    A new reproducing kernel approach for nonlinear fractional three-point boundary value problems

In this article, a new reproducing kernel approach is developed for obtaining numerical solution of nonlinear three-point boundary value problems with fractional order. This approach is based on reproducing kernel which is constructed by shifted Legendre polynomials. In considered problem, fractional derivatives with respect to α and β are defined in Caputo sense. This method has been applied to some examples which have exact solutions. In order to shows the robustness of the proposed method, some numerical results are given in tabulated forms.

Authors

10/05/2020

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1 Introduction

In this paper, a new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of nonlinear fractional three-point boundary value problem,

 a2(ξ)cDαz(ξ)+a1(ξ)cDβz(ξ)+a0(ξ)z(ξ)=g(ξ,z(ξ),z′(ξ)),ξ∈[0,1] (1)

with following boundary conditions,

 z(0)=γ0,z(θ)=γ1, z(1)=γ2, 0<θ<1, 1<α≤2, 0<β≤1. (2)

Here, and are sufficiently smooth functions and fractional derivatives are taken in Caputo sense. Without loss of generality, we pay regard 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 studies have been concerned in engineering and applied science such as dynamical systems, fluid mechanics, control theory, oil industries, heat conduction can be well-turned by fractional differential equations 8 ; 9 ; 10 . Some applications, qualitative behaviors of solution and numerical methods to find approximate solution have been investigated for differential equation with fractional order 11 ; 12 ; 13 ; 14 .

More particularly, it is not easy to directly get exact solutions to most differential equations with fractional order. Hence, numerical techniques are utilised largely. 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 , Adomian decomposition method 19 , collocation methods 20 , homotopy analysis method 21 , differential transform method 22 , variational iteration method 23 , reproducing kernel space method 24 ; 25 and so on 26 ; 27 ; 28 .

In 1908, Zaremba firstly introduced reproducing kernel concept 29 . His resarches with regard to boundary value problems which includes Dirichlet condition. 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 30 ; 31 . Recently, this RKM is applied for different type of problem. For example, fractional order nonlocal boundary value problems 32 , Riccati differential equations 33 , forced Duffing equations with nonlocal boundary conditions 34 , Bratu equations with fractional order Caputo derivative 35 , time-fractional Kawahara equation 36 , two-point boundary value problem 37 , nonlinear fractional Volterra integro-differential equations 38 .

Recently, Legendre reproducing kernel method is proposed for fractional two-point boundary value problem of Bratu Type Equations 39 . The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving nonlinear three-point boundary value problem with 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 solution with 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 are provided which will be used in this research.

Definition 2.1 Let and . Then, the order left Riemann-Liouville fractional integral operator is given as 8 ; 12 ; 13 :

 Jα0+z(ξ)=1Γ(α)ξ∫0(ξ−s)α−1z(s)ds,

here is Gamma function, and .

Definition 2.2 Let and . Then, the order left Caputo differential operator is given as 8 ; 12 ; 13 :

 cDα0+z(ξ)=1Γ(m−α)∫ξ0∂m∂ξmz(s)(ξ−s)m−α−1ds, m−1<α0.

Definition 2.3 In order to construct polynomial type reproducing kernel, the first kind shifted Legendre polynomials are defined over the interval . For obtaining these polynomials the following iterative formula can be given:

 P0(ξ) = 1, P1(ξ) = 2ξ−1, ⋮ (n+1)Pn+1(ξ) = (2n+1)(2ξ−1)Pn(ξ)−nPn−1(ξ), n=1,2,...

