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Mathematical Problems in Engineering
VolumeΒ 2012, Article IDΒ 329575, 11 pages
http://dx.doi.org/10.1155/2012/329575
Research Article

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

Er Gao,1,2Β Songhe Song,1,2Β and Xinjian Zhang1

1Department of Mathematics and Systems Science, College of Science, National University of Defense Technology, Changsha 410073, China
2State key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China

Received 28 April 2012; Accepted 11 July 2012

Academic Editor: HungΒ Nguyen-Xuan

Copyright Β© 2012 Er Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order π‘žβˆˆ(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 π‘žβˆˆ(1,2] by applying some standard fixed point theorems: π‘π·π‘žπ‘’π‘’(π‘₯)=𝑓(π‘₯,𝑒(π‘₯),(πœ™π‘’)(π‘₯),(πœ“π‘’)(π‘₯)),1<π‘žβ‰€2,ξ…ž(ξ€·πœ‚0)+π‘Žπ‘’1ξ€Έ=0,π‘π‘’ξ…ž(ξ€·πœ‚1)+𝑒2ξ€Έ=0,0<πœ‚1β‰€πœ‚2<1,(1.1) where 𝑐𝐷 is the Caputo's fractional derivative and π‘“βˆΆ[0,1]×𝑋→𝑋 is continuous.

In this paper, we consider the following nonlinear fractional integro-differential equation with four-point nonlocal boundary conditions: π‘π·π‘žπ‘’π‘’(π‘₯)+(πœ™π‘’)(π‘₯)+(πœ“π‘’)(π‘₯)=𝑓(π‘₯,𝑒(π‘₯)),1<π‘žβ‰€2,ξ…žξ€·πœ‚(0)+π‘Žπ‘’1ξ€Έ=0,π‘π‘’ξ…žξ€·πœ‚(1)+𝑒2ξ€Έ=0,0<πœ‚1β‰€πœ‚2<1,(1.2) where 𝑐𝐷 is the Caputo's fractional derivative and π‘“βˆΆ[0,1]×𝑋→𝑋 is continuous, for 𝛾,π›ΏβˆΆ[0,1]Γ—[0,1]β†’[0,+∞), ξ€œ(πœ™π‘’)(π‘₯)=π‘₯0ξ€œπ›Ύ(π‘₯,𝑑)𝑒(𝑑)𝑑𝑑,(πœ“π‘’)(π‘₯)=π‘₯0𝛿(π‘₯,𝑑)𝑒(𝑑)𝑑𝑑,(1.3) and π‘Ž,π‘βˆˆ(0,1). Here, (𝑋,β€–β‹…β€–) is a Banach space and 𝐢=𝐢([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 β€–β‹…β€–.

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 𝑑=0 and 𝑑=1 adding or removing heat according to the temperatures detected by two sensors at 𝑑=πœ‚1 and 𝑑=πœ‚2.

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 π‘”βˆΆ[0,+∞)→ℝ, the Caputo derivative of fractional order π‘ž is defined as π‘π·π‘ž1𝑔(𝑑)=ξ€œΞ“(π‘›βˆ’π‘ž)𝑑0(π‘‘βˆ’π‘ )π‘›βˆ’π‘žβˆ’1𝑔(𝑛)(𝑠)𝑑𝑠,π‘›βˆ’1<π‘žβ‰€π‘›,π‘ž>0,(2.1) where Ξ“ denotes the gamma function.

Definition 2.2. The Riemann-Liouville fractional integral of order π‘ž is defined as πΌπ‘ž1𝑔(𝑑)=ξ€œΞ“(π‘ž)𝑑0𝑔(𝑠)(π‘‘βˆ’π‘ )1βˆ’π‘žπ‘‘π‘ ,π‘ž>0,(2.2) provided the integral exists.

