On the6olutions of a&lass of1onlinear)ractional'ifferential(quations
with%oundary&onditions
$1$%(/$66,/9$
Center for Research and Development in Mathematics and Applications (CIDMA),
University of Aveiro,
Campus Universitário de Santiago, 3810-193 Aveiro,
PORTUGAL
Abstract: In this article, we consider a class of fractional boundary value problems with Caputo fractional deriva-
tive of order α∈(2,3). The existence and uniqueness of solutions are discussed and the Adomian decomposition
method is proposed to obtain an approximation of the solution. Finally, an example is given to demonstrate the
validity of results.
Key-Words: Banach contraction principle, Adomian decomposition method, Fractional differential equations,
Caputo derivative
1 Introduction
Fractional calculus is a field of mathematical analy-
sis that can be viewed as a generalization of integer
differential calculus, involving derivatives and inte-
grals of real or complex order [12].Although its ori-
gins date back to 1695, with a famous correspon-
dence between Leibniz and L’Hôpital, the scientific
interest in this area of mathematics has become evi-
dent only in the last decades. Nowadays, it is one of
the most intensively developed areas of mathemati-
cal analysis due to its numerous applications in vari-
ous sciences and engineering, such as mechanics, bi-
ology economics, control theory, image and signal
processing, etc. [13]. One of the major difficulties
that arise in dealing with fractional differential equa-
tions is the great difficulty in solving such problems
analytically. In most cases, we do not know the ex-
act solution of the problem. Some numerical and ap-
proximate methods for solving fractional differential
equations have been proposed, including the resid-
ual power series method, the homotopy perturbation
method, fractional Adams-Moulton methods, varia-
tional interaction methods, and the Adomian decom-
position method [3], [4], [9], [7], [10], [5], [15]. The
Adomian decomposition method (ADM) was intro-
duced by Adomian in the 1980s [2], [3], [4]. The
method provides an effective procedure to obtain ex-
plicit and numerical solutions for a variety of differ-
ential systems to solve physical problems [16].
Continuing the results presented in [14], we con-
sider the fractional boundary value problem with
(left) Caputo fractional derivative (FBVP)
CDα
a+x(t)−f(t, x(t), x0(t), x00(t)) = 0,
x(a)−βx0(a) = 0, x0(a) = x0(b), x00(a) = 0,
where t∈[a, b],β∈R,0≤a<b,2< α < 3
and f: [a, b]×R3→Ris continuous. First,
a result is presented that guarantees that the prob-
lem under study has a solution and, resorting to Ba-
nach’s contraction principle, sufficient conditions are
established that guarantee that the solution sought is
unique. The method of Adomian decomposition is ex-
plained in the Section 2, and applied to approximate
the solution in an illustrative example.
2 Preliminaries
In this section we introduce some notations, defini-
tions and results used in this work.
Definition 1. The Riemann-Liouville fractional inte-
gral of order α∈R+of a function xis defined by
Iα
ax(t) = 1
Γ(α)Zt
a
(t−s)α−1x(s)ds,
provided the right-hand side is pointwise defined on
(a, ∞), where Γis Euler Gamma function (given by
Γ(α) = R∞
0tα−1e−tdt, α > 0).
Definition 2. The Caputo fractional derivative of or-
der α > 0of a continuous function xis given by
CDα
ax(t) = 1
Γ(n−α)Zt
a
x(n)(s)
(t−s)α−n+1 ds,
provided that the right-hand side is pointwise defined
on (a, ∞), where n∈Nis such that n−1< α < n.
If α∈N, then CDα
ax(t) = d
dtαx(t).
The following lemma establishes an important re-
lationship between the Riemann-Liouville integral
and the Caputo derivative and will be essential for the
study of the proposed fractional boundary value prob-
lem.
5HFHLYHG$XJXVW5HYLVHG0DUFK$FFHSWHG$SULO3XEOLVKHG0D\
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Lemma 1. [12] Let n−1< α < n,n∈N. If
x∈Cn−1([a, b]) or x∈ACn−1([a, b]), then the fol-
lowing relation holds:
(Iα
aCDα
ax)(t) = x(t)−
n−1
X
k=0
x(k)(a)
k!(t−a)k.(1)
Moreover, for α > 0and x∈C([a, b]),
(CDα
a+Iα
ax)(t) = x(t).
