Initial and Boundary Value Problems for Fractional differential ...

Initial and Boundary Value Problems for Fractional differential equations involving Atangana-Baleanu

arXiv:1706.00740v1 [math.AP] 1 Jun 2017

Derivative Fatma Al-Musalhi1, Nasser Al-Salti1, and Erkinjon Karimov2 1

Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36 Al-Khoudh, Oman

2

Institute of Mathematics named after V.I.Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, Tashkent 125, Uzbekistan. June 5, 2017

Abstract Initial value problem involving Atangana-Baleanu derivative is considered. An Explicit solution of the given problem is obtained by reducing the differential equation to Volterra integral equation of second kind and by using Laplace transform. To find the solution of the Volterra equation, the successive approximation method is used and a lemma simplifying the resolvent kernel has been presented. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of series expansion using orthogonal basis obtained by separation of variables.

1 1.1

Introduction and Preliminaries Introduction and related works

Recently, two newly definitions of fractional derivative without singular kernel were suggested, namely, Caputo-Fabrizio fractional derivative [8] and Atangana-Baleanu fractional derivative [6]. These new derivatives have been applied to real life problems, for example, in the fields of thermal science, material sciences, groundwater modelling and mass-spring system [1], [2], [3], 1

[5], [6] and have been considered in a number of other recent work, see for example, [4], [7], [9], [10], [11], [14]. The main difference between these two definitions is that Caputo-Fabrizio derivative is based on exponential kernel while Atangana-Baleanu definition used Mittag-leffler function as a non-local kernel. The non-locality of the kernel gives better description of the memory within structure with different scale. These two new derivatives are defined as follows Definition 1.1. Let f ∈ H 1 (a, b), b > a, α ∈ [0, 1]. The Caputo-Fabrizio fractional derivative is defined as CF

α a Dt f (t)

B(α) = 1−α

Z

t

 −α (t − s) ds, f (s) exp 1−α 

′

a

and the Atangana-Baleanu fractional derivative is given by   Z B(α) t ′ −α α ABC α f (s)Eα (t − s) ds, a Dt f (t) = 1−α a 1−α

(1)

(2)

where B(α) denotes a normalization function such that B(0) = B(1) = 1 and Eα (z) =

∞ X k=0

zk , Γ(αk + 1)

Re(α) > 0, z ∈ C,

is the Mittag-leffler function of one parameter [12]. For properties related to these derivatives see [6], [7], [11]. In this paper, we are concerned with solutions to initial and boundary value problems for fractional differential equations involving Atangana-Baleanu derivative. We first recall the Mittag-Leffler of two parameters Eα,β (z) =

∞ X k=0

zk , Γ(αk + β)

Re(α) > 0, Re(β) > 0, z ∈ C,

and a generalized Mittag-Leffler function δ Eα,β (z) =

∞ X n=0

(δ)n z n , Γ(αn + β)n!

which is introduced by Prabhakar in [13], where α, β, δ ∈ C with Re(α) > 0, and (δ)n = Γ(δ + n) is Pochammer’s symbol. For δ = 1, it is reduced to Mittag-Leffler function. Γ(δ) Moreover, Mittag-Leffler function Eα (λtα ) is bounded(see [13]), i.e, Eα (λtα ) ≤ M,

(3)

where M denotes a positive constant. In [6], Atangana and Baleanu considered the time fractional ordinary differential equation ABC

α 0 Dt f (t)

2

= u(t),

and on using Laplace transform they found the following solution Z t α 1−α AB α u(s)(t − s)α−1 ds, u(t) + f (t) = 0 It u(t) = B(α) B(α)Γ(α) 0 where they have defined AB

α a It f (t)

1−α α = f (t) + B(α) B(α)Γ(α)

Z

t

f (s)(t − s)α−1 ds

a

to be the fractional integral associated with the fractional derivative (2). In this paper, we consider the following initial value problem (IVP) ABC

α 0 Dt u(t)

− λu(t) = f (t),

t ≥ 0,

(4)

u(0) = u0 , where λ, u0 ∈ R. The solution of this IVP is obtained by two different methods, namely, by reducing it to Volterra integral equation of the second kind and by using Laplace transform. The use of such IVP is illustrated by considering a boundary value problem in which the solution is expressed in the form of series expansion using orthogonal basis obtained by separation of variables. The rest of the paper is organized as follows: at the end of this section, we present a lemma which is important for simplifying the resolvent kernel of the Volterra equation. Then, section 2 is devoted for our main result which is the explicit solution of the IVP (4). We conclude this paper by considering a boundary value problem where we have utlized the solution of the IVP (4).

