# Certain Analysis of Solution for the Nonlinear Two-Point Boundary Value Problem with Caputo Fractional Derivative

Authors: Shayma Adil Murad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, the existence and uniqueness of solutions for a nonlinear fractional differential equation with a two-point boundary condition in a Banach space are investigated by using the contraction mapping principle and the Brouwer fixed-point theorem with Bielecki norm. The iterative scheme of the numerical solution for the nonlinear two-point boundary value problem will be discussed and illustrated by solving some problems. The well-known Ulam-Hyers and Ulam-Hyers-Rassias stability theorems are employed to establish the stability of solutions to the boundary value problem. In the end, we provided a couple of examples to support our results.


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## Body

## 1. Introduction

Fractional calculus is an important mathematical topic due to its theoretical foundation and multiple applications in physical, chemical processes, and engineering; for instance, see [1–6]. The two-point boundary value problem occurs in applied mathematics, theoretical physics, engineering, control, and optimization theory (see [7]). The existence of solutions of initial and boundary value problems of fractional differential equations by the help of different fixed-point theorems has been discussed by many mathematicians, and the readers are referred to see the monographs [8–21]. In [22], the authors give the existence results for two-point boundary value problem of fractional differential equations at resonance by means of the coincidence degree theory. Mongkolkeha and Gopal [23] proposed new common fixed-point theorems for the Ciric type generalized F-contraction in metric spaces with the w-distance. The fixed points for the monotone γ-nonexpansive and generalized ℘-nonexpansive mappings in hyperbolic space have been approximated by [24].In fact, the subject of numerical methods for solving fractional differential equations has gained prominence and has been discussed by several authors, including a series of papers [25–31] and references cited therein, which include some recent studies on the approximation method for differential equations of fractional order. El-Ajou et al. [32] extended the application of the homotopy analysis method (HAM) to provide symbolic approximate solution for two-point boundary value problems of fractional order. Lyons et al. [33] prove an extension of Picard’s iterative existence and uniqueness theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. In [34], the successive approximation method was applied to solve the temperature field based on the given Mittag-Leffler-type.Nie et al. [35] investigated the existence and numerical method of two-point boundary value problems for fractional differential equations with Caputo’s derivative or Riemann-Liouville derivative. The solutions can be deduced by the contraction mapping principle and fractional Green function. For the Caputo’s derivative case, it has the form
(1)CĐγϖt+℧t,ϖt,Đ℘ϖt=0,(2)ϖa=A,ϖb=B.For Riemann-Liouville derivative case, it has the form(3)RĐγϖt+℧t,ϖt,Đ℘ϖt=0,ϖa=0,ϖb=B.More recently, Hyers-Ulam type stability theorems for nonlinear fractional differential equations have attracted a lot of attention as an interesting field and have been investigated in many papers (see [36–40]). Murad et al. [41] studied the existence, Ulam-Hyers, and Ulam-Hyers-Rassias theorems of solutions to a differential equation of mixed Caputo-Riemann fractional derivatives.Dai et al. [42] discussed the existence and Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions for the fractional differential equation with boundary condition. Prasad et al. [43] investigated the existence and Ulam stability of the fractional-order iterative two-point boundary value problem which has the form
(4)CĐγϖt=℧t,ϖt,ϖ2t,t∈0,1,1<γ≤2,ϖ0=A,ϖ1=B,where CĐγ is the Caputo fractional derivative and 1<γ≤2 and 0≤A≤B≤1.In this paper, we consider the nonlinear fractional differential equation which has the form(5)Đγϖt=℧t,ϖt,Đ℘ϖt,I=a,b,with boundary conditions(6)ϖa=A,ϖb=B,where ℧:I×R×R⟶R, 1<γ≤2, 0<℘≤1, Đγ and Đ℘ are the Caputo fractional derivatives, and a,b,A,B are constants. The main objective is to study the existence of a solution to the boundary value problem (5) and (6). The results are based on Brouwer’s fixed-point theorem and Banach contraction mapping principle. The analytical approximate technique to obtain the solution is a part of this work, and some examples are illustrated to explain the algorithm. Furthermore, we discuss the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the boundary value problem (5) and (6). Some examples are also constructed to illustrate and validate the main results.


