Solving Two-Point Boundary Value Problems of Fractional Differential Equations Min Lia,b , Ning-Ming Niea,b1 , Salvador Jim´enezc , Yi-Fa Tanga,d, , Luis V´azqueze a
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China b
c
Graduate School of the Chinese Academy of Sciences, Beijing 100190, China
Departamento de Matem´ atica Aplicada TTII, E.T.S.I. Telecomunicaci´ on, Universidad Polit´ecnica de Madrid, 28040-Madrid, Spain d
Departamento de Matem´ atica Aplicada, Facultad de Matem´ aticas, Instituto de Matem´ atica Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain
e
Departamento de Matem´ atica Aplicada, Facultad de Inform´ atica, Instituto de Matem´ atica Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain
Abstract We solve four kinds of two-point boundary value problems of fractional differential equations with Caputo’s derivatives or Riemann-Liouville Derivatives. Analytically, we introduce the fractional Green functions and prove the existence and uniqueness of the solutions; numerically, we design the single shooting methods and get the accurate solutions. Keywords: Caputo’s derivatives, Riemann-Liouville Derivatives, Fractional differential equation, Twopoint boundary value problem, Existence and uniqueness, Single shooting method.
1
Introduction
Fractional calculus and fractional differential equations (FDEs) are as old as the classical calculus (refer to [18] or [20] for a historical survey). They have been successfully applied to many fields, such as viscoelastic materials, signal processing, controlling, quantum mechanics, meteorology, finance, life science ( see [5], [13], [17]-[21] ). Numerous publications have been focused on analytical and numerical study of the fractional initial value problems (FIVPs) (see [7]-[9], [16], [18], [20]-[21], [23]-[25]). Comparatively, little attention has been paid to the fractional two-point boundary value problems (FBVPs). In this context, the existence of solutions of the Sturm-Liouville problem of an FDE, the Dirichlet-type FBVPs, a class of FBVPs with Riemann-Liouville fractional derivatives, and some kind of FBVPs with Caputo’s derivatives are investigated by Aleroev [1] and Nakhushev [19], Kilbas and Trujillo [15], Zhang [26], and Bai and L¨ u [3] respectively; the least squares finite-element technique is employed by Roop and his coworkers to solve some kind of FBVPs [11], [12]; and the Adomian decomposition method is used by Jafari and Daftardar-Gejji to find approximate and positive solutions of a kind of FBVPs with Caputo’s fractional derivative [14]. Besides these, few more contributions are made to analytical and numerical study of the solutions for FBVPs. 1
Corresponding author.
E-mail:
[email protected]
1
In this paper, we first study the existence and uniqueness of solutions, and numerical methods for FBVPs with Caputo’s derivatives C γ a Dt y(t)
+ f (t, y(t)) = 0, a < t < b, 1 < γ ≤ 2,
y(a) = α, y(b) = β
(1.1) (1.2)
and C γ a Dt y(t)
+ g t, y(t),
C θ a Dt y(t)
y(a) = α, y(b) = β,
= 0, a < t < b, 1 < γ ≤ 2, 0 < θ ≤ 1,
(1.3) (1.4)
where y : [a, b] 7→ R, f : [a, b] × R 7→ R, g : [a, b] × R × R 7→ R are continuous and satisfy Lipschitz
conditions
|f (t, x) − f (t, y)| ≤ Kf |x − y|, |g(t, x2 , y2 ) − g(t, x1 , y1 )|
≤ Kg |x2 − x1 | + Lg |y2 − y1 |
(1.5) (1.6)
with Lipschitz constants Kf , Kg , Lg > 0. Secondly, we study the existence and uniqueness of solutions, and numerical methods for FBVPs with Riemann-Liouville derivatives RL γ a Dt y(t)
+ f (t, y(t)) = 0, a < t < b, 1 < γ ≤ 2,
y(a) = 0, y(b) = β,
(1.7) (1.8)
and RL γ a Dt y(t)
+ g t, y(t),
RL θ a Dt y(t)
y(a) = 0, y(b) = β,
= 0, a < t < b, 1 < γ ≤ 2, 0 < θ ≤ γ − 1,
(1.9) (1.10)
where y : [a, b] 7→ R, f : [a, b] × R 7→ R, g : [a, b] × R × R 7→ R are continuous and satisfy Lipschitz
conditions (1.5) and (1.6), respectively.
