Abstract
In this article, the reproducing kernel method is presented for the fractional differential equations with periodic conditions in the Hilbert space. This method gives an approximate solution to the problem. The approximate and exact solutions are displayed in the form of series in the reproduction kernel space. In addition, we provide an error analysis for this technique. The presented method is tested by some examples to show its precision.
1. Introduction
Fractional differential equations are needed to model and analyze large volumes of problems. FDEs are applied in large number of fields such as fluid mechanics, biology, chemistry, and diffusion [1–7]. Some methods for solving these equations are Laplace transforms [8], Fourier transform [9], Adomian decomposition method [10], finite difference method [11], variational iteration method [12, 13], collocation method [14], and other methods [15–19].
Many papers have worked on FDEs with periodic conditions, some of which are listed below. Belmekki et al. have discussed the existence and uniqueness of the solution in [20]. Wei et al. have reviewed the minimal and maximal solutions for periodic problems in [21]. In [22], authors have given monotone iterative techniques for existing solutions. In [23], Javidi and Saedshoar Heris have used the method fractional backward differentiation formulas forwith periodic condition .
In this work, we use reproducing kernel Hilbert space (RKHS) method to solve multiterm FDEs in the form as follows:with periodic condition as follows:where , in Caputo sense. In [24], the existence and uniqueness of the solution have been proven to this problem by using green’s function.
The reproducing kernel method was first used in research on boundary value problems in the early twentieth century. In 1907, Zarmba was the first to introduce the kernel of certain functions and to express their reproducing properties. Since 1980, with the efforts of Cui, the reproducing kernel functions of Hilbert space have been introduced in the form of very simple polynomials. They were able to use methods based on the reproducing kernel space [25–27]. Many researchers use the RKHS method to find approximate solutions to various problems [28–30], and also some new applications of reproducing kernel methods and neural networks in machine learning are found in [31–33]. Very recently, RKHS is applied on fractional differential equations [34–36]. In this paper, the reproducing kernel method is presented for the fractional differential equations with periodic conditions in the Hilbert space. The approximate solution obtained from this method is uniformly convergent to the exact solution.
This paper is arranged as follows. Section 2 provides some definitions. Analysis of the RKHS method is proposed in Section 3. The convergence of the approximate solution to the exact solution is given in Section 4. Examples are given in Section 5.
2. Basic Definitions
We describe some of the symbols and basic definitions used in this article. Let represent the Banach space of all continuous functions of into , and shows the real valued functions on where the th order derivative is continuous.
Definition 1. The fractional integral of of order is
Definition 2. The Caputo fractional derivative of of order is
Definition 3 (see [37]). Suppose is a function Hilbert space, including all real or complex value functions defined on a abstract space , with the inner product . For each fixed , if there exist a function which satisfiesthen is called the reproducing kernel of and the Hilbert space is called the reproducing kernel space.
Remark 1. The real value function spaceis a function Hilbert space withwhere denotes the set of square Lebesgue integrable functions on .
Remark 2. The reproducing kernel function in can be written asIt is easy to prove that is obtained as follows.
From (8), we haveUsing several integration by part of , we obtain thatIf , then ; also if , then . Therefore,Therefore, satisfies the following generalized differential equation:where denotes the Dirac delta function. While , is the solution of the constant differential equation:with the boundary conditionsThe characteristic equation for (15) is . Therefore, the general solution can be written as (10), where coefficients and , are obtained by solving the following equations:Therefore, the reproducing kernel function in is obtained as follows:
Remark 3. The real value function spaceis a function Hilbert space with inner productIt can be proved that is a reproducing kernel Hilbert space and
3. Solution Procedure (2) by RKHS Method
Here, we will construct a linear differential operator and an orthogonal system in . After that, the RKHS method for obtaining solution (2) with condition (3) is presented.
First, by introducing linear operator asthen problem (2) will be converted into the following form:
Theorem 1. The operator is a bounded linear operator.
