Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

1Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11942, Jordan 2Department of Mathematics, Faculty of Science and Arts, Shaqra University, Shaqra 11691, Saudi Arabia 3Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan 4School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 5Research Institute, Center for Modeling & Computer Simulation (RI/CM & CS), King Fahd University of Petroleum &Minerals, Dharan 31261, Saudi Arabia


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 [1][2][3][4][5].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 [6][7][8][9][10].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 [16][17][18].The purpose of this paper is to extend the application of the reproducing kernel Hilbert space (RKHS) method to provide where (, ) is continuous arbitrary crisp kernel function,  is linear or nonlinear continuous increasing function,  is a continuous fuzzy-valued function, and  ∈  1 2 [, ] 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 [19][20][21][22][23][24][25][26].
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  1 2 [, ].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.

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 [27][28][29][30] 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  :  → [0, 1], while () is interpreted as the degree of membership of an element  in the fuzzy set  for each  ∈ .
Definition 1 (see [28]).A fuzzy number  is a fuzzy subset of R with normal, convex, and upper semicontinuous membership function of bounded support.
where {⋅} is the closure of {⋅}.Then, it is easy to establish that  is a fuzzy number if and only is a nonempty compact interval for each  ∈ [0, 1] and any  ∈ R  , where R  is the set of fuzzy numbers on R. 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 ().
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  ∈ [0, 1].
The metric structure on R  is given by the Hausdorff distance mapping  : for arbitrary fuzzy numbers  and V.In [31], it has been proved that (R  , ) is a complete metric space.Definition 3 (see [32]).Let , V ∈ R  , if there exists  ∈ R  such that  = V + , then  is called the -difference (Hukuhara difference) of  and V, and it is denoted by  ⊝ V.
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.
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.

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 Therefore, according to the previous results, the FIE (1) can be translated into the following equivalent form: Let  : Ω → R  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.

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  1 2 [, ].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  :  ×  → R 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 Proof.Consider where (),   () ∈  1 2 [, ].Through iterative integrations by parts for (8), we obtain that Thus, it becomes then ⟨(),   ()⟩  1 2 = ().For  ̸ = , the characteristic equation of ( 11) is given by  2 = 0, and then the characteristic values are  = 0 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   ( + 0) =   ( − 0), then, by integrating (11) from  −  to  +  with respect to  and letting  → 0, we have the jump degree of     () at  =  such that     ( + 0) −     ( − 0) = −1.Hence, through the last descriptions, the unknown coefficients of ( 7) can be obtained.
The unique representation of the reproducing kernel function   () in the space  1 2 [0, 1] is provided in [26] and is given by The space  1 2 [, ] 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.
Theorem 16.Assume that () and () ∈  1  2 [, ] 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 ‖ ⋅ ‖  1  2 .Proof.Based on the previous results, it is obvious that and Consequently, the error   is monotone decreasing in the sense of ‖ ⋅ ‖  1  2 .Similarly, the error   is monotone decreasing in the sense of ‖ ⋅ ‖  1  2 .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 [0, 1], the integer , the kernel function   (), the operator , and the functions  and .
Output is the orthogonal function system   ().

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 7.0 software package.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 0 ≤  ≤ 1.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   (  ),  = 101,   = ( − 1)/( − 1),  = 1, 2, . . ., , for the corresponding CIE system have been calculated for various  and  on [0, 1] as shown in Tables 1 and 2. Here, the absolute errors in Table 1 are  given by   = |(  ) −   (  )| while in where 0 ≤  ≤ 1.The exact fuzzy solution is () = [( − 1)(sinh() + cosh() − 1), (1 − )(sinh() + cosh() − 1)],  ∈ [0, 1].
Using the RKHS method, taking   = ( − 1)/( − 1),  = 1, 2, . . ., , if we choose negative crisp kernel function (, ) = −1 on 0 ≤  ≤ 1, then the corresponding crisp IE system of (29) can be written as The numerical results for the corresponding crisp IE system of (29) for  = 0.75 and various  in [0, 1] 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 Here, it can be observed that () = √ is a continuous increasing function on [0, ∞).Then, by using Zadeh's extension principle, we get [(())]  = [√(), √()] for all  ∈ [0, 1].Furthermore, the crisp kernel function (, ) is positive on 0 ≤  ≤ 1.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  = 0.5 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  = 101 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.

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
1 2 [, ] defined as  1 2 [, ] = {() | () is absolutely continuous real-valued function,   () ∈  2 [, ]}.Meanwhile, the inner product and norm in  1 2 [, ] are defined, respectively, by 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 .

Table 1 :
The absolute errors of approximating () for (27) at various  and .

Table 2 :
The absolute errors of approximating () for (27) at various  and .