As is well known, the existence and uniqueness of solutions for two-point boundary value problems (BVPs) is a fundamental topic, and many boundary value problems can be transformed into the corresponding integral forms by means of the so-called “Green functions”, and then the existence and uniqueness of solutions can be deduced from the contractive mapping principle and the Lipschitz conditions (refer to [4]). Naturally, one may ask: does this happen to FBVPs (1.1-1.2) or (1.3-1.4)? We will give the positive answer in the sequel. There indeed exist the corresponding Green functions for the above four kinds of FBVPs (we call them “fractional Green functions”). And similar results on existence and uniqueness of solutions can be deduced by contractive mapping principle, fractional Green functions and Lipschitz conditions. It needs to be pointed out that the “fractional Green functions” we constructed later are inspiration of the work by Bai and L¨ u, and by Zhang. Bai and L¨ u [3] have shown that there exists such a fractional Green function for an FBVP with Riemann-Liouville fractional derivatives, which encourages us to find the corresponding fractional Green functions for FBVPs with Caputo’s fractional derivatives: (1.1-1.2) and (1.3-1.4). Zhang [26] has not yet, but
2
almost obtained the fractional Green functions for FBVPs (1.1-1.2). Besides theoretical analysis, we also concern numerical methods for the FBVPs. As is well known, the shooting methods are effective numerical tools for BVPs (see [2], [22]). Another problem naturally shows up: are the shooting methods applicable to FBVPs? Again, the answer is positive. In sections 4 and 5, we will show that the shooting methods completely suit for the case of FBVPs. This paper is organized as follows: first, we do some necessary preparations in section 2; then, we focus on the existence and uniqueness of solutions for FBVPs in section 3; and then, we design the shooting methods to solve FBVPs in section 4; finally, we show the results of numerical experiments in section 5.
2
Preliminaries
First of all, we introduce the definitions and properties of Caputo’s derivatives, and Riemann-Liouville fractional integrals. Definition 2.1 (see [21]) Provided γ > 0, n − 1 < γ ≤ n and let Cn [a, b] := {y(t) : [a, b] → R; y(t) has
a continuous n-th derivative}. (1) The operator
RL γ a Dt
defined by RL γ a Dt y(t)
dn 1 = n dt Γ(n − γ)
Z
t
(t − τ )n−γ−1 y(τ )dτ
a
(2.1)
for t ∈ [a, b] and y(t) ∈ Cn [a, b], is called the Riemann-Liouville differential operator of order γ. (2)The operator
C γ a Dt
defined by
C γ a Dt y(t)
Z
1 = Γ(n − γ)
a
t n−γ−1
(t − τ )
d dτ
n
y(τ )dτ
(2.2)
for t ∈ [a, b] and y(t) ∈ Cn [a, b], is called the Caputo differential operator of order γ. Definition 2.2 (see [21])Provided γ > 0, the operator Jaγ , defined on L1 [a, b] by Jaγ y(t) =
1 Γ(γ)
Z
t
a
(t − τ )γ−1 y(τ )dτ
(2.3)
for t ∈ [a, b], is called the Riemann-Liouville fractional integral operator of order γ, where L1 [a, b] := Rb {y(t) : [a, b] → R; y(t) is measurable on [a, b] and a |y(t)|dt < ∞}.
Lemma 2.3 (see [25])
(1) Let γ > 0. Then, for every f ∈ L1 [a, b], RL γ γ a Dt Ja f
=f
(2.4)
almost everywhere. (2) Let γ > 0 and n − 1 < γ ≤ n. Assume that f is such that Jan−γ f ∈ An [a, b]. Then, Jaγ
RL γ a Dt f (t)
= f (t) −
n−1 X k=0
(t − a)γ−k−1 dn−k−1 n−γ lim J f (z). + Γ(γ − k) z→a dz n−k−1 a
(2.5)
(3) Let γ > θ > 0 and f be continuous. Then C γ γ a Dt Ja f
= f,
C θ γ a Dt Ja f
3
= Jaγ−θ f.
(2.6)
(4) Let γ ≥ 0, n − 1 < γ ≤ n and f ∈ An [a, b]. Then Jaγ
C γ a Dt f (t)
= f (t) −
n−1 X k=0
Dk f (a) (t − a)k , k!
(2.7)
where An [a, b] is a set of functions with absolutely continuous derivative of order n − 1.
