A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

Abstract

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order 𝑞∈(1,2] based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

1. Introduction

In recent years, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Several authors have used fixed point theory to show the existence of solution to differential equations of fractional order, see the monographs of Bai and Liu [1], Wu and Liu [2], Hamani et al. [3] and Ahmad and Sivasundaram [4]. At the same time, there may be several methods for solving differential equations of fractional order, such as the least squares finite-element method [5], collection method [6], fractional differential transform method [7], decomposition method [8], and variational iteration method [9]. Besides these cited works, few more contributions [10, 11] have been made to the analytical and numerical study of the solutions of fractional boundary value problems.

Ahmad and Sivasundaram [4] proved the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order 𝑞∈(1,2] by applying some standard fixed point theorems: 𝑢𝐷^𝑐𝑞𝑢(𝑥)=𝑓(𝑥,𝑢(𝑥),(𝜙𝑢)(𝑥),(𝜓𝑢)(𝑥)),1<𝑞≤2,_(0)+𝑎𝑢𝜂₁=0,𝑏𝑢^(1)+𝑢𝜂₂=0,0<𝜂₁≤𝜂₂<1,(1.1) where ^𝑐𝐷 is the Caputo's fractional derivative and 𝑓∶[0,1]×𝑋→𝑋 is continuous.

In this paper, we consider the following nonlinear fractional integro-differential equation with four-point nonlocal boundary conditions: 𝑢𝐷^𝑐𝑞𝑢(𝑥)+(𝜙𝑢)(𝑥)+(𝜓𝑢)(𝑥)=𝑓(𝑥,𝑢(𝑥)),1<𝑞≤2,_(0)+𝑎𝑢𝜂₁=0,𝑏𝑢^(1)+𝑢𝜂₂=0,0<𝜂₁≤𝜂₂<1,(1.2) where ^𝑐𝐷 is the Caputo's fractional derivative and 𝑓∶[0,1]×𝑋→𝑋 is continuous, for 𝛾,𝛿∶[0,1]×[0,1]→[0,+∞), (𝜙𝑢)(𝑥)=^𝑥0𝛾(𝑥,𝑡)𝑢(𝑡)𝑑𝑡,(𝜓𝑢)(𝑥)=^𝑥0𝛿(𝑥,𝑡)𝑢(𝑡)𝑑𝑡,(1.3) and 𝑎,𝑏∈(0,1). Here, (𝑋,‖⋅‖) is a Banach space and 𝐶=𝐶([0,1],𝑋) denotes the Banach space of all continuous functions from [0,1]→𝑋 endowed with a topology of uniform convergence with the norm denoted by ‖⋅‖.

Actually, we remark that the boundary conditions in (1.2) arise in the study of heat flow problems involving a bar of unit length with two controllers at 𝑡=0 and 𝑡=1 adding or removing heat according to the temperatures detected by two sensors at 𝑡=𝜂₁ and 𝑡=𝜂₂.

The rest of the paper is organized as follows. We begin by introducing some necessary definitions and mathematical preliminaries of the fractional calculus theory which are required for establishing our results. Then we construct some special reproducing kernel spaces, and the new reproducing kernel method is introduced in Section 3. In Section 4 we present one examples to demonstrate the efficiency of the method.

2. Preliminaries

Let us recall some basic definition and lemmas on fractional calculus.

Definition 2.1. For a function 𝑔∶[0,+∞)→ℝ, the Caputo derivative of fractional order 𝑞 is defined as ^𝑐𝐷^𝑞𝑔(𝑡)=1Γ(𝑛−𝑞)^𝑡0(𝑡−𝑠)^{𝑛−𝑞−1}𝑔^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝑞≤𝑛,𝑞>0,(2.1) where Γ denotes the gamma function.

Definition 2.2. The Riemann-Liouville fractional integral of order 𝑞 is defined as 𝐼^𝑞𝑔(𝑡)=1Γ(𝑞)^𝑡0𝑔(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠,𝑞>0,(2.2) provided the integral exists.