## 2. Preliminaries

Let us give some definitions and lemmas that are basic and needed at various places in this work.Definition 1 (see [44]).
Let℧ be a function which is defined almost everywhere a.e. on a,b. If γ>0, then
(7)abIγ℧=∫ab℧sb−sγ−1Γγds,
provided that this integral (Lebesgue) exists.Definition 2 (see [5]).
For a continuous function℧:0,∞⟶ℝ, the Caputo derivative of fractional order γ is defined as
(8)CĐγ℧t=1Γn−γ∫0tt−sn−γ−1℧nsds,n−1<γ≤n,
provided that℧n exists, where n=γ+1,γ denotes the integer part of the real number γ.Lemma 3 (see [1]).
Letγ>0. If we assume ϖ∈C0,1∩L10,1, then the Caputo fractional differential equation
(9)Đγϖt=0
has the solution(10)ϖt=c0+c1t+c2t2+⋯+cn−1tn−1,where ci∈ℝ,i=0,1,2,⋯,n−1, and n=γ+1.Lemma 4 (see [1]).
Letϖ∈C0,1∩L10,1 with fractional derivative of order γ>0 that belongs to C0,1∩L10,1. Then,
(11)IγCĐγϖt=ϖt+c0+c1t+c2t2+⋯+cn−1tn−1,
forci∈ℝ,i=0,1,2,⋯,n−1, where n is the smallest integer greater than or equal to γ.Theorem 5 (see [45]) (Brouwer’s fixed-point theorem).
LetM be a nonempty compact (closed and bounded) convex set in Rn and T:M⟶M be a continuous self-mapping. Then, T has (at least) one fixed point in M.Theorem 6 (see [45]) (Banach contraction mapping principle).
LetM be a Banach space. If T:M⟶M is a contraction, then T has a unique fixed point in M.Lemma 7.
Letϖt∈CI,ℝ and 1<γ≤2, 0<℘≤1; then, the solution of the boundary value problem (5) and (6) is given by
(12)ϖt=A+B−Ab−at−a−t−ab−aΓγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ∫att−sγ−1℧s,ϖs,Đ℘ϖsds.Proof.
Byapplying the Lemma4, we may reduce equation (5) to an equivalent equation
(13)ϖt=atIγ℧t,ϖt,Đ℘ϖt+c0+c1t−a.
Using the boundary condition (6), we find that
(14)c0=Aandc1=B−ab−a−1b−aabIγ℧t,ϖt,Đ℘ϖt.
Substituting the values ofc0 and c1 in equation (13), the result is
(15)ϖt=A+B−Ab−at−a−t−ab−aΓγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ∫att−sγ−1℧s,ϖs,Đ℘ϖsds.
The derivativeĐ℘ϖs can be written as
(16)Đ℘ϖt=B−At−a1−℘b−aΓ2−℘−t−a1−℘b−aΓ2−℘Γγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ−℘∫att−sγ−℘−1℧s,ϖs,Đ℘ϖsds.