Fixed point theory is a useful tool for studying
the existence and uniqueness of solutions to boundary
value problems. In this sense, we recall the Banach’s
contraction principle.
Theorem 1. (Banach’s contraction principle) Let
(X, d)be a complete metric space and let T:X→X
be a contraction on X. Then, Thas a unique fixed
point x∈X.
In this paper, we consider C2([a, b]) with the usual
norm
kxkC2=max
t∈[a,b]{kxk∞+kx0k∞+kx00k∞},
where kxk∞=maxt∈[a,b]|x(t)|. It is known that
C2([a, b]), endowed with such norm, is a Banach
space.
2.1 Adomian decomposition method
Consider a nonlinear differential equation, which can
be decomposed in the following form
Lx +Rx +Nx =g, (2)
where Lis the highest order differential operator
which is easily or trivially invertible, Ris the remain-
ing linear part of order less than L,Nrepresents the
nonlinear part and gis given known function.
Because Lis invertible, applying L−1to both
members of equation (2), we get
x=ϕ−L−1Rx −L−1Nx +L−1g, (3)
where ϕis the integration constant and satisfies Lϕ =
0. The idea of this method is to represent the unknown
function xby the infinite series
x(t) = ∞
X
n=0
xn(t).
In this regard, the nonlinear term N x is represented
by the infinite series of Adomian polynomials
Nx =∞
X
n=0
An(x0, x1,··· , xn),
where An’s are the Adomian polynomials [1], [2], [4],
depending on x0, x1,··· , xn, that can be formulated
by
An=1
n!
dn
dλn"N ∞
X
n=0
λnxn!#λ=0
, n = 0,1,2··· .
Note that if the non linearity has the form Nx =h(x),
with has a smooth function of x, then the polynomials
Anare generated for each nonlinearity in the way that
A0depends only on x0,A1depends on x0and x1,A2
depends on x0, x1, x2, and so forth [2]. In particular,
the first five Adomian polynomials are the following
A0=h(x0),
A1=x1h0(x0),
A2=x2h0(x0) + 1
2!x2
1h00(x0),
A3=x3h0(x0) + x1x2
2h00(x0) + 1
3!x3
1h(3)(x0),
A4=x4h0(x0) + 1
2!x2
2+x1x3h00(x0)
+1
2!x2
1x2h(3)(x0) + 1
4!x4
1h(4)(x0).
Therefore, the general solution becomes
x=−L−1R∞
X
n=0
xn−L−1∞
X
n=0
An(x0, x1,··· , xn)
+L−1g+ϕ.
where
x0=ϕ+L−1g,
x1=−L−1(Rx0)−L−1(A0),
x2=−L−1(Rx1)−L−1(A1),
.
.
.
xn+1 =−L−1Rxn−L−1An, n ≥0.
Thus, using the known term x0, all components
x1,··· , xn,··· can be determined. The n-th term ap-
proximation solution for the Adomian decomposition
method is
φn=
n−1
X
k=0
xk(t), n ≥1
and the solution x(t) = limn→+∞φn(t). The con-
vergence of this method has been proved in [5], [6],
[8], [11].
3 Existence and uniqueness of
solutions
Consider the equation
CDα
a+x(t)−f(t, x(t), x0(t), x00(t)) = 0, t ∈[a, b],
(4)
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subject to the boundary conditions
x(a)−βx0(a) = 0, x0(a) = x0(b), x00(a) = 0.
In what follows, we use the notation
ϑ=x0(a).
Applying Iα
ato both members of equation (4), us-
ing Lemma 1, it yields that
x(t) = x(a)+x0(a)(t−a)+x00(a)(t−a)2+Iα
a(fx)(t).