Lemma 2.3 (see [4]). For a given 𝜎∈𝐶[0,1], the unique solution of the boundary value problem 𝑢𝐷^𝑐𝑞𝑢(𝑥)=𝜎(𝑥),0<𝑥<1,1<𝑞≤2,_(0)+𝑎𝑢𝜂₁=0,𝑏𝑢^(1)+𝑢𝜂₂=0,0<𝜂₁≤𝜂₂<1,(2.3) is given by 𝑢(𝑥)=^𝑥0(𝑥−𝑠)^𝑞−1Γ(𝑞)𝜎(𝑠)𝑑𝑠+𝑎𝑏+𝜂²−𝑥1+𝑎𝜂₁−𝜂₂−𝑏^𝜂1₀𝜂₁−𝑠^𝑞−1+𝑎𝑥−1+𝑎𝜂₁Γ(𝑞)𝜎(𝑠)𝑑𝑠1+𝑎𝜂₁−𝜂₂−𝑏𝑏¹⁰(1−𝑠)^𝑞−2Γ(𝑞−1)𝜎(𝑠)𝑑𝑠+^𝜂2₀𝜂₂−𝑠^𝑞−1Γ(𝑞)𝜎(𝑠)𝑑𝑠.(2.4)

To introduce the next lemma, we need the following assumptions.(A₁) There exist positive functions 𝐿₁(𝑡),  𝐿₂(𝑡),  𝐿₃(𝑡) such that ‖𝑓≤(𝑡𝐿,𝑢₁(𝑡),(𝜙𝑢)(𝑡),(𝜓𝑢)(𝑡))−𝑓(𝑡,𝑣(𝑡),(𝜙𝑣)(𝑡),(𝜓𝑣)(𝑡))‖(𝑡)‖𝑢−𝑣‖+𝐿₂(𝑡)‖𝜙𝑢−𝜙𝑣‖+𝐿₃(𝑡)‖𝜓𝑢−𝜓𝑣‖,∀𝑡∈[0,1],𝑢,𝑣∈𝑋.(2.5)Further, 𝛾₀=sup_[]𝑡∈0,1||||^𝑡0𝛾(𝑡,𝑠)𝑑𝑠||||,𝛿₀=sup_[]𝑡∈0,1||||^𝑡0𝐼𝛿(𝑡,𝑠)𝑑𝑠_𝑞𝐿||||,𝐼=maxsup_[]𝑡∈0,1||𝐼^𝑞𝐿₁(𝑡)||,sup_[]𝑡∈0,1||𝐼^𝑞𝐿₂(𝑡)||,sup_[]𝑡∈0,1||𝐼^𝑞𝐿₃(𝑡)_𝑞−1||,𝐼𝐿(1)=max||𝐼^𝑞−1𝐿₁(1)||,||𝐼^𝑞−1𝐿₂(1)||,||𝐼^𝑞−1𝐿₃(1)_𝑞||,𝐿𝜂_𝑖=max||𝐼^𝑞𝐿₁𝜂_𝑖||,||𝐼^𝑞𝐿₂𝜂_𝑖||,||𝐼^𝑞𝐿₃𝜂_𝑖||,𝑖=1,2.(2.6)(A₂) There exist a number 𝜅 such that Δ≤𝜅<1, 𝑡∈[0,1], where Δ𝜆=1+𝛾₀+𝛿₀𝐼^𝑞𝐿+𝜆₁𝐼^𝑞𝐿𝜂₁+𝜆₂𝑏𝐼^𝑞−1𝐿(1)+𝐼^𝑞𝐿𝜂₂₁,=sup_[]𝑡∈0,1||||𝑎𝑏+𝜂₂−𝑡1+𝑎𝜂₁−𝜂₂−𝑏||||,𝜆²=sup_[]𝑡∈0,1||||𝑎𝑡−1+𝑎𝜂₁1+𝑎𝜂₁−𝜂₂−𝑏||||.(2.7)

Lemma 2.4 (see [4]). Assume that 𝑓∶[0,1]×𝑋×𝑋×𝑋→𝑋 is a jointly continuous function and satisfies assumption (A₁). Then the boundary value problem (1.1) has an unique solution provided Δ<1, where Δ is given in assumption (A₂).

For more information on the mathematical properties of fractional derivatives and integrals one can consult the mentioned references.