## 3. The Existence of Solution

This section deals with the existence and uniqueness of solution for the fractional differential equation (5) with boundary condition (6). Let Ca,b,R be the Banach space endowed with a Bielecki norm
(17)ϖB=∫abe−Ntϖtdt.Let the spaceΘ=ϖt∈Ca,b:Đ℘ϖt∈Ca,b, equipped with the norm
(18)ϖΘ=maxt∈a,b∫abe−Ntϖtdt+maxt∈a,b∫abe−NtĐ℘ϖtdt,which is a Banach space, whereN>0 is a fixed constant. Consider the following assumptions:(H1) There exists a functiona1t∈L1I such that ∣℧t,x,ϖ∣≤a1t+γ1∣x∣+γ2∣ϖ∣, where γ1,γ2≥0.(H2) There exists constantsΩ1,Ω2>0 such that
(19)℧t,ϖ1t,Đ℘ϖ1t−℧t,ϖ2t,Đ℘ϖ2t≤Ω1ϖ1−ϖ2+Ω2Đ℘ϖ1−Đ℘ϖ2,for eacht∈I and all ϖ1,ϖ2∈ℝ. Let us set the following notation for convenience:
(20)ℶ1=1Γγ∫abe−NtGt,sdt+e−Na−e−NbBΓγ+Db−aNΓγ,ℶ2=b−ae−Na−e−NbNΓγb−a+1γ,ζ1=1Γγ−℘∫abe−NtGt,sdt+e−Na−e−NbB−AΓγ+b−aDNΓ2−℘Γγ,ζ2=e−Na−e−Nbb−aNb−aΓ2−℘Γγ+1Γγ−℘+1.Our results are based on the Brouwer’s fixed-point theorem and Banach contraction principle.Theorem 8.
Suppose that (H1) holds. Then, the boundary value problem (5) and (6) has at least one solution on CI,ℝ.Proof.
Define the operatorJ:CI,ℝ⟶CI,ℝ as
(21)Jϖt=A+B−Ab−at−a−t−ab−aΓγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ∫att−sγ−1℧s,ϖs,Đ℘ϖsds,
and the operatorĐ℘Jϖt can be written as
(22)Đ℘Jϖt=B−At−a1−℘b−aΓ2−℘−t−a1−℘b−aΓ2−℘Γγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ−℘∫att−sγ−℘−1℧s,ϖs,Đ℘ϖsds.
Firstly, we will prove thatJHd⊂Hd, where Hd=ϖ∈C:ϖθ≤d, and choose d≥η1/1−η2γ1+γ2, for ϖ∈Hd, and the following is obtained:
(23)∫abe−NtJϖtdt≤∫abe−NtA+B−Ab−at−adt+1b−aΓγ∫abt−ae−Nt∫abb−sγ−1℧s,ϖs,Đ℘ϖsdsdt+1Γγ∫abe−Nt∫att−sγ−1℧s,ϖs,Đ℘ϖsdsdt,
and by using (H1), then(24)∫abe−NtJϖtdt≤B∫abe−Ntdt+b−aΓγ∫abe−Nt∫aba1s+γ1ϖs+γ2Đ℘ϖsdsdt+1Γγ∫abe−Nt∫att−sγ−1a1s+γ1ϖs+γ2Đ℘ϖsdsdt,∫abe−NtJϖtdt≤1Γγ∫abe−NtGt,sdt+e−Na−e−NbBΓγ+Db−aNΓγ+b−ae−Na−e−NbNΓγb−a+1γγ1d+γ2d,where
(25)D=∫aba1sds,Gt,s=∫att−sγ−1a1sds.
Thus, we have(26)Ω1∫abe−Ntϖtdt≤Ω1ℶ1+Ω1ℶ2γ1d+γ2d.
Now to establish the bounded of equation (22) by (H1), get
(27)∫abe−NtĐ℘Jϖtdt≤B−AΓ2−℘∫abe−Ntdt+b−aΓ2−℘Γγ∫abe−Nt∫aba1s+γ1ϖs+γ2Đ℘ϖsdsdt+1Γγ−℘∫abe−Nt∫att−sγ−℘−1a1s+γ1ϖs+γ2Đ℘ϖsdsdt,∫abe−NtĐ℘Jϖtdt≤1Γγ−℘∫abe−NtGt,sdt+e−Na−e−NbB−AΓγ+b−aDNΓ2−℘Γγ+e−Na−e−Nbb−aNb−aΓ2−℘Γγ+1Γγ−℘+1γ1d+γ2d,Ω2∫abe−NtĐ℘Jϖtdt≤Ω2ζ1+Ω2ζ2γ1d+γ2d,Jϖθ≤η1+dη2γ1+γ2≤d,where η1=Ω2ζ1+Ω1ω1 and η2=Ω1ζ2+Ω1ω2, and we have Jyθ≤d. Hence, J:Hd⟶Hd.
According to the Brouwer fixed-point theorem, the boundary value problem (5) and (6) has at least one solution.Theorem 9.
Assume that (H2) holds. If(28)λ2Ω1+λ3Ω2<1,where
(29)λ1=b−a2−e−Na−b−e−Nb−aN2Γγ,λ2=λ1+1Nγ,λ3=λ1Γ2−℘+1Nγ−℘,
then the boundary value problem (5) and (6) has a unique solution.Proof.
We prove thatJ is a contraction. Let ϖ1,ϖ2∈CI,ℝ. Then,
(30)Jϖ1t−Jϖ2t≤t−ab−aΓγ∫abb−sγ−1℧s,ϖ1s,Đ℘ϖ1s−℧s,ϖ2s,Đ℘ϖ2sds+1Γγ∫att−sγ−1℧s,ϖ1s,Đ℘ϖ1s−℧s,ϖ2s,Đ℘ϖ2sds,
and by the condition (H2), the result is(31)Jϖ1t−Jϖ2t≤t−ab−aΓγ∫abb−sγ−1Ω1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds+1Γγ∫att−sγ−1Ω1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds,(32)∫abe−NtJϖ1t−Jϖ2tdt≤b−aΓγ∫abe−Nt−s∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sdsdt+1Γγ∫abe−Nt−s∫att−sγ−1e−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sdsdt,
and using Holder inequality, the first term of equation (32) becomes
(33)∫abe−NtJϖ1t−Jϖ2tdt≤λ1∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds+1Γγ∫ab∫sbe−Nt−st−sγ−1dte−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds,
and for the second term, letz=Nt−s,dz=Ndt, and it follows
(34)∫abe−NtJϖ1t−Jϖ2tdt≤λ1∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds+1NγΓγ∫ab∫0Nb−se−Nzzγ−1dze−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds,≤λ1∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds+1Nγ∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds,(35)Ω1∫abe−NtJϖ1t−Jϖ2tdt≤Ω1λ2∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds.
By the same way,(36)∫abe−NtĐ℘Jϖ1t−Đ℘Jϖ2tdt≤b−aΓ2−℘Γγ∫ab∫abe−Nt−se−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sdsdt+1Γγ−℘∫ab∫ate−Nt−st−sγ−℘−1e−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sdsdt,Ω2∫abe−NtD℘Jϖ1t−Đ℘Jϖ2tdt≤Ω2λ3∫abe−NsΩ1ϖ1s−ϖ2s+Ω2Đ℘ϖ1s−Đ℘ϖ2sds.
Then, we have(37)Jϖ1t−Jϖ2tθ≤λ2Ω1+λ3Ω2ϖ1−ϖ2.
Hence, we conclude that the problem (5) and (6) has a unique solution by the contraction mapping principle.Example 1.
Consider the following fractional boundary value problem:(38)Đ1.2ϖ=et101+etϖ+56+tĐ0.3ϖ,t∈0,1,ϖ0=ϖ1=1.
Here,γ=1.2, ℘=0.3, and N=0.47; then (H2) is satisfied with Ω1=0.073105857 and Ω2=0.833333, and one can arrive at the following results:
(39)λ1=b−a2−e−Na−b−e−Nb−aN2Γγ=−1.109321597426112,λ2Ω1=λ1+1NγΩ1=0.099800501185966,λ3Ω2=λ1Γ2−℘+1Nγ−℘Ω2=0.626725087609827,λ2Ω1+λ3Ω2=0.726525588795793<1.
Hence, by Theorem9, the boundary value problem (38) has a unique solution on 0,1.Example 2.
Consider the following fractional boundary value problem:(40)Đ3/2ϖ=t+1e−tt+32ϖ2+ϖ+ettet+5Đ1/2ϖ5+Đ1/2ϖ,t∈0,1,ϖ0=1ϖ1=1.
Here,γ=1.5, ℘=0.5, and N=1/2, and according to the Lipschitz condition, we have
(41)℧t,ϖ1t,Đ℘ϖ1t−℧t,ϖ2t,Đ℘ϖ2t≤t+1e−tt+32ϖ1−ϖ2+ettet+5Đ℘ϖ1−Đ℘ϖ2,Ω1=0.1111111,Ω2=0.352187428,λ1=−1.152083842554695.
Finally, the following obtainedλ2Ω1+λ3Ω2=0.432795980868753<1. Therefore, from Theorem 9, we conclude that boundary value problem (40) has a unique solution.