Lemma 2.3 (see [4]). For a given 𝜎∈𝐢[0,1], the unique solution of the boundary value problem π‘π·π‘žπ‘’π‘’(π‘₯)=𝜎(π‘₯),0<π‘₯<1,1<π‘žβ‰€2,ξ…žξ€·πœ‚(0)+π‘Žπ‘’1ξ€Έ=0,π‘π‘’ξ…žξ€·πœ‚(1)+𝑒2ξ€Έ=0,0<πœ‚1β‰€πœ‚2<1,(2.3) is given by ξ€œπ‘’(π‘₯)=π‘₯0(π‘₯βˆ’π‘ )π‘žβˆ’1π‘Žξ€·Ξ“(π‘ž)𝜎(𝑠)𝑑𝑠+𝑏+πœ‚2ξ€Έβˆ’π‘₯ξ€·πœ‚1+π‘Ž1βˆ’πœ‚2ξ€Έξ€œβˆ’π‘πœ‚10ξ€·πœ‚1ξ€Έβˆ’π‘ π‘žβˆ’1+ξ€·Ξ“(π‘ž)𝜎(𝑠)π‘‘π‘ π‘Žπ‘₯βˆ’1+π‘Žπœ‚1ξ€Έξ€·πœ‚1+π‘Ž1βˆ’πœ‚2ξ€Έξƒ¬π‘ξ€œβˆ’π‘10(1βˆ’π‘ )π‘žβˆ’2ξ€œΞ“(π‘žβˆ’1)𝜎(𝑠)𝑑𝑠+πœ‚20ξ€·πœ‚2ξ€Έβˆ’π‘ π‘žβˆ’1ξƒ­.Ξ“(π‘ž)𝜎(𝑠)𝑑𝑠(2.4)

To introduce the next lemma, we need the following assumptions.(A1) There exist positive functions 𝐿1(𝑑),  𝐿2(𝑑),  𝐿3(𝑑) such that ‖𝑓(𝑑,𝑒(𝑑),(πœ™π‘’)(𝑑),(πœ“π‘’)(𝑑))βˆ’π‘“(𝑑,𝑣(𝑑),(πœ™π‘£)(𝑑),(πœ“π‘£)(𝑑))‖≀𝐿1(𝑑)β€–π‘’βˆ’π‘£β€–+𝐿2(𝑑)β€–πœ™π‘’βˆ’πœ™π‘£β€–+𝐿3[](𝑑)β€–πœ“π‘’βˆ’πœ“π‘£β€–,βˆ€π‘‘βˆˆ0,1,𝑒,π‘£βˆˆπ‘‹.(2.5)Further, 𝛾0=sup[]π‘‘βˆˆ0,1||||ξ€œπ‘‘0𝛾||||(𝑑,𝑠)𝑑𝑠,𝛿0=sup[]π‘‘βˆˆ0,1||||ξ€œπ‘‘0𝛿||||,𝐼(𝑑,𝑠)π‘‘π‘ π‘žπΏξƒ―=maxsup[]π‘‘βˆˆ0,1||πΌπ‘žπΏ1||(𝑑),sup[]π‘‘βˆˆ0,1||πΌπ‘žπΏ2||(𝑑),sup[]π‘‘βˆˆ0,1||πΌπ‘žπΏ3||ξƒ°,𝐼(𝑑)π‘žβˆ’1𝐿||𝐼(1)=maxπ‘žβˆ’1𝐿1||,||𝐼(1)π‘žβˆ’1𝐿2||,||𝐼(1)π‘žβˆ’1𝐿3||ξ€Ύ,𝐼(1)π‘žπΏξ€·πœ‚π‘–ξ€Έξ€½||𝐼=maxπ‘žπΏ1ξ€·πœ‚π‘–ξ€Έ||,||πΌπ‘žπΏ2ξ€·πœ‚π‘–ξ€Έ||,||πΌπ‘žπΏ3ξ€·πœ‚π‘–ξ€Έ||ξ€Ύ,𝑖=1,2.(2.6)(A2) There exist a number πœ… such that Ξ”β‰€πœ…<1, π‘‘βˆˆ[0,1], where ξ€·Ξ”=1+𝛾0+𝛿0πΌξ€Έξ€½π‘žπΏ+πœ†1πΌπ‘žπΏξ€·πœ‚1ξ€Έ+πœ†2ξ€·π‘πΌπ‘žβˆ’1𝐿(1)+πΌπ‘žπΏξ€·πœ‚2,πœ†ξ€Έξ€Έξ€Ύ1=sup[]π‘‘βˆˆ0,1||||π‘Žξ€·π‘+πœ‚2ξ€Έβˆ’π‘‘ξ€·πœ‚1+π‘Ž1βˆ’πœ‚2ξ€Έ||||βˆ’π‘,πœ†2=sup[]π‘‘βˆˆ0,1||||ξ€·π‘Žπ‘‘βˆ’1+π‘Žπœ‚1ξ€Έξ€·πœ‚1+π‘Ž1βˆ’πœ‚2ξ€Έ||||.βˆ’π‘(2.7)