Then we introduce the contractive mapping principle. Let S be a Banach space, T : S 7−→ S be a
mapping, and k · k denote the norm of S.
Definition 2.4 (see [4]) If there exists a constant ρ (0 ≤ ρ < 1), such that kT x − T yk ≤ ρkx − yk
(2.8)
for any x, y ∈ S, then T is said to be a contractive mapping of S.
Lemma 2.5 (Contractive Mapping Principle) (see [4]) If T is a contractive mapping of Banach space S, then there exists a unique fixed point y ∈ S satisfying y = T y.
And then, it is worth pointing out that without loss of generality we only need to consider the
case of homogeneous boundary conditions α = β = 0 in the above four kinds of FBVPs. When |α| + |β| = 6 0, we set q(t) =
(b − t)α + (t − a)β , u(t) = y(t) − q(t), b−a
(2.9)
for (1.1-1.2) and (1.3-1.4), then there is an FDE for u(t), in the same form of (1.1-1.2) and (1.3-1.4) satisfying the Lipschitz conditions (the same or almost the same as (1.5) or (1.6)) and homogeneous boundary conditions u(a) = u(b) = 0. The case for (1.7-1.8) and (1.9-1.10) is similar. We let p(t) =
β(t − a)γ , u(t) = y(t) − p(t) (b − a)γ
(2.10)
Thus hereafter we always assume that y(a) = α = β = y(b) = 0.
(2.11)
Finally in this section, we introduce the basic steps for the shooting method to solve FBVPs (1.7-1.8) and (1.9-1.10), (1.1-1.2), and (1.3-1.4). According to the idea of the shooting method in classical case, one always turns a BVP into an initial value problem (IVP) which can be solved by some suitable numerical method (see Chapter 4 in [2]). And, it is already known that there exist successful numerical methods for FIVPs (see [7]-[8], [16], [25]). We write down the corresponding procedure for FBVPs. Denote the corresponding initial value conditions of FBVPs(1.1-1.2) and (1.3-1.4) as y(a) = a0 , y ′ (a) = a1 , a0 , a1 ∈ R.
(2.12)
Then FBVPs(1.1-1.2) and (1.3-1.4) can turn into FIVPs(1.1,2.12) and (1.3,2.12), respectively. Usually, not all ak (k = 0, 1) are equal to zero, in other words, the initial value conditions (2.12) are inhomogeneous. Setting z(t) = y(t) − a0 − a1 (t − a),
(2.13)
(1.1,2.12) and (1.3-2.12) can be transformed into another FIVPs with homogeneous initial value conditions.
4
For a given equispaced mesh a = t0 < t1 < · · · < tN = b with stepsize h = (b − a)/N , we give a
fractional backward difference scheme of order one (refer to [25]) to solve FIVPs (1.1) and (1.3) with homogeneous initial value conditions y(a) = 0, y ′ (a) = 0 : zm
γ
= −h f (tm , zm ) −
m X
ωk zm−k ,
(2.14)
k=1
and zm
=
−hγ g(tm , zm ,
m m X 1 X ω ˜ z ) − ωk zm−k , i m−i hθ i=0
(2.15)
k=1
where ω0
=
ω ˜0
=
γ+1 )ωk−1 , k = 1, 2, · · · , N, k θ+1 1, ω ˜ i = (1 − )˜ ωi−1 , i = 1, 2, · · · , N. i
1, ωk = (1 −
(2.16) (2.17)
The case for (1.7-1.8) and (1.9-1.10) is similar. The reader should note that the initial values take the following form RL γ−1 y(a) a Dt
= b1 ∈ R,
lim Ja2−γ y(t) = b2 ∈ R.
t→a+
(2.18)
It is easy to check that b2 = lim+ Ja2−γ y(t) = 0. t→a
Usually, b1 6= 0 and let z(t) = y(t) −
b1 (t − a)γ−1 . Γ(γ)
(2.19)
(2.20)
in (1.7,2.18) and (1.9,2.18). Then they are transformed into FIVPs with homogeneous initial conditions. We also employ (2.14) or (2.15) to simulate them.