1.2

Preliminaries

As metioned earlier, one way to solve the IVP (4) is to reduce it to a Volterra integral equation and in order to simplify our calculations, namely the resolvent kernel of the Volterra equation, we have established the following Lemma: Lemma 1.2. Let λ ∈ R and 0 < α < 1 with λ 6= ∞  X

B(α) , then 1−α

i  i   B(α) α −α α iα−1 i (t − ξ) (t − ξ) Eα,iα B(α) − λ(1 − α) 1−α 1−α i=1   B(α) α αλ = (t − ξ)α−1 Eα,α (t − ξ)α . B(α) − λ(1 − α) 1 − α B(α) − λ(1 − α)

3

(5)

Proof. We begin by expanding the following series: i  i   ∞  X α −α B(α) α iα−1 i (t − ξ) (t − ξ) Eα,iα B(α) − λ(1 − α) 1−α 1−α i=1     B(α) α −α α−1 α = (t − ξ) Eα,α (t − ξ) + 1 −α  B(α) − λ(1 − α) 2 1 − α 2  B(α) α −α 2 (t − ξ)2α + (t − ξ)2α−1 Eα,2α B(α) − λ(1 − α) 1 − α 1 − α  3  3   B(α) α −α 3α 3α−1 3 +··· . (t − ξ) (t − ξ) Eα,3α B(α) − λ(1 − α) 1−α 1−α Using the definition of Mittag-Leffler functions, it can be written as follows:  n −α α    (t − ξ) ∞ (1)n X B(α) α 1−α α−1 (t − ξ) + B(α) − λ(1 − α) 1−α Γ(αn + α)n! n=0  n −α α 2  2  (t − ξ) ∞ (2)n X α B(α) 1−α 2α−1 (t − ξ) + B(α) − λ(1 − α) 1−α Γ(αn + 2α)n! n=0  n −α α 3  3  (t − ξ) ∞ (3)n X α B(α) 1−α 3α−1 +··· , (t − ξ) B(α) − λ(1 − α) 1−α Γ(αn + 3α)n! n=0 and expanding the series representations of Mittag-Leffler functions, gives " #    2    2α α 1 (t − ξ) α (t − ξ) α α B(α) (t − ξ)α−1 − + +··· B(α) − λ(1 − α) 1−α Γ(α) 1−α Γ(2α) 1−α Γ(3α) 2  2      α (t − ξ)α 1 B(α) α 2α−1 (t − ξ) −2 +··· + Γ(3α)  B(α) − λ(1 − α) 3  1 − α 3  Γ(2α) 1 − α B(α) α 1 + +··· . (t − ξ)3α−1 B(α) − λ(1 − α) 1−α Γ(3α) Now, combining like terms, we have    2   α (t − ξ)α−1 α (t − ξ)2α−1 B(α) B(α) + α Γ(α) 1 −α Γ(2α)  B(α) − λ(1 − α) B(α)  − λ(1 3 − α) 3α−1 1 − B(α) (t − ξ) α B(α) B(α) 1−2 −1 + B(α) − λ(1 − α) B(α) − λ(1 − α) 1−α Γ(3α) B(α) − λ(1 − α) 2 #  B(α) +··· , + B(α) − λ(1 − α) and simplifying further gives    B(α) α (t − ξ)α−1 1−α # " B(α) − λ(1 − α) 2  2α α (t − ξ) αλ αλ (t − ξ) 1 + + +··· Γ(α) B(α) − λ(1 − α) Γ(2α) B(α) − λ(1 − α) Γ(3α)      α αλ B(α) (t − ξ)α−1 Eα,α tα . = B(α) − λ(1 − α) 1−α B(α) − λ(1 − α)

4

2

Main Result

2.1

Initial value problem

Here, we consider the following problem: Find a solution u(t) ∈ H 1 (0, T ) that satisfies the following equation ABC

α 0 Dt u(t)