## 4. Iterative Numerical Schema

For the solution of the boundary value problem (5) and (6), an iterative schema is provided. Starting with ϖ0t and Đ℘ϖ0t, the width h=b−a/N, and ti=a+ih,i=0,1,⋯N, the integral equations (21) and (22) are numerically evaluated to obtain the sequence ϖnt, a≤t≤b. The trapezoidal rule is most convenient for estimating ϖntk, k=0,1..⋯N, and the approximation for (21) and (22) becomes
(42)ϖn+1tk=A+B−AkN−kNℵa,b,n+ℵa,tk,n,(43)Đ℘ϖn+1tk=B−Ak1−℘h−℘NΓ2−℘−k1−℘h−℘NΓ2−℘ℵa,b,n+Ja,tk,n,where
(44)ℵa,b,n=abIγ℧s,ϖ,Đ℘ϖ,ℵa,tk,n=atkIγ℧s,ϖ,Đ℘ϖ,Ja,tk,n=atkIγ−℘℧s,ϖ,Đ℘ϖ,and the process of iteration may be terminated by setting a criterion(45)maxϖn+1tk−ϖntk≤ε,a≤t≤b,where ε is a constant which is to be taken smaller than the discretization error Oh2 in ϖtk for the solution of the boundary value problem (5) and (6).