The orthogonality requirement is

 ⟨Pn,Pm⟩=∫10ρ[0,1](ξ)Pn(ξ)Pm(ξ)dξ=⎧⎪ ⎪⎨⎪ ⎪⎩0,n≠m,1,n=m=0,12n+1,n=m≠0, (3)

here, weighted function is taken as,

 ρ[0,1](ξ)=1. (4)

Legendre basis functions can be established so that this basis function system satisfy the homogeneous boundary conditions as:

 z(0)=0and z(1)=0. (5)

Eq. (5) has a advantageous feature for solving boundary value problems. Therefore, these basis functions for can be defined as;

 ϕj(ξ)={Pj(ξ)−P0(ξ),j is even,Pj(ξ)−P1(ξ),j is odd. (6)

such that this system satisfy the conditions

 ϕj(0)=ϕj(1)=0. (7)

It is worth noting that the basis functions given in Eq. (6) are complete system. For more information about orthogonal polynomials, please see 41 ; 42 ; 43 .

Definition 2.4 Let , and with its inner product be a Hilbert space of real-valued functions on . Then, the reproducing kernel of is iff

1. .

The last condition is known as reproducing property. Especially, for any , ,

 R(x,ξ)=⟨R(⋅,x),R(⋅,ξ)⟩H.

If a Hilbert space satisfies the above two conditions then is called reproducing kernel Hilbert space. Uniqueness of the reproducing kernel can be shown by use of Riesz representation theorem 40 .

Theorem 2.1 Let be an orthonormal basis of -dimensional Hilbert space , then

 R(x,ξ)=Rx(ξ)=n∑j=1¯ej(x)ej(ξ) (8)

is reproducing kernel of 30 ; 31 .

Definition 2.5 Let polynomials space be pre-Hilbert space over with real coefficients and its degree and inner product as:

 ⟨z,v⟩Wmρ=∫10ρ[0,1](ξ)z(ξ)v(ξ)dξ, ∀z,v∈Wmρ[0,1] (9)

with described by Eq. (4), and the norm

 ∥z∥Wmρ=√⟨z,z⟩Wmρ, ∀z∈Wmρ[0,1]. (10)

With the aid of definiton of Hilbert space, for any fixed , is a subspace of and ,

Theorem 2.2 Hilbert space is a reproducing kernel space.

Proof. From Definition 2.5, it is quite apparent that functions space is a finite-dimensional. It is well known that all finite-dimensional pre-Hilbert space is a Hilbert space. Herewith, using this consequence and Theorem 2.1, is a reproducing kernel space.

For solving problem (1)-(2), it is required to describe a closed subspace of so that satisfy homogeneous boundary conditions.

Definition 2.6 Let

 0Wmρ[0,1]={z | z∈Wmρ[0,1], z(0)=z(1)=0}.

One can easily demonstrate that is a reproducing kernel space using Eq. (6). From Theorem 2.1, the kernel function of can be written as

 Rmx(ξ)=m∑j=2hj(ξ)hj(x). (11)

Here, is complete system which is easily obtained from basis functions in Eq. (6) with the help of Gram-Schmidt orthonormalization process. Eq. (11) is very useful for implementation. In other words, and can readily re-calculated by increasing .

3 Main Results

In this section, some important results related to reproducing kernel method with shifted Legendre polynomials are presented. In the first subsection, generation of reproducing kernel which is satify three-point boundary value problems is presented. In the second subsection, representation of solution is given . Then, we will construct an iterative process for nonlinear problem in 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 .

is defined as .

Obviously, reproducing kernel space is a closed subspace of . The reproducing kernel of can be given with the following theorem.

Theorem 3.1 The reproducing kernel of ,

 θRmx(ξ)=Rmx(ξ)−Rmx(θ)Rmθ(ξ)Rmθ(θ). (12)

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

 θWmρ[0,1]=−Rmx(α)z(θ)Rmθ(θ)=z(ξ).

Namely, is of reproducing kernel of . This completes the proof.

3.2 Representation of solution in θWmρ[0,1] Hilbert space

In this subsection, reproducing kernel method with Legendre polyomials is established for obtaining numerical solution of three-point boundary value problem. For Eqs. (1)-(2), the approximate solution shall be constructed in . Firstly, we will define linear operator as follow,

 L:θWmρ[0,1]→L2ρ[0,1]

such that

 Lz(ξ):=a2(ξ)cDαz(ξ)+a1(ξ)cDβz(ξ)+a0(ξ)z(ξ).