3. Reproducing Kernel Method

3.1. Some Reproducing Kernel Spaces

Firstly, inner space 𝑊¹²[0,1] is defined as 𝑊¹²[0,1]={𝑢(𝑥)∣𝑢 is absolutely continuous real-valued functions, 𝑢′∈𝐿²[0,1]}. The inner product in 𝑊¹²[0,1] is given by (𝑓,ℎ)_𝑊¹²=𝑓(0)ℎ(0)+¹⁰𝑓^(𝑡)ℎ^(𝑡)𝑑𝑡,𝑓,ℎ∈𝑊¹²[0,1],(3.1) and the norm ‖𝑢‖_𝑊¹² is denoted by ‖𝑢‖_𝑊¹²=(𝑢,𝑢)_𝑊¹². From [12], 𝑊¹²[0,1] is a reproducing kernel Hilbert space and the reproducing kernel is 𝐾₁(𝑡,𝑠)=1+min{𝑡,𝑠}.(3.2)

In order to solve (1.2) using RKM, we construct a reproducing kernel space 𝐻³²[0,1] in which every function satisfies the boundary conditions of (1.2). Inner space 𝐻³²[0,1] is defined as 𝐻³²[0,1]={𝑢(𝑥)|𝑢,𝑢^,𝑢^ are absolutely continuous real valued functions, 𝑢^∈𝐿²[0,1], and 𝑢^(0)+𝑎𝑢(𝑐)=0, 𝑏𝑢^(1)+𝑢(𝑑)=0}, and the inner product is defined as follows: (𝑓,ℎ)_𝐻³²=𝑓(0)ℎ(0)+¹⁰𝑓^(𝑡)ℎ^(𝑡)𝑑𝑡,𝑓,ℎ∈𝐻³²[0,1].(3.3)

Theorem 3.1 3.1. 𝐻³²[0,1] is a Hilbert reproducing kernel space.

Proof. Suppose {𝑣_𝑛(𝑥)}^∞𝑛=1 is a Cauchy sequence in 𝐻³²[0,1], that means ‖‖𝑣_𝑛+𝑝−𝑣_𝑛‖‖²=𝑣_𝑛+𝑝(0)−𝑣_𝑛(0)²+¹⁰𝑣^(3)𝑛+𝑝(𝑥)−𝑣^𝑛(3)(𝑥)²𝑑𝑥⟶0,𝑛⟶∞.(3.4) Therefore, we have 𝑣_𝑛+𝑝(0)−𝑣_𝑛(0)→0 and ∫¹⁰[𝑣^(3)𝑛+𝑝(𝑥)−𝑣^𝑛(3)(𝑥)]²𝑑𝑥→0, which shows that {𝑣_𝑛(0)}^∞𝑛=1 is a Cauchy sequence in ℝ and {𝑣^𝑛(3)(𝑥)}^∞𝑛=1 is a Cauchy sequence in space 𝐿²[0,1]. So, we have lim_𝑛→∞𝑣_𝑛(0)⟶𝜆,¹⁰𝑣^𝑛(3)(𝑥)−ℎ(𝑥)²𝑑𝑥⟶0,𝑛⟶∞,(3.5) where 𝜆 is a real constant and ℎ(𝑥)∈𝐿²[0,1].
Let 𝑔(𝑥)=𝜆+12^𝑥0(𝑥−𝑡)²ℎ(𝑡)𝑑𝑡+𝑎₁𝑥+𝑎₂𝑥²,(3.6) where 𝑎₁,𝑎₂ are determined by 𝑔^(0)+𝑎𝑔(𝑐)=0, and 𝑏𝑔^(1)+𝑔(𝑑)=0.
From ℎ(𝑥)∈𝐿²[0,1], 𝑔^(𝑥)=∫^𝑥0ℎ(𝑡)𝑑𝑡+2𝑎₂ is absolutely continuous in [0,1] and 𝑔^(𝑥)=ℎ(𝑥)∈𝐿²[0,1] is almost true everywhere in [0,1]. Consequently, 𝑔(𝑥)∈𝐻³²[0,1]. Moreover, ‖‖𝑣_𝑛−𝑔(𝑥)‖‖²=𝑣_𝑛(0)−𝜆²+¹⁰𝑣^𝑛(3)(𝑥)−ℎ(𝑥)²𝑑𝑥⟶0,𝑛⟶∞.(3.7) That means that, 𝐻³²[0,1] is complete.
Similar to [13], we can prove that the point-evaluation functional 𝑥^∗(𝑥^∗(𝑥)=𝑢(𝑥), 𝑥∈[0,1]) of 𝐻³²[0,1] is bounded. So 𝐻³²[0,1] is a Hilbert reproducing kernel space.

