1. Introduction
In this paper, a new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of a multi-order fractional nonlinear three-point boundary value problem as follows:
with the following boundary conditions,
Here, ∈ and are sufficiently smooth functions, is a weighted function and it will be taken as for Legendre polynomials. Fractional derivatives are also taken in the Caputo sense. Without loss of generality, we pay regards to , and . Because , and , boundary conditions can be easily reduced to , and .
Nonlinear fractional multi-point boundary value problems appear in a different area of applied mathematics and physics ([
1,
2,
3,
4,
5,
6,
7] and references therein). Many important phenomena have been concerned in engineering and applied science, such as dynamical systems, fluid mechanics, control theory, oil industries, and heat conduction, and can be well-turned by fractional differential equations [
8,
9,
10]. Some applications, qualitative behaviors of solutions and numerical methods to find approximate solutions have been investigated for differential equations with fractional order in [
11,
12,
13,
14].
More specfically, it is not easy to directly get exact solutions to most differential equations with fractional order. Hence, numerical techniques are largely utilized. Actually, in recent times, many efficient and convenient methods have been developed, such as the finite difference method [
15], finite element method [
16], homotopy perturbation method [
17], Haar wavelet methods [
18], collocation methods [
19], homotopy analysis method [
20], differential transform method [
21], variational iteration method [
22], reproducing kernel space method [
23,
24] and so on [
25,
26,
27,
28,
29,
30,
31,
32].
In 1908, Zaremba firstly introduced the reproducing kernel concept [
33]. His researches regarded boundary value problems, which include the Dirichlet condition. The reproducing kernel method (RKM) produces a solution in convergent series form for many differential, partial and integro-differential equations. For more information, we refer to [
34,
35]. Recently, the RKM has been applied for a different type of problem. For example, fractional order nonlocal boundary value problems [
36], Riccati differential equations [
37], forced Duffing equations with a nonlocal boundary conditions [
38], Burgers’ equation with a fractional order Caputo derivative [
39], time-fractional Kawahara equation [
40], fractional order Boussinesq equation [
41], nonlinear fractional Volterra integro-differential equations [
42].
The Legendre reproducing kernel method is proposed for the fractional two-point boundary value problem of Bratu type equations [
43]. The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving multi-order fractional nonlinear three-point boundary value problems with a Caputo derivative.
The remainder part of the paper is prepared as follows: some fundamental definitions of fractional calculus and the theory of reproducing kernel with Legendre basis functions are given in
Section 2. The structure of the solution with a Legendre reproducing kernel is demonstrated in
Section 3. In order to show the effectiveness of the proposed method, some numerical findings are reported in
Section 4. Finally, the last section contains some conclusions.
2. Preliminaries
In this section, several significant concepts, definitions, theorems, and properties that will be used in this research are provided.
Definition 1 ([
8,
12,
13])
. Let and . Then, the α order Riemann–Liouville fractional integral operator is given as:here is a Gamma function, and . Definition 2 ([
8,
12,
13])
. Let and . Then, the α order Caputo differential operator is given as: Definition 3 ([
26,
43]).
In order to a construct polynomial type reproducing kernel, the first kind of shifted Legendre polynomials are defined over the interval . For obtaining these polynomials, the following iterative formula can be given:The orthogonality requirement ishere, the weighted function is taken as, Legendre basis functions can be established so that this basis function system satisfies the homogeneous boundary conditions as: Equation (5) has an advantageous feature for solving boundary value problems. Therefore, these basis functions for can be defined as;such that this system satisfies the conditions It is worth noting that the basis functions given in Equation (6) are a complete system. For more information about orthogonal polynomials, please see [44,45,46]. Definition 4. Let , and with its inner product be a Hilbert space of real-valued functions on Ψ. Then, the reproducing kernel of is iff
- (1)
- (2)
.
The last condition is known as a reproducing property. Especially, for any x, ξ∈Ψ, If a Hilbert space satisfies the above two conditions then it is called a reproducing kernel Hilbert space. The uniqueness of the reproducing kernel with respect to the inner product can be shown by the use of the Riesz representation theorem [47]. Theorem 1. Let be an orthonormal basis of n-dimensional Hilbert space , thenis a reproducing kernel of [34,35]. Definition 5. Let polynomials space be the pre-Hilbert space over with real coefficients and its degree and inner product as:with described by Equation (4), and the norm With the aid of definiton of Hilbert space, for any fixed m, is a subspace of and , .
