsequence such that as . But, on the other aspect as well, we have known that and . Hence, let , and, by the continuity of and , we have and . That is, and are the solutions to system (13). So, the proof of the theorem is complete.

The reliability of the numerical result will depend on an error estimate; therefore, the analysis of error and as well as the sources of error in numerical methods is also a critically important part of the study of numerical technique.

Theorem 16. Assume that and are given by (16) and (17) and and are the errors in the approximate solutions and , respectively, where and are given in (18). Then, the errors are monotone decreasing in the sense of .

Proof. Based on the previous results, it is obvious that and . Clearly, . Consequently, the error is monotone decreasing in the sense of . Similarly, the error is monotone decreasing in the sense of . So, the proof of the theorem is complete.

Software packages have great capabilities for solving mathematical, physical, and engineering problems. The aim of the next algorithm is to implement a procedure to solve FIE (1) in numeric form in terms of its grid nodes based on the use of RKHS method.

Algorithm 17. To approximate the solution to system (5), we do the following steps.
Input. Consider the endpoints of , the integer , the kernel function , the operator , and the functions and .
Output. Consider approximate solutions and .
Step  1. Fix in and set :If , set .Else set .For , do the following.Set .Set .Output is the orthogonal function system .Step  2. For and , do the following:If , then set .Else set .Else set .Output is the orthogonalization coefficients .Step  3. For , do the following:Set .Output is the orthonormal function system .Step  4. For , do the following:Set .Set .Set .Set .Set .Set .Output is the approximate solutions and and then stop.

5. Experiment Results

The proposed method provides analytical as well approximate solutions in terms of a rapidly convergent series with easily computable components. However, there is a practical need to evaluate these solutions and to obtain numerical values from these series. The consequent series truncation and the practical procedure are conducted to accomplish this task. In this section, we consider three examples to illustrate the efficiency and performance of the RKHS in finding approximate series solution for both linear and nonlinear FIEs. On the other hand, results obtained are compared with the exact solution to each example and are found to be in good agreement with each other. In the process of computation, all symbolic and numerical computations are performed by Mathematica software package.

Example 1. Consider the following fuzzy Volterra integral equation:where The exact fuzzy solution is , .

For the conduct of proceedings in the fuzzy solution, we have the following system of CIEs taking into account that the crisp kernel function is positive on . Thus, by considering the parametric form of (27), one can write

To illustrate the fuzzy behaviors of the approximate solutions at some specific certain computed nodes, the absolute errors of numerically approximating , , , , for the corresponding CIE system have been calculated for various and on as shown in Tables 1 and 2. Here, the absolute errors in Table 1 are given by while in Table 2 they are given by . Anyhow, it is clear from the tables that the approximate solutions are in close agreement with the exact solutions as well; we see that at . The fuzzy solution to (27) has been plotted as shown in Figure 1. On the other aspect as well, one can see that the numerical result satisfies the convex symmetric triangular fuzzy number.

Example 2. Consider the following fuzzy Volterra integral equation:where . The exact fuzzy solution is , .

Using the RKHS method, taking , , if we choose negative crisp kernel function on , then the corresponding crisp IE system of (29) can be written as

The numerical results for the corresponding crisp IE system of (29) for and various in are given in Tables 3 and 4. As we mentioned earlier, it is possible to pick any point in the independent interval of and as well the approximate solutions will be applicable. The fuzzy solution to (29) has been plotted as shown in Figure 2.

Example 3. Consider the following nonlinear fuzzy Volterra integral equation:where . The exact solution is , .

Here, it can be observed that is a continuous increasing function on . Then, by using Zadeh’s extension principle, we get for all . Furthermore, the crisp kernel function is positive on . So, the FIE (31) can be converted into nonlinear crisp integral equations system as

The absolute errors of numerically approximating and for the corresponding crisp integral equations system (32) have been calculated at and various as shown in Tables 5 and 6. It is clear from the tables that the approximate solutions are in close agreement with the analytic solutions, by using only in our algorithm. Indeed, we can conclude that higher accuracy can be achieved by computing further RKHS steps for this nonlinear example. Finally, the fuzzy solution to FIE (31) has been plotted as shown in Figure 3.

6. Conclusion

The aim of present analysis is to propose a relatively recent numerical method for solving a class of fuzzy Volterra integral equations using the concept of Riemann integrability. The method is applied directly without using linearization, transformation, or restrictive conditions. Numerical results show that the RKHS is of higher precision and powerful technique and applicable to solve different types of fuzzy integral equations. Moreover, the accuracy of the solution can be improved by selecting large value of .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.