Lemma 2.4 (see [4]). Assume that π‘“βˆΆ[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 Ξ”<1, where Ξ” 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 π‘Š12[0,1] is defined as π‘Š12[0,1]={𝑒(π‘₯)βˆ£π‘’ is absolutely continuous real-valued functions, π‘’β€²βˆˆπΏ2[0,1]}. The inner product in π‘Š12[0,1] is given by (𝑓,β„Ž)π‘Š12ξ€œ=𝑓(0)β„Ž(0)+10π‘“ξ…ž(𝑑)β„Žξ…ž(𝑑)𝑑𝑑,𝑓,β„Žβˆˆπ‘Š12[],0,1(3.1) and the norm β€–π‘’β€–π‘Š12 is denoted by β€–π‘’β€–π‘Š12=(𝑒,𝑒)π‘Š12. From [12], π‘Š12[0,1] is a reproducing kernel Hilbert space and the reproducing kernel is 𝐾1(𝑑,𝑠)=1+min{𝑑,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]={𝑒(π‘₯)|𝑒,π‘’ξ…ž,π‘’ξ…žξ…ž are absolutely continuous real valued functions, π‘’ξ…žξ…žξ…žβˆˆπΏ2[0,1], and π‘’ξ…ž(0)+π‘Žπ‘’(𝑐)=0, π‘π‘’ξ…ž(1)+𝑒(𝑑)=0}, and the inner product is defined as follows: (𝑓,β„Ž)𝐻32ξ€œ=𝑓(0)β„Ž(0)+10π‘“ξ…žξ…žξ…ž(𝑑)β„Žξ…žξ…žξ…ž(𝑑)𝑑𝑑,𝑓,β„Žβˆˆπ»32[].0,1(3.3)