From x00(a) = 0 and x(a) = βx0(a) = βϑ, it follows
that
x(t) = βϑ +ϑ(t−a) + Iα
a(fx)(t),(5)
where fx(t) = f(t, x(t), x0(t), x00(t)).
3.1 Existence of solutions
The next theorem was proved in [14, Theorem 3] and
establishes sufficient conditions for the existence of
solutions to the fractional boundary value problem
(1). This result was obtained by applying Mawhin’s
coincidence theory.
Theorem 2. Let f: [a, b]×R3→Rbe continuous,
and suppose the following conditions are verified:
(H1) There exist nonnegative constants p1, p2, p3and
qsatisfying η·p∗<1, with
η=(b−a)α
Γ(α+ 1) +(b−a)α−1
Γ(α)+(b−a)α−2
Γ(α−1) (6)
and p∗=maxt∈[a,b]{p1, p2, p3}such that for all
(u, v, w)∈R3
|f(t, u, v, w)| ≤ p1|u(t)|+p2|v(t)|
+p3|w(t)|+q, t ∈[a, b].
(H2) There exists a constant R > 0such that for x∈
domL, if |x0(t)|> R for all t∈[a, b], then
Zb
a
(b−s)α−2f(s, x(s), x0(s), x00(s))ds6= 0.
(H3) There exists a positive constant R∗such that for
c1∈R, if |c1|> R∗for t∈[a, b], either
c1f(t, c1(t−a+β), c1,0) >0, t ∈[a, b],
or
c1f(t, c1(t−a+β), c1,0) <0, t ∈[a, b].
Then the fractional boundary value problem FBVP
has at least one solution in C2([a, b]).
3.2 Uniqueness of solution
The following theorem establishes sufficient condi-
tions for the uniqueness of solutions.
Theorem 3. Suppose that (H1)–(H3) are verified and
assume that the following condition is satisfied:
(H4) There exists nonnegative constants d1, d2and d3
such that
|f(t, u, v, w)−f(t, u, v, w)| ≤ d1|u−u|
+d2|v−v|+d3|w−w|,
for every t∈[a, b],(u, v, w)∈R3,(u, v, w)∈
R3.
If
η·d∗<1(7)
with ηas defined in (6) and d∗=max{d1, d2, d3},
then the fractional boundary value problem FBVP has
a unique solution in C2([a, b]).
Proof. Let us prove that we have a unique solution
in C2([a, b]). For this purpose, and since the so-
lution of the problem can be rewritten in terms of
the integral equation (5), we consider the operator
T:C2([a, b]) →C2([a, b]) defined by
(T x)(t) = βϑ +ϑ(t−a) + Iα
a(fx)(t)
=βϑ+ϑ(t−a)+ 1
Γ(α)Zt
a
(t−s)α−1fx(s)ds.
Let BR={x∈C2(R) : kXkC2≤R}and
choose
R≥(|ϑ|+b−a+ 1)|ϑ|+ηq
1−ηp∗,
with p∗=max{p1, p2, p3}. Note that 1−ηp∗>1
according to (H1). We have that
|T x(t)|
≤ |ϑ|(|β|+t−a) + 1
Γ(α)Zt
a
(t−s)α−1|fx(s)|ds
≤ |ϑ|(|β|+t−a) + 1
Γ(α)Zt
a
(t−s)α−1(p1|x(s)|
+p2|x0(s)|+p3|x00(s)|+q)ds
≤ |ϑ|(|β|+b−a) + 1
Γ(α)Zt
a
(t−s)α−1(p∗(|x(s)|
+|x0(s)|+|x00(s)|) + qds
≤ |ϑ|(|β|+b−a) + (b−a)α
Γ(α+ 1)q+(b−a)α
Γ(α+ 1)p∗kxkC2.
Moreover, we have
(T x)0(t) = ϑ+1
Γ(α−1) Zt
a
(t−s)α−2fx(s)ds.
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Thus, we obtain
|(T x)0(t)|
≤ |ϑ|+1
Γ(α−1) Zt
a
(t−s)α−2|fx(s)|ds
≤ |ϑ|+1
Γ(α−1) Zt
a
(t−s)α−2(p1|x(s)|
+p2|x0(s)|+p3|x00(s)|+q)ds
≤ |ϑ|+(b−a)α−1
Γ(α)q+(b−a)α−1
Γ(α)p∗kxkC2.
