Numerical solution of fractional differential equations with temporal two-point BVPs using reproducing kernal Hilbert space method

In this paper, the reproducing kernel Hilbert space method had been extended to model a numerical solution with two-point temporal boundary conditions for the fractional derivative in the Caputo sense, convergent analysis and error bounds are discussed to verify the theoretical results. Numerical examples are given to illustrate the accuracy and efficiency of the presented approach.


Introduction
In recent years, the study of fractional derivatives has been an important topic. It has been used to model many phenomena in numerous fields such as science and engineering. There are many interpretations for fractional derivatives, such as the definition of Caputo [1], the definition of Riemann-Liouville [2], the definition of Grunwald-Letnikov [3], and most recently, Conformable [4], Atangana-Baleanu [5], Wallström [6], Jumarie [7], Klimek [8] and others.
In practice, where quantitative results are needed for given real-world problems, numerically approximate solutions can often be demonstrably better, more reliable, more detailed, efficient and cost-effective than analytical ones for certain fractional structures. A number of studies [9][10][11][12][13][14] were therefore involved in developing approaches for providing estimated solutions. One of these approaches is the Hilbert space kernel reproduction (RKHS) method used for the first time by S. Zaremba for the harmonic and biharmonic functions at the beginning of the 20th century to find solutions for boundary value problems (BVPs).
The RKHS precede the Dirac delta function in many ways, among which we mention providing an important structure for random distribution of multi-round data and, providing accurate approximation of multi-dimensional general functions and the possibility to pick any point in the integration interval.
Moreover, the numerical solution of the different groups of BVP can be found in [40][41][42]. The two-point BVPs has a strong interest in applied mathematics, this kind of problems arise directly from mathematical models or by turning partial differential equations into ordinary differential equations. As this type of problems does not have an exact solution, many special techniques have been used to solve it, including the shooting method [43][44], the collocation method [45][46], the finite difference method [47,48], and the quasilinearization method [49,50]. The continuous genetic algorithm approach was used to solve these schemes in [51][52][53].
The present paper is structured as follows: in Section 2, we set out some basic concepts and results from fractional calculus theory. In Section 3, the iterative form of the reproducing kernel algorithm is used to build and measure the solution of the fractional differential method with temporal two points. In Section 4 and 5, the convergence and error estimator are discussed to provide a number of numerical results to demonstrate the efficiency and accuracy of the reproducing kernel Hilbert space method. At last in section 6, a conclusion of the results is made.

Preliminaries
In applied mathematics and mathematical analysis, there are several definitions of fractional derivatives, Riemann-Liouville and Caputo are the most popular of all [54]. In this section, we list some of these definitions in addition to reproducing kernel spaces on finite domain [t 0 , t f ].
for t 0 ≤ x ≤ t f , is called the Riemann-Liouville fractional integral operator of order n. For n = 0, we set J 0 t 0 := I, the identity operator. Definition 2.2. [55] Let n ∈ R + and m = [n]. The operator D n t 0 defined by is called the Riemann-Liouville fractional differential operator of order n. For n = 0, we set D 0 t 0 := I, the identity operator. Definition 2.3. [55] Let α ∈ R + and n − 1 < α < n. The operator D α * t 0 defined by for t 0 ≤ x ≤ t f , is called the Caputo differential operator of order α.
Definition 2.4. [35] Let M be nonempty set, the function K : M × M −→ C is a reproducing kernel of the Hilbert space H if the following conditions are met: the reproducing property: ∀t ∈ M, ∀z ∈ H : z(.), K(., t) = z(t).
The second condition means that the value of z at the point t is reproduced by the inner product of z with K.
Note: The reproducing kernel is unique, symmetric and positive definite.
The inner product and its norm are given by: The inner product and its norm are given by: The inner product and its norm in W 3 2 [t 0 , t f ] are given by: (1) In [56], W 1 2 is RKHS and its reproducing kernel is: (2) In [57], W 2 2 is RKHS and its reproducing kernel is: t(−t 2 + 3s(2 + t)) t ≤ s, s(−s 2 + 3t(2 + s)) t > s.

Reproducing kernel Hilbert space method (RKHSM)
In this section, we develop an iterative method for constructing and calculating fractional differential equations with a temporal two-point solution. In order to emphasize the idea, we start by considering the general form of the BVP: Subject to BC's: where: δ, β ∈ R, and D α denotes the Caputo fractional derivative of order α and We use the RKHS method to obtain a solution of BVPs (3.1) and (3.2) based on the following methodology: • To attain a problem with homogenous BC's, we first assume that: Y(t 0 ) = γ, (γ arbitrary) and We get: (3.4) Subject to: • Then, we construct the reproducing kernel space W 2 2 [t 0 , t f ] in which each function satisfies the homogeneous boundary conditions of (3.5) using the space W 1 2 [t 0 , t f ]. Take K t (τ) and R t (τ) to be the reproducing kernel functions of the spaces W 2 2 [t 0 , t f ] and W 1 2 [t 0 , t f ] respectively. • Next, we define the invertible bounded linear operator L : (3.6) The BVPs (3.4), (3.5) can therefore be transformed to the following form: Applying Riemann-Liouville fractional integral operator J α t 0 to both sides using U(t 0 ) = 0 and V(t 0 ) = 0, we get: Thus, we can notice that: LU(t) = U(t), and so the BVPs are transformed to the equivalent form: , and L * is the adjoint operator of L.
can be written on the following form: Proof. It is clear that: ik are positive orthogonalization coefficients such that: then the solutions of (3.8) given by: (3.10) The same for finding V(t): The theorem is proved.
Since W 2 2 is Hilbert space we get: (3.11) are convergent in the sense of . W 2 2 and (3.11) represents the numerical solution of (3.8).
(1) If the system (3.7) is linear, then the exact solutions can be found directly from (3.10).