3
Existence and uniqueness of the solutions for FBVPs
In this section, we concentrate on the existence and uniqueness of the solutions for FBVPs (1.7-1.8), (1.9-1.10), (1.1-1.2) and (1.3-1.4). Lemma 3.1 (1) FBVP (1.1-1.2) is equivalent to y(t) =
Z
b
G(t, s)f (s, y(s))ds,
(3.1)
a
where G(t, s) is called the fractional Green function defined as follows: (t − a)(b − s)γ−1 (t − s)γ−1 − , a ≤ s ≤ t ≤ b, (b − a)Γ(γ) Γ(γ) G(t, s) = (t − a)(b − s)γ−1 , a ≤ t ≤ s ≤ b. (b − a)Γ(γ)
(3.2)
(2) FBVP (1.7-1.8) is equivalent to
y(t) =
Z
a
b
b s)f (s, y(s))ds, G(t,
5
(3.3)
b s) is the fractional Green function defined as follows: where G(t, γ−1 (t − a)γ−1 b − s (t − s)γ−1 − , Γ(γ) b−a Γ(γ) b s) = G(t, γ−1 (t − a)γ−1 b − s , Γ(γ) b−a Proof:
a ≤ s ≤ t ≤ b,
(3.4)
a ≤ t ≤ s ≤ b.
We only give the proof for (2). The case for (1) is similar to that for (2). According to
Lemma 2.3 (2), acting the operator Jaγ to both sides of the equation in (1.1) yields y(t) −
(t − a)γ−1 Γ(γ)
RL γ−1 y(z) z=a a Dz
−
(t − a)γ−2 lim J 2−γ y(z) + Jaγ f (t, y(t)) = 0. Γ(γ − 1) z→a+ a
(3.5)
Let t → b and consider (2.11): y(b) = 0 and (2.19): lim+ Ja2−γ y(z) = 0, we obtain z→a
RL γ−1 y(z) z=a a Dz
=
Z b a
b−s b−a
γ−1
f (s, y(s))ds.
(3.6)
Substituting (3.6) into (3.5), we have y(t)
(Z ) γ−1 b (t − a)γ−1 b−s = · f (s, y(s)) ds − Jaγ f (t, y(t)) Γ(γ) b−a a Z b b s)f (s, y(s))ds. = G(t, a
Conversely, it is easy to verify directly that (3.3) is the solution of (1.7-1.8). 0
0
Let P = C [a, b], where C [a, b] denote the space of all continuous functions on [a, b]. We define the norm k · kf and the operator Tf , Tbf as follows: kykf := Kf max |y(t)|, a≤t≤b
Tf y(t) := Tbf y(t) :=
Z
b
G(t, s)f (s, y(s))ds,
a
Z
a
b
b s)f (s, y(s))ds, G(t,
∀ y(t) ∈ P.
Obviously, P is a complete norm space with respect to k · kf and Tf , Tbf are continuous operators. Now (3.1) and (3.3) can be rewritten as y = T y and y = Tbf y, for y ∈ P, respectively. And, “finding a sufficient condition for existence and uniqueness of the solution for FBVP (1.1-1.2) ( or (1.7-1.8) )” is equivalent to “finding a sufficient condition under which Tf ( or Tbf ) is a contractive mapping of P”.
Theorem 3.2 Let f be a continuous function on [a, b] × R and satisfy the Lipschitz condition (1.5). (1) If
2Kf (b − a)γ < 1, Γ(γ + 1)
(3.7)
then there exists a unique solution for FBVP (1.1-1.2) in P. (2) If Kf
(γ − 1)γ−1 (b − a)γ < 1, γ γ Γ(γ + 1)
6
(3.8)
then there exists a unique solution for FBVP (1.7-1.8) in the space P. Proof for (1): For u(t), v(t) ∈ P, and (t, s) ∈ [a, b]× [a, b], according to the definition for the operator
“T ” we have
|T u(t) − T v(t)| ≤ ≤ ≤
Z
b
|G(t, s)| · |f (s, u(s)) − f (s, v(s))|ds (Z ) Z t b (t − a)(b − s)γ−1 (t − s)γ−1 ku − vkf ds + ds (b − a)Γ(γ) Γ(γ) a a a
2(b − a)γ ku − vkf . Γ(γ + 1)
Thus kT u − T vkf = max Kf |T u(t) − T v(t)| ≤ a≤t≤b
2Kf (b − a)γ ku − vkf . Γ(γ + 1)
Considering (3.7), we finish the proof for (1) according to Lemma 2.5. Proof for (2) is similar to that for (1), we omit it here. Lemma 3.3 (1) FBVP (1.3-1.4) is equivalent to Z b y(t) = G(t, s)g s, y(s),
C θ a Ds y(s)
a
where G(t, s) is the fractional Green function defined in (3.2). (2) FBVP (1.9-1.10) is equivalent to Z b b s)g s, y(s), y(t) = G(t,
ds,
RL θ a Ds y(s)
a
b s) is the fractional Green function defined in (3.4). where G(t,
(3.9)
ds,
(3.10)
The proof for this lemma is very similar to that for Lemma 3.1. We omit it. Again, let
P1 = C1 [a, b] := y(t)| y(t), y ′ (t) ∈ C0 [a, b] , θ 0 P2 = Cθ [a, b] := y(t)| y(t) ∈ C0 [a, b], RL a Dt y(t) ∈ C [a, b] .