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

Proof. Suppose {𝑣𝑛(π‘₯)}βˆžπ‘›=1 is a Cauchy sequence in 𝐻32[0,1], that means ‖‖𝑣𝑛+π‘βˆ’π‘£π‘›β€–β€–2=𝑣𝑛+𝑝(0)βˆ’π‘£π‘›ξ€Έ(0)2+ξ€œ10𝑣(3)𝑛+𝑝(π‘₯)βˆ’π‘£π‘›(3)ξ‚„(π‘₯)2𝑑π‘₯⟢0,π‘›βŸΆβˆž.(3.4) Therefore, we have 𝑣𝑛+𝑝(0)βˆ’π‘£π‘›(0)β†’0 and ∫10[𝑣(3)𝑛+𝑝(π‘₯)βˆ’π‘£π‘›(3)(π‘₯)]2𝑑π‘₯β†’0, which shows that {𝑣𝑛(0)}βˆžπ‘›=1 is a Cauchy sequence in ℝ and {𝑣𝑛(3)(π‘₯)}βˆžπ‘›=1 is a Cauchy sequence in space 𝐿2[0,1]. So, we have limπ‘›β†’βˆžπ‘£π‘›ξ€œ(0)βŸΆπœ†,10𝑣𝑛(3)ξ‚„(π‘₯)βˆ’β„Ž(π‘₯)2𝑑π‘₯⟢0,π‘›βŸΆβˆž,(3.5) where πœ† is a real constant and β„Ž(π‘₯)∈𝐿2[0,1].
Let 1𝑔(π‘₯)=πœ†+2ξ€œπ‘₯0(π‘₯βˆ’π‘‘)2β„Ž(𝑑)𝑑𝑑+π‘Ž1π‘₯+π‘Ž2π‘₯2,(3.6) where π‘Ž1,π‘Ž2 are determined by π‘”ξ…ž(0)+π‘Žπ‘”(𝑐)=0, and π‘π‘”ξ…ž(1)+𝑔(𝑑)=0.
From β„Ž(π‘₯)∈𝐿2[0,1], π‘”ξ…žξ…žβˆ«(π‘₯)=π‘₯0β„Ž(𝑑)𝑑𝑑+2π‘Ž2 is absolutely continuous in [0,1] and π‘”ξ…žξ…žξ…ž(π‘₯)=β„Ž(π‘₯)∈𝐿2[0,1] is almost true everywhere in [0,1]. Consequently, 𝑔(π‘₯)∈𝐻32[0,1]. Moreover, β€–β€–π‘£π‘›β€–β€–βˆ’π‘”(π‘₯)2=𝑣𝑛(0)βˆ’πœ†2+ξ€œ10𝑣𝑛(3)ξ‚„(π‘₯)βˆ’β„Ž(π‘₯)2𝑑π‘₯⟢0,π‘›βŸΆβˆž.(3.7) That means that, 𝐻32[0,1] is complete.