## 5. Convergence ofϖntk and Đγϖntk

To prove the convergence of the iteration schema (42) and (43), let us suppose that ℧:I×ℝ×ℝ⟶ℝ be continuous function that satisfies the conditions.(H3) There exists a constantχ,χ1,χ2>0 such that ℧t,ϖt,Đ℘ϖt≤χ,
(46)d2ds2b−sγ−1℧s,ϖs,Đ℘ϖs≤χ1,d2ds2b−sγ−℘−1℧s,ϖs,Đ℘ϖs≤χ2,for eacht,s∈I and all ϖ,ϖ1,ϖ2,⋯∈R. After simplifying equations (42) and (43), we have
(47)ϖn+1tk=A+B−AkN+hγ2Γγkγ−1−kNNγ−1℧na+khγΓγ∑j=1k−1k−jγ−1−kNN−jγ−1℧ntj−khγNΓγ∑j=kN−1N−jγ−1℧ntj−h3k12Γγtk−sγ−1℧nt″,(48)Đ℘ϖn+1tk=B−Ak1−℘h−℘NΓ2−℘−hγ−℘∑j=1kk−j1−℘Γγ−℘−k1−℘N−jγ−1NΓ2−℘Γγ℧ntj+hγ−℘2Γ2−℘Γγ−℘ΓγNkγ−℘−1Γ2−℘Γγ−Nγ−1k1−℘Γγ−℘℧na+k1−℘hγ−℘NΓ2−℘Γγ∑j=k+1N−1N−jγ−1℧ntj−h3k12Γγ−℘tk−sγ−℘−1℧nt″,(49)ϖ0,k=A+B−Ab−atk−a,Đ℘ϖ0,k=A+B−Atk−a1−℘b−aΓ2−℘.Now, Computingϖn+1,k−ϖn,k and Đ℘ϖn+1,k−Đ℘ϖn,k from equations (47) and (48) and by using (H2) and (H3), the result is
(50)ϖ1,ktk−ϖ0,ktk≤khγNγ−1−kγ−1Γγ+1χ+Δ1,Đ℘ϖ1,ktk−Đ℘ϖ0,ktk≤hγ−℘k1−℘Γγ+1Γ2−℘kγ−1−Γγ−℘+1Nγ−1Γγ+1Γ2−℘Γγ−℘+1χ+Δ2.Let(51)P=khγNγ−1−kγ−1Γγ+1,Q=hγ−℘k1−℘Γγ+1Γ2−℘kγ−1−Γγ−℘+1Nγ−1Γγ+1Γ2−℘Γγ−℘+1,Δ1=kh36Γγχ1,Δ2=kh36Γγ−℘χ2,ϖ2,ktk−ϖ1,ktk≤PΩ1ϖ1−ϖ0+Ω2Đ℘ϖ1−Đ℘ϖ0+Δ1,Đ℘ϖ2,ktk−Đ℘ϖ1,ktk≤QΩ1ϖ1−ϖ0+Ω2Đ℘ϖ1−Đ℘ϖ0+Δ2,and so on, in general(52)ϖn+1,ktk−ϖn,ktk≤PPΩ1+QΩ2n−1Ω1ϖ1−ϖ0+Ω2Đ℘ϖ1−Đ℘ϖ0+P∑i=0n−2PΩ1+QΩ2iΩ1Δ1+Ω2Δ2+Δ1,Đ℘ϖn+1,ktk−Đ℘ϖn,ktk≤QPΩ1+QΩ2n−1Ω1ϖ1−ϖ0+Ω2Đ℘ϖ1−Đ℘ϖ0+P∑i=0n−2PΩ1+QΩ2iΩ1Δ1+Ω2Δ2+Δ2.IfPΩ1+QΩ2<1, then the process of iteration is convergent and the bound of truncation error is given by
(53)Entk=ϖtk−ϖntk=∑i=n∞ϖi+1,k−ϖi,k≤Δ1≤ψ1h2,Đ℘Entk=Đ℘ϖtk−Đ℘ϖntk=∑i=n∞Đ℘ϖi+1,k−Đ℘ϖi,k≤Δ2≤ψ2h2.