From [12, 14], we have the following.

Theorem 3.2. The reproducing kernel of 𝐻³²[0,1] is 𝑅(𝑡,𝑠)=1120𝑅¹(𝑡,𝑠)Δ²+𝑅²(𝑡,𝑠)+𝑅₃(𝑡,𝑠)+𝑅₂(𝑠,𝑡)+𝑅₃(𝑠,𝑡)+⎧⎪⎨⎪⎩1Δ120𝑠³𝑠²1−5𝑠𝑡+10𝑡²,𝑡≥𝑠,120𝑡³10𝑠²−5𝑠𝑡+𝑡²,𝑡<𝑠,(3.8) where 𝑅Δ=𝑏(−2+𝑎(−2+𝑐)𝑐)−𝑑(𝑑+𝑎𝑐(−𝑐+𝑑)),₁=(𝑡−,𝑠𝑠)5𝑏(−4+𝑑)𝑑₃+𝑎𝑐₃−6𝑑⁵(𝑠+𝑎𝑐(−𝑐+𝑠))−𝑏−𝑐4²0−𝑏+5𝑐5𝑑(+−410+𝑑²𝑑)(𝑑𝑏₃(−2+𝑠)+𝑑(−𝑑+𝑠))+5𝑎(−4+𝑐)𝑐₃(𝑎𝑐(𝑐−𝑠)−𝑠)+𝑎𝑐₃(𝑏(−2+𝑠)+𝑑(−𝑑+𝑠))𝑡(𝑡+𝑎𝑐(−𝑐+𝑡))𝑠+−65𝑎𝑏𝑐₂(−4+𝑐)(𝑎𝑐(𝑐−𝑠)−𝑠)+𝑐²−5𝑐𝑑+10𝑑²(𝑎𝑐(𝑐−𝑠)−𝑠)+120(𝑑−𝑠)(𝑏(−(−𝑑2++𝑎𝑠()𝑐+−𝑑𝑑)(−(𝑐𝑑−+𝑠𝑠))−)𝑠𝑡)(+𝑏(𝑏−2−+2𝑡+)+𝑎𝑑−(2−𝑐𝑑++𝑐𝑡₂))×(𝑑−𝑡)(−𝑑+𝑎(𝑐−𝑑)(𝑐−𝑡)−𝑡)+𝑏−2+𝑎−2𝑐+𝑐₂−(−2+𝑠)𝑠𝑅−(−2+𝑡)𝑡₂,(𝑡,𝑠)=𝑡(𝑡+𝑎𝑐(−𝑐+𝑡))⎛⎜⎜⎜⎝−124𝑏(−4+𝑠)𝑠³+⎧⎪⎨⎪⎩1120𝑠³10𝑑²1−5𝑑𝑠+𝑠²,𝑑≥𝑠120𝑑³𝑑²𝑅−5𝑑𝑠+10𝑠²,𝑑<𝑠₃⎞⎟⎟⎟⎠,(𝑡,𝑠)=−𝑎𝑡(𝑏(−2+𝑡)+𝑑(−𝑑+𝑡))=⎧⎪⎨⎪⎩1120𝑠³10𝑐²1−5𝑐𝑠+𝑠²,𝑐≥𝑠,120𝑐³𝑐²−5𝑐𝑠+10𝑠²,𝑐<𝑠.(3.9)

Actually, it is easy to prove that for every 𝑥∈[0,1] and 𝑢(𝑦)∈𝐻³²[0,1], 𝑅(𝑥,𝑦)∈𝐻³²[0,1] and (𝑢(𝑦),𝑅(𝑥,𝑦))=𝑢(𝑥) holds, that is, 𝑅(𝑥,𝑦) is the reproducing kernel of 𝐻³²[0,1].