Similar to [13], we can prove that the point-evaluation functional π‘₯βˆ—(π‘₯βˆ—(π‘₯)=𝑒(π‘₯), π‘₯∈[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 1𝑅(𝑑,𝑠)=𝑅1201(𝑑,𝑠)Ξ”2+𝑅2(𝑑,𝑠)+𝑅3(𝑑,𝑠)+𝑅2(𝑠,𝑑)+𝑅3(𝑠,𝑑)Ξ”+⎧βŽͺ⎨βŽͺ⎩1𝑠1203𝑠2βˆ’5𝑠𝑑+10𝑑2ξ€Έ1,𝑑β‰₯𝑠,𝑑1203ξ€·10𝑠2βˆ’5𝑠𝑑+𝑑2ξ€Έ,𝑑<𝑠,(3.8) where 𝑅Δ=𝑏(βˆ’2+π‘Ž(βˆ’2+𝑐)𝑐)βˆ’π‘‘(𝑑+π‘Žπ‘(βˆ’π‘+𝑑)),1(𝑑,𝑠)=βˆ’π‘ ξ€·ξ€·5𝑏(βˆ’4+𝑑)𝑑3βˆ’6𝑑5ξ€Έ(𝑠+π‘Žπ‘(βˆ’π‘+𝑠))+π‘Žπ‘3𝑐2βˆ’5𝑐𝑑+10𝑑2ξ€Έ(𝑏(βˆ’2+𝑠)+𝑑(βˆ’π‘‘+𝑠))βˆ’π‘ξ€·ξ€·βˆ’40𝑏+5(βˆ’4+𝑑)𝑑3ξ€Έ(π‘Žπ‘(π‘βˆ’π‘ )βˆ’π‘ )+5π‘Ž(βˆ’4+𝑐)𝑐3(𝑏(βˆ’2+𝑠)+𝑑(βˆ’π‘‘+𝑠))𝑑(𝑑+π‘Žπ‘(βˆ’π‘+𝑑))+π‘Žπ‘3π‘ ξ€·ξ€·π‘βˆ’5𝑏(βˆ’4+𝑐)(π‘Žπ‘(π‘βˆ’π‘ )βˆ’π‘ )+2βˆ’5𝑐𝑑+10𝑑2ξ€Έ(π‘Žπ‘(π‘βˆ’π‘ )βˆ’π‘ )+6π‘Žπ‘2ξ€Έξ€·ξ€·ξ€·(𝑏(βˆ’2+𝑠)+𝑑(βˆ’π‘‘+𝑠))𝑑(𝑏(βˆ’2+𝑑)+𝑑(βˆ’π‘‘+𝑑))+120(π‘‘βˆ’π‘ )(βˆ’π‘‘+π‘Ž(π‘βˆ’π‘‘)(π‘βˆ’π‘ )βˆ’π‘ )+π‘βˆ’2+π‘Žβˆ’2𝑐+𝑐2βˆ’Γ—ξ€·ξ€·ξ€·(βˆ’2+𝑠)𝑠(π‘‘βˆ’π‘‘)(βˆ’π‘‘+π‘Ž(π‘βˆ’π‘‘)(π‘βˆ’π‘‘)βˆ’π‘‘)+π‘βˆ’2+π‘Žβˆ’2𝑐+𝑐2,π‘…βˆ’(βˆ’2+𝑑)𝑑2(βŽ›βŽœβŽœβŽœβŽβˆ’1𝑑,𝑠)=𝑑(𝑑+π‘Žπ‘(βˆ’π‘+𝑑))24𝑏(βˆ’4+𝑠)𝑠3+⎧βŽͺ⎨βŽͺ⎩1𝑠1203ξ€·10𝑑2βˆ’5𝑑𝑠+𝑠2ξ€Έ1,𝑑β‰₯𝑠𝑑1203𝑑2βˆ’5𝑑𝑠+10𝑠2ξ€ΈβŽžβŽŸβŽŸβŽŸβŽ ,𝑅,𝑑<𝑠3⎧βŽͺ⎨βŽͺ⎩1(𝑑,𝑠)=βˆ’π‘Žπ‘‘(𝑏(βˆ’2+𝑑)+𝑑(βˆ’π‘‘+𝑑))=𝑠1203ξ€·10𝑐2βˆ’5𝑐𝑠+𝑠2ξ€Έ1,𝑐β‰₯𝑠,𝑐1203𝑐2βˆ’5𝑐𝑠+10𝑠2ξ€Έ,𝑐<𝑠.(3.9)