Moreover, we get that
(T x)00(t) = 1
Γ(α−2) Zt
a
(t−s)α−3fx(s)ds
and
|(T x)00(t)|
≤1
Γ(α−2) Zt
a
(t−s)α−3|fx(s)|ds
≤1
Γ(α−2) Zt
a
(t−s)α−3(p1|x(s)|
+p2|x0(s)|+p3|x00(s)|+q)ds
≤(b−a)α−2
Γ(α−1) q+(b−a)α−2
Γ(α−1) p∗kxkC2.
Finally, we can conclude
kT xkC2=max kT xk∞+kT x0k∞+kT x00k∞
≤(|β|+b−a+ 1)|ϑ|+ηq +ηp∗kxkC2
≤(|β|+b−a+ 1)|ϑ|+ηq +ηp∗R
≤R,
which shows that T(BR)⊂BR.
Let us now take x, y ∈C2([a, b]). Note that T y is
defined by
(T y)(t)=βϑ+ϑ(t−a)+ 1
Γ(α)Zt
a
(t−s)α−1fy(s)ds.
Applying (H3), it follows that, for any t∈[a, b],
kT x −T yk
≤1
Γ(α)Zt
a
(t−s)α−1|fx(s)−fy(s)|ds
+1
Γ(α−1) Zt
a
(t−s)α−2|fx(s)−fy(s)|ds
+1
Γ(α−2) Zt
a
(t−s)α−3|fx(s)−fy(s)|ds
≤1
Γ(α)Zt
a
(t−s)α−1(d1|x(s)−y(s)|
+d2|x0(s)−y0(s)|+d3|x00(s)−y00(s)||ds
+1
Γ(α−1) Zt
a
(t−s)α−2|(d1|x(s)−y(s)||
+d2|x0(s)−y0(s) + d3|x00(s)−y00(s)||ds
+1
Γ(α−2) Zt
a
(t−s)α−3|(d1|x(s)−y(s)|
+d2|x0(s)−y0(s)|+d3|x00(s)−y00(s)||ds
≤d∗kx−ykC2
Γ(α)Zt
a
(t−s)α−1ds
+d∗kx−ykC2
Γ(α−1) Zt
a
(t−s)α−2ds
+d∗kx−ykC2
Γ(α−2) Zt
a
(t−s)α−3|ds
=ηd∗kx−ykC2.
Since η·d∗<1, by Banach’s contraction principle, T
has a unique fixed point which is the unique solution
of the problem (1), and the proof is complete.
4 Illustrative example
Consider the following fractional boundary value
CD
5
2
1+x(t) = 1
10t+1
15 sin(x(t)) + 2
5x0(t)
x(1) = x0(1) = x0(2), x00(1) = 0
,(8)
t∈[1,2], where α=5
2,β= 1,a= 1,b= 0 and
f(t, x(t), x0(t), x00(t)) = 1
10t+1
15 sin(x(t))+ 2
5x0(t)
is a continuous function. It follows that
|f(t, x(t), x0(t), x00(t))| ≤ 1
5+1
15|x(t)|+2
5|x0(t)|
with p1=1
15,p2=2
5,p3= 0 and q=1
5. It
follows that p∗=max{p1, p2, p2}=2
5. Moreover,
η=58
15√πand consequently, pη =116
75√π<1. Thus,
condition (H1) is verified.
Let R= 1, and for any x∈domL, assume
|x0(t)|> R holds for t∈[1,2]. From the continuity
of x0, either x0(t)> R or x0(t)<−Rfor t∈[1,2].
If x0(t)>1, one has
Z2
1
(2 −s)3
21
10t+1
15 sin(x(t)) + 2
5x0(t)ds
>1
10 −1
15 +2
5Z2
1
(2 −s)3
2ds=13
75 >0.