## 6. Numerical Illustrations

To show the efficiency of this method, we will approximate the solutions for some fractional differential equations of order1<γ≤2 and 0<℘≤1, using a prepared program in Matlab. To solve these problems, we used equations (42) and (43) to obtain the sequences yntk,Đ℘yntk. By using the exact solutions, we computed the error at each pivotal point. The partial output of these error terms is presented in Tables 1 and 2.Example 3.
Consider the fractional boundary value problem(54)Đ3/2ϖ+13ϖ+14Đ1/2ϖ=−7t0.52π+t3−2t1.53π−t23,t∈0,1,
ϖ0=0,ϖ1=0. The exact solution is ϖt=t1−t.Table 1
Absolute error for the numerical solutions of Examples3–5.
tExample3 iteration 8Example4 iteration 6Example5 iteration 7h=1/20h=1/40h=1/20h=1/40h=0.01h=0.0050.1.243e-2.876e-3.109e-2.393e-3.362e-3.185e-30.2.278e-2.996e-3.228e-2.822e-3.677e-3.347e-30.3.276e-2.986e-3.352e-2.126e-2.925e-3.473e-30.4.255e-2.910e-3.469e-2.168e-2.108e-2.556e-30.5.224e-2.797e-3.564e-2.202e-2.115e-2.592e-30.6.185e-2.659e-3.618e-2.221e-2.113e-2.577e-30.7.142e-2.505e-3.610e-2.218e-2.999e-3.509e-30.8.964e-3.342e-3.520e-2.185e-2.765e-3.390e-30.9.486e-3.172e-3.324e-2.115e-2.430e-3.219e-3Table 2
Error for the numerical solution of Example6.
tExample6 iteration 8h=0.01h=0.0051.1.194e-6.208e-71.2.262e-6.566e-71.3.299e-6.560e-71.4.338e-6.103e-71.5.376e-6.603e-71.6.392e-6.125e-61.7.367e-6.160e-61.8.290e-6.150e-61.9.164e-6.935e-7A comparison of the absolute errors, computed by the proposed method withh=1/20, h=1/40, and N=8 at t∈0,1, is calculated in Table 1. The approximate solutions obtained with the exact solution of corresponding fractional-order equation when γ=3/2, ℘=1/2, and h=1/20 are given in Figure 1(a).Example 4.
Consider the fractional boundary value problem(55)Đ3/2ϖ+ϖ=t5−t4+128t3.57π−64t2.55π,t∈0,1,
ϖ0=0,ϖ1=0. The exact solution is ϖt=t5−t4.Figure 1
A comparison between exact and approximate solutions: (a) Example3 when γ=3/2, ℘=1/2, and h=1/20. (b) Example 4 when γ=3/2 and h=1/40.
(a)(b)In Figure1(b), the approximate solutions obtained with h=1/40 together with the exact solution of this problem are plotted. Furthermore, by considering h=1/20, h=1/40, and N=6, the absolute errors at some selected points are reported in Table 1.Example 5.
Consider the fractional boundary value problem(56)ϖ′′+ϖ+Đ1/2ϖ=t2+t+2t1.5Γ5/2+t0.5Γ3/2+3,t∈0,1,
ϖ0=1,ϖ1=3. The exact solution is ϖt=t2+t+1.Table1 shows the absolute error between exact and numerical solutions when h=0.01, h=0.005, and N=7. Figure 2(a) compares both the exact and numerical solutions for the fractional differential equation (56) with γ=2, ℘=1/2, and h=0.01 in some points t∈0,1.Example 6.
Consider the boundary value problem(57)ϖ′′=−tϖ′2e−2ϖ,t∈1,2,Figure 2
Exact and approximate solutions withh=0.01. (a) Example 5 when γ=2 and ℘=1/2. (b) Example 6 when γ=2 and ℘=1.
(a)(b)ϖ1=0,ϖ2=ln2. The exact solution is ϖt=lnt.The numerical results of Example6 for the values γ=2,℘=1 and h=0.01 are shown in Figure 2(b). In addition, the absolute error is presented in Table 2 when h=0.01, h=0.005, and N=8.