Actually, it is easy to prove that for every π‘₯∈[0,1] and 𝑒(𝑦)∈𝐻32[0,1], 𝑅(π‘₯,𝑦)∈𝐻32[0,1] and (𝑒(𝑦),𝑅(π‘₯,𝑦))=𝑒(π‘₯) holds, that is, 𝑅(π‘₯,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 𝐿𝑒=π‘π·π‘žπ‘’(π‘₯)+(πœ™π‘’)(π‘₯)+(πœ“π‘’)(π‘₯)+𝛽(π‘₯)𝑒(π‘₯) and 𝐹(π‘₯,𝑒(π‘₯))=𝑓(π‘₯,𝑒(π‘₯))+𝛽(π‘₯)𝑒(π‘₯). We can convert (1.2) into an equivalent equation 𝐿𝑒(π‘₯)=𝐹(π‘₯,𝑒(π‘₯)). It is clear that 𝐿∢𝐻32[0,1]β†’π‘Š12[0,1] is a bounded linear operator.

Put πœ‘π‘–(π‘₯)=𝐾1(π‘₯𝑖,π‘₯), Ψ𝑖(π‘₯)=πΏβˆ—πœ‘π‘–(π‘₯), where πΏβˆ— is the adjoint operator of 𝐿. Then Ψ𝑖𝐿(π‘₯)=βˆ—πœ‘π‘–ξ€Έ=ξ€·πœ‘(𝑦),𝑅(π‘₯,𝑦)𝑖(𝑦),𝐿𝑦=𝑅(π‘₯,𝑦)𝐿𝑦𝑅(π‘₯,𝑦),πœ‘π‘–ξ€Έ(π‘₯)=𝐿𝑦𝑅(π‘₯,𝑦)|𝑦=π‘₯𝑖.(3.10)

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

Lemma 3.3. Under the previous assumptions, if {π‘₯𝑖}βˆžπ‘–=1 is dense on [0,1], then {Ψ𝑖(π‘₯)}βˆžπ‘–=1 is the complete basis of 𝐻32[0,1].

The orthogonal system {Ψ𝑖(π‘₯)}βˆžπ‘–=1 of 𝐻32[0,1] can be derived from Gram-Schmidt orthogonalization process of {Ψ𝑖(π‘₯)}βˆžπ‘–=1, and Ψ𝑖(π‘₯)=𝑖𝑗=1𝛽𝑖𝑗Ψ𝑗(π‘₯).(3.11)

We also can prove the following theorem.

Theorem 3.4. If {π‘₯𝑖}βˆžπ‘–=1 is dense on [0,1] and the solution of (1.2) is unique, the solution can be expressed in the form 𝑒(π‘₯)=βˆžξ“π‘–π‘–=1ξ“π‘˜=1π›½π‘–π‘˜πΉξ€·π‘₯π‘˜ξ€·π‘₯,π‘’π‘˜ξ€Έξ€ΈΞ¨π‘–(π‘₯).(3.12)

The approximate solution of the (1.2) is 𝑒𝑛(π‘₯)=𝑛𝑖𝑖=1ξ“π‘˜=1π›½π‘–π‘˜πΉξ€·π‘₯π‘˜ξ€·π‘₯,π‘’π‘˜ξ€Έξ€ΈΞ¨π‘–(π‘₯).(3.13)

If (1.2) is linear, that is 𝐹(π‘₯,𝑒(π‘₯))=𝐹(π‘₯), 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: 𝑒0𝑒(π‘₯)=0,𝑛+1(π‘₯)=𝑛+1𝑖=1𝐡𝑖Ψ𝑖(π‘₯),(3.14) where 𝐡𝑖=βˆ‘π‘–π‘˜=1π›½π‘–π‘˜πΉ(π‘₯π‘˜,𝑒𝑛(π‘₯π‘˜)).

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 |𝑒(π‘₯)|≀𝑀‖𝑒‖𝐻32, for all 𝑒(π‘₯)∈𝐻32[0,1].

Proof. For all the π‘₯∈[0,1] and π‘’βˆˆπ»32[0,1], there are ||𝑒||=||𝑒(π‘₯)(β‹…),𝐾3ξ€Έ||≀‖‖𝐾(β‹…,π‘₯)3β€–β€–(β‹…,π‘₯)𝐻32⋅‖𝑒‖𝐻32(4.1)𝐾3(β‹…,π‘₯)∈𝐻32[0,1], and note that 𝑀=max[]π‘₯∈0,1‖‖𝐾3β€–β€–(β‹…,π‘₯)𝐻32.(4.2) That is, |𝑒(π‘₯)|≀𝑀‖𝑒‖𝐻32.

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

Lemma 4.2. If π‘’π‘›β€–β‹…β€–β†’βˆ’π‘’ (π‘›β†’βˆž), ‖𝑒𝑛‖ is bounded, π‘₯𝑛→𝑦  (π‘›β†’βˆž) and 𝐹(π‘₯,𝑒(π‘₯)) is continuous, then 𝐹(π‘₯𝑛,π‘’π‘›βˆ’1(π‘₯𝑛))→𝐹(𝑦,βˆ’π‘’(𝑦)).