We define the norm k · kg , k · kgˆ and the operator Tg , Tbg as follows: Z b θ kykg := max Kg |y(t)| + Lg C D y(t) , T y(t) := G(t, s)g s, y(s), g a t a≤t≤b
kykgˆ := max
a≤t≤b
a
θ , Tbg y(t) := Kg |y(t)| + Lg RL a Dt y(t)
Z
b
a
b s)g s, y(s), G(t,
C θ a Ds y(s)
ds,
RL θ a Ds y(s)
∀ y(t) ∈ P1 .
ds,
∀ y(t) ∈ P2 .
Obviously, P1 , P2 are complete norm space with respect to k · kg and k · kgˆ, respectively. The operators Tg , Tbg are continuous. Now (3.9), (3.10) can be rewritten as y = Tg y and y = Tbg y, respectively. And,
“finding a sufficient condition for existence and uniqueness of the solution of FBVP (1.3-1.4) ( or (1.9-1.10) )” is equivalent to “finding a sufficient condition under which T ( or Tbg ) is a contractive mapping of P1 ( or P2 )”. Theorem 3.4
Let g be a continuous function on [a, b] × R × R and satisfy the Lipschitz condition (1.6).
(1) If
Kg
(b − a)γ−θ (b − a)γ−θ 2(b − a)γ + Lg + < 1, Γ(γ + 1) Γ(γ + 1)Γ(2 − θ) Γ(γ + 1 − θ)
7
(3.11)
then there exists a unique solution for FBVP (1.3-1.4) in P1 . (2) If Kg
(γ − 1)γ−1 (b − a)γ (2γ − θ)(b − a)γ−θ + Lg < 1, γ γ Γ(γ + 1) γΓ(γ − θ + 1)
(3.12)
then there exists a unique solution for FBVP (1.9-1.10) in the space P2 . Proof for (1): On the one hand, we have |Tg u(t) − Tg v(t)|
2(b − a)γ ku − vkg Γ(γ + 1)
≤
for u(t), v(t) ∈ P1 , and (t, s) ∈ [a, b] × [a, b], which is similar to Theorem 3.2. On the other hand, according to Lemma 2.3, we have C θ a Dt Tg u(t)
=
Z
(t − a)1−θ Γ(2 − θ)
b
a
(b − s)γ−1 g s, u(s), Γ(γ)(b − a)
C θ a Ds u(s)
And then C θ a Dt Tg u −
C θ a Dt Tg v
ds − Jaγ−θ g t, u(t),
C θ a Dt u(t)
.
(b − a)γ−θ (b − a)γ−θ ≤ ku − vkg · + . Γ(2 − θ)Γ(γ + 1) Γ(γ − θ + 1)
Combined with the definition of k · kg and Lemma 2.5 , (3.11) holds.
Proof for (2) is similar to that for (1). We omit it here.
4
Shooting methods for FBVPs
In this section, we process solving the FBVPs (1.1-1.2), (1.3-1.4), (1.7-1.8) and (1.9-1.10) by single shooting methods. Since the procedure of shooting method for FBVP (1.3-1.4) and (1.9-1.10) are very similar to that for FBVP (1.1-1.2) and (1.7-1.8), we only show the details for FBVP (1.1-1.2) and (1.7-1.8) in the following subsections 4.1 and 4.2.
4.1
Shooting method for linear problems
We consider the linear case of fractional two-point boundary value problem (1.1) with homogeneous boundary value conditions C γ a Dt y(t)
+ c(t)y(t) + d(t) = 0, a < t < b, 1 < γ ≤ 2,
y(a) = 0, y(b) = 0 where c(t), d(t) ∈ C0 [a, b]. According to Theorem 3.2, if b−a