15)
and so: (3.16) We can guarantee that the approximations U n (t), V n (t) satisfies the conditions enjoined by (3.7) through the iterative process of (3.16).

Error estimation and convergence
In this section, we present some convergence theories to emphasize that the approximate solution we got is close to the exact solution. Indeed, this finding is very powerful and efficient to RKHS theory and its applications.
, η = 1, 2 are the complete orthonormal systems in the space W 2 2 [t 0 , t f ] then we have: Thus U n (t) W 2 2 , V n (t) W 2 2 are monotone increasing. Lemma 4.2. As n → ∞, the approximate solutions U n (t), V n (t) and its derivatives U n (t), V n (t) are uniformly convergent to the exact solutions U(t), V(t) and its derivatives U (t), V (t) respectively.
Proof. For any t ∈ [t 0 , t f ]: and F(t, U(t), V(t)), G(t, U(t), V(t)) are continuous in [t 0 , t f ], then: as n −→ ∞. Proof. For the first part, we will prove that: it is easy to see that: , by reproducing property of K t (τ) we have: and from the symmetric property of K t (τ) we get: thus: in the sense of the . W 2 2 as t n −→ t and n −→ ∞ for any t ∈ [t 0 , t f ]. Moreover, since F and G are continuous, we obtain: as n −→ ∞.
, then the approximate solutions U n (t), V n (t) in Eq (3.14) convergent to the exact solutions U(t), V(t) of Eq (3.7) in the space W 2 2 [t 0 , t f ] and U(t), V(t) given by (3.12). Proof. We first start by proving the convergence of U n (t) and V n (t) from Eq (3.14) we conclude that: , (η) = 1, 2 we get: U n (t) W 2 2 , V n (t) W 2 2 are monotone increasing by Lemma (4.1). From the assymption that (3) and (4) in [25] we have: let j goes to ∞, by theorem (4.1) and the continuity of F and G we have: Proof. From the extension form of U(t), V(t) and U n (t), V n (t) in Eqs (3.12), (3.14) and (3.15) we can write: Clearly: ξ n ∞ n=1 , ξ n ∞ n=1 are decreasing in a sense of . W 2 2 from theorem (3.2) the series Theorem 4.4. The approximate solutions U n (t), V n (t) of (3.7) converge to its exact solutions U(t), V(t) with not less than the second order convergence. That is: |U n − U| ≤ Mk 2 and |V n − V| ≤ Nk 2 , where k = t f −t 0 n . Proof. See [36].

Numerical examples and algorithm
Numerical examples are conducted in order to verify the accuracy of this method. Computations are performed using Mathematica 11.0.
Algorithm 1: Use the following stages to approximate the solutions of BVPs (3.4) and (3.5) based on RKHS method.
Then we implement the above algorithm using numerical simulations. We arrange the resulting data in tables and graphs for examples discussed on [t 0 , t f ] as follows: Example 5.1. Consider the following system: subject to: with exact solution when α = 1 is: After the initial conditions have been homogenised and algorithm 1 used, apply t i = 0.5i n , i = 1, n and n = 40, the tables 1 and 2 describe the exact solutions of ω(t) and Θ(t) and approximate solutions for different values of α.   Graphs of the approximate solutions of ω(t) are plotted in Figure 1 (a), for different values of α. It is obvious from Figure 1 (a) that the approximate solutions are in reasonable alignment with the exact solution when α = 1 and the solutions are continuously based on a fractional derivative. The graph in Figure 1 (b) represent the absolute errors of θ(t).
Example 5.2. Consider the following system: with conditions: when α = 1 the exact solution is: After homogenizing the initial conditions and using algorithm 1, apply t i = i n , i = 1, n and n = 35, the tables 3 and 4 describe the exact solutions of ω(t) and Θ(t) and approximate solutions for different values of α.   Graphs of the approximate solutions of θ(t) are plotted in Figure 2 (b) for different values of α. The graph in Figure 2 (a) represent the absolute errors of ω(t).
Example 5.3. Consider the following fractional system: with exact solution: After the initial conditions have been homogenised and algorithm 1 used, apply t i = i n , i = 1, n and n = 30, the tables 5-7 describe the exact solutions of ω(t), Θ(t) and ρ and approximate solutions for different values of α.   Graphs of the approximate solutions of ω(t) and θ(t) are plotted in Figure 3 (a), Figure 3 (b) for different values of α. The graph in Figure 3 (c) represent the absolute errors of ρ(t). Now, we consider the following tables where the RKHS method has been applied in order to give numerical approximations with other values of n, and then compare it with finite difference and collocation methods.

Conclusions
In this article, we effectively utilize the RKHSM to develop an approximate solution of differential fractional equations with temporal two-point BVP. The results of examples demonstrate reliability and consistency of the method. In the future, we recommend further research on the RKHS method, as solving the temporal two-point boundary value problems with the conformable and the Atangana-Baleanu derivatives. We expect to achieve better results and good approximations for the solutions.

Conflict of interest
The authors state that they have no conflict of interest. All authors have worked in an equal sense to find these results.