Theorem 4.3. Suppose that ‖𝑒𝑛‖ is bounded in (3.13) and (1.2) has a unique solution. If {π‘₯𝑖}βˆžπ‘–=1 is dense on [0,1], then the n-term approximate solution 𝑒𝑛(π‘₯) derived from the above method converges to the analytical solution 𝑒(π‘₯) of (1.2).

Proof. First, we will prove the convergence of 𝑒𝑛(π‘₯).
From (3.14), we infer that 𝑒𝑛+1(π‘₯)=𝑒𝑛(π‘₯)+𝐡𝑛+1Ψ𝑛+1(π‘₯).(4.3) The orthonormality of {Ψ𝑖}βˆžπ‘–=1 yields that ‖‖𝑒𝑛+1β€–β€–2=‖‖𝑒𝑛‖‖2+𝐡𝑛+1ξ€Έ2=β‹―=𝑛+1𝑖=1𝐡𝑖2.(4.4) That means ‖𝑒𝑛+1β€–β‰₯‖𝑒𝑛‖. Due to the condition that ‖𝑒𝑛‖ is bounded, ‖𝑒𝑛‖ is convergent and there exists a constant β„“ such that βˆžξ“π‘–=1𝐡𝑖2=β„“.(4.5) If π‘š>𝑛, then β€–β€–π‘’π‘šβˆ’π‘’π‘›β€–β€–2=β€–β€–π‘’π‘šβˆ’π‘’π‘šβˆ’1+π‘’π‘šβˆ’1βˆ’π‘’π‘šβˆ’2+β‹―+𝑒𝑛+1βˆ’π‘’π‘›β€–β€–2.(4.6) In view of (π‘’π‘šβˆ’π‘’π‘šβˆ’1)βŸ‚(π‘’π‘šβˆ’1βˆ’π‘’π‘šβˆ’2)βŸ‚β‹―βŸ‚(𝑒𝑛+1βˆ’π‘’π‘›), it follows that β€–β€–π‘’π‘šβˆ’π‘’π‘›β€–β€–2=β€–β€–π‘’π‘šβˆ’π‘’π‘šβˆ’1β€–β€–2+β€–β€–π‘’π‘šβˆ’1βˆ’π‘’π‘šβˆ’2β€–β€–2‖‖𝑒+β‹―+𝑛+1βˆ’π‘’π‘›β€–β€–2=π‘šξ“π‘–=𝑛+1𝐡𝑖2⟢0asπ‘›βŸΆβˆž.(4.7) The completeness of 𝐻32[0,1] shows that π‘’π‘›β†’βˆ’π‘’ as π‘›β†’βˆž in the sense of ‖⋅‖𝐻32.
Secondly, we will prove that βˆ’π‘’ is the solution of (1.2).
Taking limits in (3.12), we get βˆ’π‘’(π‘₯)=βˆžξ“π‘–=1𝐡𝑖Ψ𝑖(π‘₯).(4.8) So πΏβˆ’π‘’(π‘₯)=βˆžξ“π‘–=1𝐡𝑖𝐿Ψ𝑖(𝐿π‘₯),βˆ’π‘’ξ‚ξ€·π‘₯𝑛=βˆžξ“π‘–=1𝐡𝑖𝐿Ψ𝑖,πœ‘π‘›ξ‚=βˆžξ“π‘–=1𝐡𝑖Ψ𝑖,πΏβˆ—πœ‘π‘›ξ‚=βˆžξ“π‘–=1𝐡𝑖Ψ𝑖,Ψ𝑛.(4.9) Therefore, 𝑛𝑖=1π›½π‘›π‘—ξ‚€πΏβˆ’π‘’ξ‚ξ€·π‘₯𝑛=βˆžξ“π‘–=1𝐡𝑖Ψ𝑖,𝑛𝑗=1𝛽𝑛𝑗Ψ𝑗ξƒͺ=βˆžξ“π‘–=1𝐡𝑖Ψ𝑖,Ψ𝑛=𝐡𝑛.(4.10) If 𝑛=1, then πΏβˆ’π‘’ξ€·π‘₯1ξ€Έξ€·π‘₯=𝐹1,𝑒0ξ€·π‘₯1.ξ€Έξ€Έ(4.11) If 𝑛=2, then 𝛽21πΏβˆ’π‘’ξ€·π‘₯1ξ€Έ+𝛽22πΏβˆ’π‘’ξ€·π‘₯2ξ€Έ=𝛽21𝐹π‘₯1,𝑒0ξ€·π‘₯1ξ€Έξ€Έ+𝛽22𝐹π‘₯2,𝑒1ξ€·π‘₯2.ξ€Έξ€Έ(4.12) It is clear that ξ‚€πΏβˆ’π‘’ξ‚ξ€·π‘₯2ξ€Έξ€·π‘₯=𝐹2,𝑒1ξ€·π‘₯2.ξ€Έξ€Έ(4.13) Moreover, it is easy to see by induction that ξ‚€πΏβˆ’π‘’ξ‚ξ€·π‘₯𝑗π‘₯=𝐹𝑗,π‘’π‘—βˆ’1ξ€·π‘₯𝑗,𝑗=1,2,….(4.14) Since {π‘₯𝑖}βˆžπ‘–=1 is dense on [0,1], for all π‘Œβˆˆ[0,1], there exists a subsequence {π‘₯𝑛𝑗}βˆžπ‘—=1 such that π‘₯π‘›π‘—βŸΆπ‘Œasπ‘—βŸΆβˆž.(4.15) It is easy to see that (πΏβˆ’π‘’)(π‘₯𝑛𝑗)=𝐹(π‘₯𝑛𝑗,π‘’π‘›π‘—βˆ’1(π‘₯𝑛𝑗)). Let π‘—β†’βˆž; by the continuity of 𝐹(π‘₯,𝑒(π‘₯)) and Lemma 4.2, we have ξ‚€πΏβˆ’π‘’ξ‚ξ‚€(π‘Œ)=πΉπ‘Œ,βˆ’π‘’ξ‚.(π‘Œ)(4.16) At the same time, βˆ’π‘’βˆˆπ»32[0,1]; clearly, 𝑒 satisfies the boundary conditions of (1.2).
That is, βˆ’π‘’ is the solution of (1.2).
The proof is complete.