## 7. Stability Theorems

In this section, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of the boundary value problem (5) and (6). For the definitions of Ulam-Hyers stability and Ulam-Hyers-Rassias stability, see [46]. For ℧:I×ℝ×ℝ⟶ℝ, the norm is defined as
(58)ϖ=maxt∈a,bΩ1ϖt+maxt∈a,bΩ2Đ℘ϖt.Definition 10.
Equation (5) is Ulam-Hyers stability if there exists a real number cf>0 such that for each ε>0 and for each solution z∈C1I,ℝ of the inequality
(59)cĐγzt−℧t,zt,Đ℘zt≤ε,t∈I,
there exists a solutionϖ∈C1I,ℝ of equation (5) with
(60)zt−ϖt≤cfε,t∈I.Definition 11.
Equation (5) is Ulam-Hyers-Rassias stability with respect to φ∈C1I,ℝ+ if there exists a real number cf>0 such that for each ε>0 and for each solution z∈CI,ℝ of the inequality
(61)cĐγzt−℧t,zt,Đ℘zt≤εφt,t∈I,there exists a solutionϖ∈C1I,ℝ of equation (5) with
(62)zt−ϖt≤cfεφt,t∈I.For proving, we need the following hypothesis.(H4) There exists an increasing functionφ∈CI,ℝ+, and there exists Λφ,λφ>0 such that for any t∈I, we have
(63)∫att−sγ−1Γγφsds≤Λφφt,∫att−sγ−℘−1Γγ−℘φsds≤λφφt.Theorem 12.
Assume that℧:I×R×R⟶R is a continuous function and (H2) holds with PΩ1+QΩ2<1. Then, the problem (5) and (6) is Ulam-Hyers stability.Proof.
Letztk∈CI,ℝ be a solution of the inequality (59), and there exists a solution ϖ∈CI,ℝ of equation (5). Then, we have
(64)ϖtk=A+B−Ab−at−a−t−ab−aΓγ∫abb−sγ−1℧s,ϖs,Đ℘ϖsds+1Γγ∫atktk−sγ−1℧s,ϖs,Đ℘ϖsds.
From inequality (21), for each t∈I, we get
(65)ztk−A−B−AkN+kNabIγ℧tk,ztk,Đ℘ztk−atkIγ℧tk,ztk,Đ℘ztk−Δ1≤khγεΓγ+1,Đ℘ztk−B−Ak1−℘h−℘NΓ2−℘+k1−℘h−℘NΓ2−℘abIγ℧tk,ztk,Đ℘ztk−atkIγ−℘℧tk,ztk,Đ℘ztk−Δ2≤khγ−℘εΓγ−℘+1,
and by (H2), for eacht∈I, one can arrive at the following results:
(66)ztk−ϖtk≤khγεΓγ+1+PΩ1z−ϖ+Ω2Đ℘z−Đ℘ϖ,(67)Đ℘ztk−Đ℘ϖtk≤khγ−℘εΓγ−℘+1+QΩ1z−ϖ+Ω2Đ℘z−Đ℘ϖ,(68)ztk−ϖtk=maxt∈IΩ1ztk−ϖtk+maxt∈IΩ2Đ℘ztk−Đ℘ϖtk.
Then, from equation (68), we conclude that
(69)ztk−ϖtk≤cfε,t∈I,wherecf=Γγ−℘+1khγΩ1+Γγ+1khγ−℘Ω2Γγ+1Γγ−℘+11−PΩ1+QΩ2.
Thus, the problem (5) and (6) is Ulam-Hyers stability.Theorem 13.
Suppose that conditions (H2) and (H4) are satisfied. Then, the problem (5) and (6) is Ulam-Hyers-Rassias stability.Proof.
Letztk∈CI,ℝ be a solution of the inequality (61), and there exists a solution y∈CI,ℝ of equation (5). From inequality (61), for each t∈I, we have
(70)ztk−A−B−AkN+kNabIγ℧tk,ztk,Đ℘ztk−atkIγ℧tk,ztk,Đ℘ztk−Δ1≤εΛφφtk,Đ℘ztk−B−Ak1−℘h−℘NΓ2−℘+k1−℘h−℘NΓ2−℘abIγ℧tk,ztk,Đ℘ztk−atkIγ−℘℧tk,ztk,Đ℘ztk−Δ2≤ελφφtk,
and by using the hypothesis (H2), for eacht∈I, the result is
(71)ztk−ϖtk≤εΛφφtk+PΩ1z−ϖ+Ω2Đ℘z−Đ℘ϖ,Đ℘ztk−Đ℘ϖtk≤ελφφtk+QΩ1z−ϖ+Ω2Đ℘z−Đ℘ϖ.Then, the use of equation (68) implies that
(72)ztk−ϖtk≤cfεφtk,t∈I,wherecf=Ω1Λφ+Ω2λφ1−PΩ1+QΩ2.Then, the problem (5) and (6) is Ulam-Hyers-Rassias stability.