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If x0(t)<−1, one has
Z2
1
(2 −s)3
21
10t+1
15 sin(x(t)) + 2
5x0(t)ds
<1
5+1
15 −2
5Z2
1
(2 −s)3
2ds=−4
75 <0.
Thus, for |x0(t)|>1,
Z2
1
(2 −s)3
2f(s, u(s), u0(s), u00(s))ds6= 0
and condition (H2) is verified.
Finally, we observe that
ft, c1t−1
2, c1,0
=1
10t+2
5c1+1
15 sin c1t−1
22!.
Take R∗=1
2and assume |c1|>1
2. Thus, if c1>1
2,
c1ft, c1t−1
2, c1,0>1
21
10 −1
15 +1
5
=7
60 >0
and if c1<−1, one has
c1ft, c1t−1
2, c1,0<−1
21
5+1
15 −1
5
=−1
30 <0.
Therefore, condition (H3) is verified.
It follows from Theorem 2 that fractional bound-
ary value problem (8) has at least one solution.
According to the Theorem 3, we have that d1=
1
15, d2=2
5and d3= 0. Thus, d∗=2
5and η·d∗<1
and the solution of the problem (8) is unique.
4.1 Numerical part
The integral equation (8) we can be identified with
Lx +Rx +Nx =gwhere
(Rx)(t) = −2
5x00(t)
(Nx)(t) = −1
15 sin(x(t))
g(t) = 1
10t.
As before, let x0(1) = ϑ. Applying L−1=Iα
ato
both members of equation in the problem (8) and used
ADM method presented in the Section 2, we get
x0(t) = ϑt +1
10I
5
2
1+t
=ϑt+4
3√π2(t−1)5/2
7+4(t−1)5/2t
35 ,
x1(t) = −I
5
2
1+Rx0−I
5
2
1+A0,
xn+1(t) = −I
5
2
1+Rxn−I
5
2
1+An, n = 1,2,··· .
The first few Adomian polynomials Anthat represent
the nonlinear term −1
15 sin(x(t)) are defined as
A0=−1
15 sin(x0),
A1=−1
15x1cos(x0),
A2=−1
15x2cos(x0) + 1
30x2
1sin(x0),
A3=−1
15x3cos(x0) + 1
15x1x2
2sin(x0),
+1
90x3
1cos(x0).
Expressing the n-term approximation solution of
fractional boundary value problem (1) as φn(t) =
Pn
k=0 xk(t), the exact solution can be obtained by
x(t) = lim
n→∞ φn(t).
5 Conclusions
In this paper, we continue the study of a class of
boundary value problems, using fractional derivative
of Caputo of order α∈(2,3). Following the results
obtained in [14], it is presented a result of existence of
solutions [14] and conditions are obtained to guaran-
tee its uniqueness by applying the Banach contraction
theorem. From a numerical point of view, it is con-
sidered the Adomian decomposition method, which
provides a numerical approximation to the solution.
Finally, an example of application of the previously
presented theory is given.
We emphasize that it is presented a possible ap-
proach to the study of solutions of a class of non-
linear fractional differential equations under certain
boundary conditions. The same problem can be stud-
ied from a different point of view, obtaining other suf-
ficient conditions for the existence and uniqueness of
solutions as well as other numerical techniques can
be used. On the other hand, the approaches proposed
here can also be applied to similar problems.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Anabela S. Silva has written, reviewed, and
actively participated in all the
publication stages of this manuscript.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the Center for Re-
search and Development in Mathematics and Ap-
plications (CIDMA) through the Portuguese Foun-
dation for Science and Technology (FCT - Fun-
dação para a Ciência e a Tecnologia), reference
UIDB/04106/2020, and by national funds (OE),
through FCT, I.P., in the scope of the framework
contract foreseen in the numbers 4, 5 and 6 of the
article 23, of the Decree-Law 57/2016, of August
29, changed by Law 57/2017, of July 19.
Conflict of Interest
The author declares no conflict of interest.
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International , CC BY 4.0)
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.33
Anabela S. Silva
E-ISSN: 2224-2880
284
Volume 22, 2023