3.2. The Reproducing Kernel Method

In recent years, there has been a growing interest in using a reproducing kernel to solve the operator equation. In this section, the representation of analytical solution of (1.2) is given in the reproducing kernel space 𝐻³²[0,1].

Note 𝐿𝑢=^𝑐𝐷^𝑞𝑢(𝑥)+(𝜙𝑢)(𝑥)+(𝜓𝑢)(𝑥)+𝛽(𝑥)𝑢(𝑥) and 𝐹(𝑥,𝑢(𝑥))=𝑓(𝑥,𝑢(𝑥))+𝛽(𝑥)𝑢(𝑥). We can convert (1.2) into an equivalent equation 𝐿𝑢(𝑥)=𝐹(𝑥,𝑢(𝑥)). It is clear that 𝐿∶𝐻³²[0,1]→𝑊¹²[0,1] is a bounded linear operator.

Put 𝜑_𝑖(𝑥)=𝐾₁(𝑥_𝑖,𝑥), Ψ_𝑖(𝑥)=𝐿^∗𝜑_𝑖(𝑥), where 𝐿^∗ is the adjoint operator of 𝐿. Then Ψ_𝑖(𝑥)==𝐿𝜑^∗𝜑_𝑖(𝑦),𝑅(𝑥,𝑦)_𝑖=(𝑦),𝐿_𝑦𝑅(𝑥,𝑦)𝐿_𝑦𝑅(𝑥,𝑦),𝜑_𝑖(𝑥)=𝐿_𝑦𝑅(𝑥,𝑦)|_{𝑦=𝑥𝑖}.(3.10)

Similar to [15], we can prove the following.

Lemma 3.3. Under the previous assumptions, if {𝑥_𝑖}^∞𝑖=1 is dense on [0,1], then {Ψ_𝑖(𝑥)}^∞𝑖=1 is the complete basis of 𝐻³²[0,1].

The orthogonal system {Ψ_𝑖(𝑥)}^∞𝑖=1 of 𝐻³²[0,1] can be derived from Gram-Schmidt orthogonalization process of {Ψ_𝑖(𝑥)}^∞𝑖=1, and Ψ_𝑖(𝑥)=^𝑖_𝑗=1𝛽_𝑖𝑗Ψ_𝑗(𝑥).(3.11)

We also can prove the following theorem.

Theorem 3.4. If {𝑥_𝑖}^∞𝑖=1 is dense on [0,1] and the solution of (1.2) is unique, the solution can be expressed in the form 𝑢(𝑥)=^∞_𝑖𝑖=1_𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥).(3.12)

The approximate solution of the (1.2) is 𝑢_𝑛(𝑥)=^𝑛_𝑖𝑖=1_𝑘=1𝛽_𝑖𝑘𝐹𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥).(3.13)

If (1.2) is linear, that is 𝐹(𝑥,𝑢(𝑥))=𝐹(𝑥), then the approximate solution of (1.2) can be obtained directly from (3.13). Else, the approximate process could be modified into the following form: 𝑢𝑢₀(𝑥)=0,_𝑛+1(𝑥)=^𝑛+1_𝑖=1𝐵_𝑖Ψ_𝑖(𝑥),(3.14) where 𝐵_𝑖=∑^𝑖𝑘=1𝛽_𝑖𝑘𝐹(𝑥_𝑘,𝑢_𝑛(𝑥_𝑘)).

4. Convergent Theorem of the Numerical Method

In this section, we will give the following convergent theorem of our algorithm.

Lemma 4.1. There exists a constant 𝑀, satisfied |𝑢(𝑥)|≤𝑀‖𝑢‖_𝐻³², for all 𝑢(𝑥)∈𝐻³²[0,1].