In fact, 𝑒𝑛(π‘₯) is just the orthogonal projection of exact solution βˆ’π‘’(π‘₯) onto the space Span{Ψ𝑖}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: 𝑐𝐷3/21𝑒(𝑑)=5ξ€œπ‘‘0π‘’βˆ’(π‘ βˆ’π‘‘)+π‘’βˆ’(π‘ βˆ’π‘‘)/25𝑒(𝑠)𝑑𝑠+𝑒2(𝑑)βˆ’2𝑑2𝑒(𝑑)+20𝑑17𝑒(𝑑)+454[],𝑒153𝑒(𝑑)+𝑓(𝑑),π‘‘βˆˆ0,1ξ…ž(10)+2𝑒131=0,4π‘’ξ…ž(ξ‚€21)+𝑒3=0,(5.1) where 𝑓(𝑑)=1674244/585225βˆ’2354𝑒𝑑/2/3825βˆ’169π‘’π‘‘βˆš/3825+4βˆšπ‘‘/πœ‹βˆ’3316𝑑/2601βˆ’162647𝑑2/65025+20𝑑3/17+𝑑4. According to Lemma 2.4, the boundary value problem (5.1) has a unique solution on [0,1]. 𝑒(𝑑)=𝑑2+10𝑑/17βˆ’227/153 is the solution of (5.1), so it is the one and the only one solution. Using our method, taking π‘₯𝑖=(π‘–βˆ’1)/(π‘βˆ’1), 𝑖=1,2,…,𝑁,𝑁=11,101, the numerical results are given in Table 1.

tab1
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 π‘žβˆˆ(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.

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