The Eqs.(1)-(2) can be stated as follows

 {Lz=g(ξ,z(ξ),z′(ξ))z(0)=z(θ)=z(1)=0. (13)

Easily can be shown that linear operator is bounded. We will obtain the representation solution of Eq. (13) in the space. Let be the polynomial form of reproducing kernel in space.

Theorem 3.2 Let be any distinct points in open interval for Eqs. (1)-(2), then
Proof. For any fixed , put

 ψmj(ξ) = L∗ θRmξj(ξ)=⟨L∗% θRmξj(ξ),θRmξ(x)⟩θWmρ (14) = ⟨θRmξj(ξ),Lx θRmξ(x)⟩L2ρ=Lx θRmξ(x)|x=ξj.

It is quite obvious that . Therefore . Here, shows the adjoint operator of . For any fixed and , .

Theorem 3.3 Let be any distinct points in open interval for , then is complete in .

Proof. For every fixed , let

 ⟨z(ξ),ψmj(ξ)⟩θWmρ=0,

this result shows, for ,

 ⟨z(ξ),ψmj(ξ)⟩θWmρ = ⟨z(ξ),L∗ θRmξj(ξ)⟩θWmρ (15) = ⟨Lz(ξ),θRmξj(ξ)⟩L2ρ = Lz(ξj)=0.

In Eq. (15), by use of inverse operator, it is decided that . Thus, is complete in . This completes the proof.

Theorem 3.3 indicates that in Legendre reproducing kernel approach, using a finite distinct points are enough. But, in traditional reproducing kernel method need to dense sqeuence on the interval. Namely, this new approach is vary from traditional method in 28 ; 32 ; 33 ; 34 ; 35 ; 38 .

The orthonormal system of can be derived with the help of the Gram-Schmidt orthogonalization process using ,

 ¯ψmj(ξ)=j∑k=0βmjkψmk(ξ), (16)

here show the coefficients of orthogonalization.

Theorem 3.4 Suppose that is the exact solution of Eqs. (1)-(2) and shows any distinct points in open interval , in that case

 zm(ξ)=m−2∑j=0j∑k=0βmjkg(ξk,zm(ξk),z′m(ξk))¯ψmj(ξ). (17)

Proof. Since from Theorem 3.3 can be written

 zm(ξ)=m−2∑i=0⟨zm(ξ),¯ψmj(ξ)⟩θWmρ¯ψmj(ξ).

On the other part, using Eq. (14) and Eq. (16), we obtain which is the precise solution of Eq. (10) in as,

 zm(ξ) = m−2∑j=0⟨zm(ξ),¯ψmj(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0⟨zm(ξ),j∑k=0βmjkψmk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨zm(ξ),ψmk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨zm(ξ),L∗θRmξk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨Lzm(ξ),θRmξk(ξ)⟩L2ρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨g(ξ,zm(ξ),z′m(ξ)),θRmξk(ξ)⟩L2ρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjkg(ξk,zm(ξk),z′m(ξk))¯ψmj(ξ).

The proof is completed.

Theorem 3.5 If , then for , where is a constant.

Proof. We have for any , From the expression of , it pursue that
So,

 |z(s)m(ξ)| = |⟨zm(ξ),∂sξ{\ }θRmξ(x)⟩θWmρ| ≤ ∥zm(ξ)∥θWmρ[0,1]∥∂sξ{\ }θRmξ(ξ)∥θWmρ ≤ Fs∥zm(ξ)∥θWmρ,s=0,…,m−1.

Therefore, .

Theorem 3.6 and its derivatives are respectively uniformly converge to and ().

Proof By using Theorem 3.5 for any we get

 |z(s)m(ξ)−z(s)(ξ)| = |⟨zm(ξ)−z(ξ),∂sξ θRmξ(ξ)⟩|θWmρ ≤ ∥∂sξ θRmξ(ξ)∥θWmρ∥zm(ξ)−z(ξ)∥θWmρ ≤ Fs∥zm(ξ)−z(ξ)∥θWmρ, s=0,…,m−1.

where are positive constants. Therefore, if in the meaning of the norm of as , and its derivatives are respectively uniformly converge to and its derivatives . This completes the proof.