Proof. For all the 𝑥∈[0,1] and 𝑢∈𝐻³²[0,1], there are ||𝑢(𝑥)||=||𝑢(⋅),𝐾₃(⋅,𝑥)||≤‖‖𝐾₃(⋅,𝑥)‖‖_𝐻32⋅‖𝑢‖_𝐻³²(4.1)𝐾₃(⋅,𝑥)∈𝐻³²[0,1], and note that 𝑀=max_[]𝑥∈0,1‖‖𝐾₃(⋅,𝑥)‖‖_𝐻32.(4.2) That is, |𝑢(𝑥)|≤𝑀‖𝑢‖_𝐻³².

By Lemma 4.1, it is easy to obtain the following lemma.

Lemma 4.2. If 𝑢→_{𝑛‖⋅‖}⁻𝑢 (𝑛→∞), ‖𝑢_𝑛‖ is bounded, 𝑥_𝑛→𝑦  (𝑛→∞) and 𝐹(𝑥,𝑢(𝑥)) is continuous, then 𝐹(𝑥_𝑛,𝑢_𝑛−1(𝑥_𝑛))→𝐹(𝑦,⁻𝑢(𝑦)).

Theorem 4.3. Suppose that ‖𝑢_𝑛‖ is bounded in (3.13) and (1.2) has a unique solution. If {𝑥_𝑖}^∞𝑖=1 is dense on [0,1], then the n-term approximate solution 𝑢_𝑛(𝑥) derived from the above method converges to the analytical solution 𝑢(𝑥) of (1.2).

Proof. First, we will prove the convergence of 𝑢_𝑛(𝑥).
From (3.14), we infer that 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+𝐵_𝑛+1Ψ_𝑛+1(𝑥).(4.3) The orthonormality of {Ψ_𝑖}^∞𝑖=1 yields that ‖‖𝑢_𝑛+1‖‖²=‖‖𝑢_𝑛‖‖²+𝐵_𝑛+1²=⋯=^𝑛+1_𝑖=1𝐵_𝑖².(4.4) That means ‖𝑢_𝑛+1‖≥‖𝑢_𝑛‖. Due to the condition that ‖𝑢_𝑛‖ is bounded, ‖𝑢_𝑛‖ is convergent and there exists a constant ℓ such that ^∞_𝑖=1𝐵_𝑖²=ℓ.(4.5) If 𝑚>𝑛, then ‖‖𝑢_𝑚−𝑢_𝑛‖‖²=‖‖𝑢_𝑚−𝑢_𝑚−1+𝑢_𝑚−1−𝑢_𝑚−2+⋯+𝑢_𝑛+1−𝑢_𝑛‖‖².(4.6) In view of (𝑢_𝑚−𝑢_𝑚−1)⟂(𝑢_𝑚−1−𝑢_𝑚−2)⟂⋯⟂(𝑢_𝑛+1−𝑢_𝑛), it follows that ‖‖𝑢_𝑚−𝑢_𝑛‖‖²=‖‖𝑢_𝑚−𝑢_𝑚−1‖‖²+‖‖𝑢_𝑚−1−𝑢_𝑚−2‖‖²+⋯+‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²=^𝑚_𝑖=_𝑛+1𝐵_𝑖²⟶0as𝑛⟶∞.(4.7) The completeness of 𝐻³²[0,1] shows that 𝑢_𝑛→⁻𝑢 as 𝑛→∞ in the sense of ‖⋅‖_𝐻³².
Secondly, we will prove that ⁻𝑢 is the solution of (1.2).
Taking limits in (3.12), we get ⁻𝑢(𝑥)=^∞_𝑖=1𝐵_𝑖Ψ_𝑖(𝑥).(4.8) So 𝐿⁻𝑢(𝑥)=^∞_𝑖=1𝐿𝐵_𝑖𝐿Ψ_𝑖(𝑥),₋𝑢𝑥_𝑛=^∞_𝑖=1𝐵_𝑖𝐿Ψ_𝑖,𝜑_𝑛=^∞_𝑖=1𝐵_𝑖Ψ_𝑖,𝐿^∗𝜑_𝑛=^∞_𝑖=1𝐵_𝑖Ψ_𝑖,Ψ_𝑛.(4.9) Therefore, ^𝑛_𝑖=1𝛽_𝑛𝑗𝐿⁻𝑢𝑥_𝑛=^∞_𝑖=1𝐵_𝑖Ψ_𝑖,^𝑛_𝑗=1𝛽_𝑛𝑗Ψ_𝑗=^∞_𝑖=1𝐵_𝑖Ψ_𝑖,Ψ_𝑛=𝐵^𝑛.(4.10) If 𝑛=1, then 𝐿⁻𝑢𝑥₁=𝐹𝑥₁,𝑢₀𝑥₁.(4.11) If 𝑛=2, then 𝛽₂₁𝐿⁻𝑢𝑥₁+𝛽₂₂𝐿⁻𝑢𝑥₂=𝛽₂₁𝐹𝑥₁,𝑢₀𝑥₁+𝛽₂₂𝐹𝑥₂,𝑢₁𝑥₂.(4.12) It is clear that 𝐿⁻𝑢𝑥₂=𝐹𝑥₂,𝑢₁𝑥₂.(4.13) Moreover, it is easy to see by induction that 𝐿⁻𝑢𝑥_𝑗=𝐹𝑥_𝑗,𝑢_𝑗−1𝑥_𝑗,𝑗=1,2,….(4.14) Since {𝑥_𝑖}^∞𝑖=1 is dense on [0,1], for all 𝑌∈[0,1], there exists a subsequence {𝑥_𝑛𝑗}^∞𝑗=1 such that 𝑥_𝑛𝑗⟶𝑌as𝑗⟶∞.(4.15) It is easy to see that (𝐿⁻𝑢)(𝑥_𝑛𝑗)=𝐹(𝑥_𝑛𝑗,𝑢_𝑛𝑗−1(𝑥_𝑛𝑗)). Let 𝑗→∞; by the continuity of 𝐹(𝑥,𝑢(𝑥)) and Lemma 4.2, we have 𝐿⁻𝑢(𝑌)=𝐹𝑌,⁻𝑢(𝑌).(4.16) At the same time, ⁻𝑢∈𝐻³²[0,1]; clearly, 𝑢 satisfies the boundary conditions of (1.2).
That is, ⁻𝑢 is the solution of (1.2).
The proof is complete.