Theorem 2 ([
43]).
Hilbert space is a reproducing kernel space. Definition 6. One can easily demonstrate that is a reproducing kernel space using Equation (6). From Theorem 1, the kernel function of can be written ashere, is complete system, which is easily obtained from the basis functions in Equation (6) with the help of the Gram-Schmidt orthonormalization process. Equation (11) is very useful for implementation. In other words, and can be readily updated and re-calculated by increasing m. 3. Main Results
In this section, some important results related to the reproducing kernel method with shifted Legendre polynomials are presented. In the first subsection, the generation of reproducing kernel that satisfies three-point boundary value problems is presented. In the second subsection, the representation of a solution is given in . Then, we will construct an iterative process for a nonlinear problem in the third subsection.
3.1. Generation of Reproducing Kernel for Three-Point Boundary Value
Problems
In this subsection, we shall generate a reproducing kernel Hilbert space in which every functions satisfies , and for . Namely, is defined as .
Obviously, reproducing kernel space is a closed subspace of . The reproducing kernel of can be given with the following theorem.
Theorem 3. The reproducing kernel of is : Proof. Frankly, not all elements of
vanish at
. This shows that
0. Hence, it can be easily seen that
and therefore
. For
, clearly,
, it follows that
Namely, is a reproducing kernel of . This completes the proof. □
3.2. Representation of Solution in Hilbert Space
In this subsection, the reproducing kernel method with Legendre polynomials is established for obtaining a numerical solution of a three-point boundary value problem. For Equations (
1) and (
2), the approximate solution shall be constructed in
. Firstly, we will define the linear operator
L as follows,
such that
Therefore, Equations (
1) and (
2) can be stated as follows
It can easily be shown that the linear operator
L is bounded. We will obtain the representation solution of Equation (
13) in the
space. Let
be the polynomial form of the reproducing kernel in
space.
Theorem 4 ([
43]).
Let be any distinct points in open interval for Equations (1) and (2), then for . Theorem 5 ([
43]).
Let be any distinct points in open interval for , then is complete in . Theorem 5 indicates that in the Legendre reproducing kernel approach, using finite distinct points is enough. However, in the traditional reproducing kernel method needs a dense sequence on the interval. So, this new approach varies from the traditional method in [
27,
36,
37,
38,
39,
42].
The orthonormal system
of
can be derived with the help of the Gram-Schmidt orthogonalization process using
,
here,
shows the coefficients of orthogonalization.
Theorem 6. Suppose that is the exact solution of Equations (1) and (2) and shows any distinct points in open interval for ; in that case, the approximate solution can be expressed as Proof. Since
, from Theorem 5, the following equality can be written
On the other part, using Theorem 4 and Equation (
14), we obtain
, which is the precise solution of Equation (
10) in
as,
The proof is completed. □
Theorem 7. If , then for and , where F is a constant.
Proof. We have for any , From the expression of , it pursues that
Therefore, . □
Theorem 8. The approximate solution and its derivatives , respectively, uniformly converge to the exact solution z and its derivatives ().
Proof. By using Theorem 7 for any
we get
where
are positive constants. Therefore, if
in the norm of
as
,
and its derivatives
, respectively, uniformly converge to
z and its derivatives
. This completes the proof. □
If the considered problem is linear, a numerical solution can be obtained directly from (
15). However, for a nonlinear problem, the following iterative procedure can be constructed.
3.3. Construction of Iterative Procedure
In this subsection, we will use the following iterative sequence to overcome the nonlinearity of the problem,
, inserting,
here, an orthogonal projection operator is defined as
and
shows the
n-th iterative numerical solution of (
16). Then, the following important theorem will be given for the iterative procedure.
Theorem 9. If are distinct points in open interval for , then Proof. Since
,
is the complete orthonormal system in
,
This completes the proof. □
Taking
and define the iterative sequence
Remark 1. For obtaining homogeneous boundary conditions in Equation (2), if the transformation is done, then coefficients can be found as , , . Here, is a new unknown variable, and homogeneous boundary conditions are also satisfied.