If considered problem is linear, numerical solution can be directly get from (17). But, for nonlinear problem the following iterative procedure can be construct.

3.3 Construction of iterative procedure

In this subsection, we will use the following iterative sequence to overcome the nonlinearity of the problem, , inserting,

 ⎧⎨⎩Lym,n(ξ)=g(ξ,zm,n−1(ξ),z′m,n−1(ξ))zm,n(ξ)=Pm−1ym,n(ξ) (18)

here, orthogonal projection operator is defined as and shows the -th iterative numerical solution of (18). Then, the following important theorem will be given for iterative procedure.

Theorem 3.7 If is distinct points in open interval , then

 ym,n(ξ)=m−2∑j=0j∑k=0βmjkg(ξk,zm,n−1(ξk),z′m,n−1(ξk))¯ψmj(ξ) (19)

Proof. Since , is the complete orthonormal system in ,

 ym,n(ξ) = m−2∑j=0⟨ym,n(ξ),¯ψmj(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0⟨ym,n(ξ),j∑k=0βmjkψmk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨ym,n(ξ),ψmk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨ym,n(ξ),L∗ θRmξk(ξ)⟩θWmρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨Lym,n(ξ),θRmξk(ξ)⟩L2ρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjk⟨g(ξ,zm,n−1(ξ),z′m,n−1(ξ)),θRmξk(ξ)⟩L2ρ¯ψmj(ξ) = m−2∑j=0j∑k=0βmjkg(ξk,zm,n−1(ξk),z′m,n−1(ξk))¯ψmj(ξ)

This completes the proof.
Taking and define the iterative sequence

 zm,n(ξ)=Pm−1ym,n(ξ)=m−2∑j=0j∑k=0βmjkg(ξk,zm,n−1(ξk),z′m,n−1(ξk))¯ψmj(ξ), n=1,2,… (20)

4 Numerical applications

In this section, some nonlinear three-point boundary value problems are considered to exemplify the accuracy and efficiency of proposed approach. Numerical results which is achieved by L-RKM are shown with tables.

Example 4.1 We consider the following fractional order nonlinear three-point boundary value problem with Caputo derivative:

 cDαz(ξ)+(ξ+1)cDβz(ξ)+ξz(ξ)−z2(ξ)=f(ξ),1<α≤2.0<β≤1. (21)
 z(0)=z(12)=z(1)=0. (22)

Here, a known function such that the exact solution of this problem is .

By using proposed approach for Eqs. (21)-(22), and choosing nodal points as , the approximate solution is computed by Eq. (20). For (21)-(22), comparison of absolute errors for different , values are demonstrated in Table 1 and Table 2 and comparison of exact solution and numerical solution for and is given in Table 3.

Example 4.2 We take care of the following nonlinear three-point boundary value problem with Caputo derivative

 ξ2cDαz(ξ)+(ξ2−1)cDβz(ξ)+ξ3z(ξ)−z(ξ)z′(ξ)−z3(ξ)=f(ξ),1<α≤2.0<β≤1. (23)
 z(0)=z(35)=z(1)=0. (24)

Here, a known function such that the exact solution of this problem is .

By using proposed approach for Eqs. (23)-(24), and choosing nodal points as , the approximate solution is computed by Eq. (20). For (23)-(24), comparison of absolute errors for different , values are demonstrated in Table 4 and Table 5 and comparison of exact solution and numerical solution for and is given in Table 6.

5 Conclusion

In this research, a novel numerical approach which is called L-RKM has been proposed and successfully implemented to find the approximate solution of nonlinear three-point boundary value problems with Caputo derivative. For nonlinear problem, a new iterative process is proposed. Numerical findings show that the present approach is efficient and convenient for solving three-point boundary value problems with fractional order.

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