In fact, 𝑢_𝑛(𝑥) is just the orthogonal projection of exact solution ⁻𝑢(𝑥) onto the space Span{Ψ_𝑖}^𝑛𝑖=1.

5. Numerical Example

To give a clear overview of the methodology as a numerical tool, we consider one example in this section. We apply the reproducing kernel method and results obtained by the method are compared with the analytical solution of each example and are found to be in good agreement with each other. Also, the numerical results obtained are compared with the corresponding experimental results obtained by the methods presented in [8, 9].

Example 5.1. Consider the following boundary value problem: ^𝑐𝐷^3/2𝑢(𝑡)=15^𝑡0𝑒^{−(𝑠−𝑡)}+𝑒^{−(𝑠−𝑡)/2}5𝑢(𝑠)𝑑𝑠+𝑢²(𝑡)−2𝑡²𝑢(𝑡)+20𝑡𝑢17𝑢(𝑡)+145543𝑢(𝑡)+𝑓(𝑡),𝑡∈[0,1_],(0)+12𝑢13=0,14𝑢^(1)+𝑢23=0,(5.1) where 𝑓(𝑡)=1674244/585225−2354𝑒^𝑡/2/3825−169𝑒^𝑡/3825+4√𝑡/√𝜋−3316𝑡/2601−162647𝑡²/65025+20𝑡³/17+𝑡⁴. According to Lemma 2.4, the boundary value problem (5.1) has a unique solution on [0,1]. 𝑢(𝑡)=𝑡²+10𝑡/17−227/153 is the solution of (5.1), so it is the one and the only one solution. Using our method, taking 𝑥_𝑖=(𝑖−1)/(𝑁−1), 𝑖=1,2,…,𝑁,𝑁=11,101, the numerical results are given in Table 1.

Table 1 

Absolute errors for Example 5.1.

6. Conclusion

In this paper, RKM is presented to solve four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order 𝑞∈(1,2]. The results of numerical examples demonstrate that the present method is more accurate than the existing methods.

Acknowledgment

The work is supported by NSF of China under Grant no. 10971226.