## 8. Examples

In this section, we give some examples to illustrate the usefulness of our main results.Example 7.
Consider the following fractional boundary value problem:(73)Đ3/2ϖ+13ϖ+14Đ1/2ϖ=−7t0.52π+t3−2t1.53π−t23,t∈0,1,ϖ0=0,ϖ1=0.Here,γ=3/2, and ℘=1/2. By Lipschitz condition, we obtain Ω1=1/3 and Ω2=1/4. To estimate the Ulam stability, let ϖ=1 and h=0.1, and by Theorem 12, we have
(74)Đ3/2ϖ+13ϖ+14Đ1/2ϖ+7t0.52π−t3+2t1.53π+t23≤1.2648822308≤ε,ztk−1≤ζε,ζ=Ω1kh3/2+khΓ5/2Ω2Γ5/21−PΩ1+QΩ2,which shows the problem (73) is Ulam-Hyers stability. Using Matlab, the function pt=z−ϖ is computed and depicted in Figure 3(a), while the function zt for t∈0,1 is given in Figure 3(b).Example 8.
Consider the following fractional boundary value problem:(75)Đ1.2ϖ=et101+etϖ+56+tĐ0.3ϖ,t∈0,1,ϖ0=1,ϖ1=1.Figure 3
(a) The functionpt for t∈0,1. (b) The function zt for t∈0,1.
(a)(b)Here,γ=1.2, and ℘=0.3. By Lipschitz condition, we obtain Ω1=0.073105857 and Ω2=0.833333. Now, we investigate the Ulam-Hyers-Rassias stability for equation (75), let ϖ=t,φ=sint and h=0.1, and by Theorem 13, the following is obtained:
(76)Đ1.2ϖ−et101+etϖ−56+tĐ0.3ϖ≤0.778612319543994≤ε,ztk−ϖtk≤Ω1tk2.2E2,3.2−tk2+Ω2tk1.9E2,2.9−tk2ε1−PΩ1+QΩ2.The functionpt=z−ϖ is depicted in Figure 4(a) for t∈0,1, while the function pt for t∈0,1 is given in Figure 4(b), and the function pt for t∈0,10 is plotted in Figure 4(c).Figure 4
(a) The value ofpt for t∈0,1. (b) The value pt for t∈0,1. (c) The value of pt for t∈0,10.
(a)(b)(c)Moreover, Figures5(a) and 5(b) show the function zt for t∈0,1 and t∈0,10, respectively.Example 9.
Consider the following fractional boundary value problem:(77)Đ2ϖ=e−t20ϖ+910Đ0.5ϖ,t∈0,1,ϖ0=ϖ1=1.Figure 5
The graphic of the functionzt: (a) for t∈0,1 and (b) for t∈0,10.
(a)(b)Here,γ=2, and ℘=0.5. By Lipschitz condition, we obtain Ω1=0.05 and Ω2=0.9. Now, we investigate the Ulam-Hyers-Rassias stability for equation (77), let ϖ=et+1, and φ=et, and by Theorem 13, the results are
(78)Đ2ϖ−e−t20ϖ−57Đ0.5ϖ≤ε,zt−ϖt≤εΩ1t2E1,3t+Ω2t1.5E1,2.5t1−PΩ1+QΩ2.The behavior of the functionpt=z−ϖ is depicted in Figure 6(a) for t∈0,1. Furthermore, the behavior of zt for t∈0,1 and for t∈0,5 is plotted in Figures 6(b) and 6(c), respectively.Example 10 (see [47, 48]).
Consider the following Bagley-Torvik equation:(79)D2ϖ+θD℘ϖ=−1−et−1,t∈0,1,ϖ0=ϖ1=0.
The exact solution of Example10 is not known for general values of ℘. Only, the exact solution is
(80)ϖt=t1−et−1,
forθ=−1 and ℘=1. This example was solved for different values of 0<℘≤1, and the numerical solutions are shown in (Figures 7 and 8).Figure 6
(a) Functionpt for t∈0,1. (b) Function zt for t∈0,1. (c) Function zt for t∈0,5.
(a)(b)(c)Figure 7
Exact and approximate solution for Example10, when θ=−1.Figure 8
Approximate solution for Example10 when (a) θ=−0.5 and (b) θ=0.5.
(a)(b)


## 9. Conclusions

In this article, the author analyzed the existence, uniqueness, and stability of solutions for nonlinear two-point boundary value problems in the sense of Caputo fractional derivative. Banach’s fixed-point theorem was used to prove the uniqueness of the solution, while Brouwer’s fixed-point theorem was employed to study the existence results. In addition, the numerical solution for the fractional differential equation with boundary condition was studied by using successive approximation method. Discussion on the convergence and error analysis of the proposed method is presented. Some numerical examples are given to show the accuracy of the suggested method. Finally, the Hyers-Ulam and Hyers-Ulam-Rassias stability of the problem (5) and (6) are studied. Examples are presented to illustrate our theoretical results.



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