Proof. It can be easily shown that L is a linear operator. So, we only prove the boundary of L. From (20), we haveBy reproducing property of , we haveBy Schwarz inequality, we getwhere are positive constants and the proof is completed.
Thus, and That is,where . We will construct a complete system of of by setting and , where is dense on [0, T] and is conjugate operator of .
Lemma 1 (see [37]). If is dense on [0, T], then is a complete system of and .
By using the Gram–Schmidt process of is obtained the orthonormal basis of space , which satisfies
The coefficients are positive and given bywhere .where are unknown and we will obtain by using . So, suppose and is given bywhere of is given by
Theorem 2 (see [37]). Let be dense set in and the exact solution of (23) in space be unique, thenand for this problem, approximate solution th order as follows:
Theorem 3. If , then there exists constant such thatProof. For each , we obtainand we also havewhere and are positive constants and .
Corollary 1. The approximate solutions and uniformly converge to the exact solutions and , respectively.
Proof. From Theorem 3, for each , we obtainwhere are positive constants. Then, if as , the approximate solutions and converge uniformly to and , respectively.
Remark 4. We apply the following two cases to solve equations (2) and (3) by using the RKHS method.
Case 1. Let (2) be linear and (30) and (31) denote the exact and approximate solutions, respectively.
Case 2. Let (2) be nonlinear; in this case, the solution of (2) is as follows:
We can guarantee that in equation (37) satisfies condition (3).
4. Convergence Analysis
In this section, we will show that approximate solution of equation (37) is convergent to the exact solution of equation (2). First, we express the following lemma.
Lemma 2. If and is continuous function with respect to and , then .
Proof. Observe thatReproducing property of yields thatFrom the symmetry of , result is . Therefore, as . From Corollary 1, it holds that . Then, . Because is continuous functions, then .
Lemma 3. For in equation (37), we haveProof. Suppose , therefore,By using orthogonality of , we obtainIf thenBesides if thenthat is, . By the same manner, it yields thatHence, is obtained by taking the limit of equation (37). Therefore, is an orthogonal projector of to span . Then,
Theorem 4. Let be bounded and is dense on , then -term approximate solutions in equation (37) converge to
Proof. Firstly, we prove that the convergence of in equation (37) is convergent in the sense of . From equation (37), it is inferred that . Since is orthogonal, hence,where . It holds that .
Because is bounded, is convergent as . Therefore, there exists constant such that . The implies that .
Let then from the orthogonality of , it follows thatbecause . Consequently,Hence, is complete, and then , as .
Now, we prove that is the solution of equation (23). Because is dense on , for each , there exists a subsequence such that as .
Since is dense on , thus for all , there exists a subsequence such that as . By Lemma 3, it follows that . Hence, let , from Lemma 2 and the continuity of , we have .
Theorem 5. Assume , where is derived from the RKHS method. Therefore, is decreasing in .
Proof. Note thattherefore, .
5. Numerical Tests
We provide three examples to explain the content given, and we realize the validity and accuracy of the RKHS method.
Example 1. In this example, we consider FDE with periodic condition:for . The exact solution is . We choose points in , and by using the proposed method, the approximate solution is obtained. Absolute error values are reported in Table 1 for and . The graphs of the absolute error and the numerical solution are plotted in Figure 1. Here,andfinally and are obtained from (36) and (37), respectively.
Example 2. We consider FDE with periodic condition:for . The exact solution is . We choose 20 points in , and by using the proposed method, we obtained the approximate solution Absolute error values are reported in Table 2 for and . The graphs of the absolute error and the numerical solution are plotted in Figure 2.
Example 3. We consider FDE with periodic conditionfor . The exact solution is . Using this method, taking , for and , the numerical results are given in Table 3 for . Three graphs of the approximate solution for are drawn in Figure 3.
6. Conclusion
In this paper, we have proposed the RKHS method to solve fractional differential equations with periodic conditions. This method is a powerful technique for finding approximate solutions. The approximation error and convergence analysis are obtained in the RKHS. We illustrate the efficiency of the method with a few examples.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.