− λu(t) = f (t),

0 ≤ t ≤ T,

(6)

and the initial condition u(0) = u0 ,

(7)

where λ, u0 ∈ R. The solution of this initial value problem is formulated in the following theorem: B(α) , f (t) ∈ C(0, T ) and f (0) = −λu0 , then the solution of the initial 1−α value problem (6) − (7) is given by   1−α αλ B(α)u0 α Eα t + f (t) u(t) = B(α) − λ(1 − α) Z B(α) − λ(1 − α)  B(α) − λ(1 − α)  (8) t αB(α) αλ α α−1 (t − ξ) dξ. + f (ξ)(t − ξ) Eα,α B(α) − λ(1 − α) (B(α) − λ(1 − α))2 0

Theorem 2.1. If λ 6=

Proof. Using the definition of Atangana-Baleanu fractional derivative, we have   Z −α B(α) t ′ α u (s)Eα (t − s) ds − λu(t) = f (t), 1−α 0 1−α which on integrating by parts leads to      Z B(α) −α B(α) t d α Eα u(s)ds = − λ u(t) − (t − s) 1−α 1 − α 0 ds  1 − α −α α B(α)u0 Eα t . f (t) + 1−α 1−α For λ 6=

B(α) , one can write the above equation as a Volterra integral equation of the second 1−α

kind u(t) −

Z

0

where

and

t

u(s)K(t, s)ds = fb(t),

  B(α) d −α α K(t, s) = Eα (t − s) , B(α) − λ(1 − α) ds 1−α   1−α −α B(α)u 0 α Eα t + f (t). fb(t) = B(α) − λ(1 − α) 1−α B(α) − λ(1 − α) 5

(9)

To solve equation (9), we use successive approximation method starting with u0 (t) = fb(t). Then,

Z

u1 (t) = fb(t) +

u2 (t) = fb(t) +

Z0

= fb(t) +

and similarly we obtain u2 (t) Z

t

u0 (s) K(t, s)ds t

0

t

u1 (s) K(t, s)ds  Z0 t  Z s b b b = f (t) + f (s) + f (ξ)K(s, ξ)dξ K(t, s) ds 0 Z0 t Z t Z t = fb(t) + fb(s)K(t, s)ds + fb(ξ)dξ K(s, ξ) K(t, s) ds 0

Set K2 (t, ξ) =

Z

fb(s) K(t, s)ds,

t

0

ξ

K(t, s) K(s, ξ)ds, so that ξ

u2 (t) = fb(t) + = fb(t) +

Z

t

Z0 t 0

fb(s)K(t, s)ds +

Z

t 0

fb(ξ)K2 (t, ξ)dξ

fb(ξ) [K1 (t, ξ) + K2 (t, ξ)] dξ.

Continuing the same process, the nth term will have the following form un (t) = fb(t) +

where

K1 (t, ξ) = K(t, ξ),

Ki (t, ξ) =

Z

t

fb(ξ)

0

Z

n X

Ki (t, ξ)dξ,

i=1

t

K(t, s)Ki−1 (s, ξ)ds,

i = 2, 3, · · · ,

ξ

which can be derived using mathematical induction. To obtain the general expression for the kernel Ki (t, ξ), we substitute for K(t, s) and start with Z t K2 (t, ξ) = K(t, s)K1 (s, ξ)ds ξ 2 Z t        d −α −α d B(α) α α Eα Eα ds (t − s) (s − ξ) = B(α) − λ(1 − α) ds 1−α dξ 1−α ξ 2  2 Z t    −α −α B(α) α−1 α (t − s) Eα,α (t − s) = B(α) − λ(1 − α) 1−α ξ 1 − α  −α α−1 α (s − ξ) Eα,α (s − ξ) ds, 1−α whereupon using Theorem 5. in [13], K2 (t, ξ) reduces to K2 (t, ξ) =



B(α) B(α) − λ(1 − α)

2 

α 1−α

6

2

(t − ξ)

2α−1

2 Eα,2α



 −α α (t − ξ) . 1−α

Repeating the same procedure for K3 (t, ξ), we have Z t K3 (t, ξ) = K(t, s)K2 (s, ξ)ds ξ 3  2 Z t     α −α d B(α) α Eα (t − s) = B(α) − λ(1 − α) 1−α 1−α  ξ ds  −α 2α−1 2 α (s − ξ) Eα,2α (s − ξ) ds 1 − α  3  3 Z t   B(α) α −α α−1 α = (t − s) Eα,α (t − s) B(α) − λ(1 − α) 1−α 1 − α ξ  −α 2α−1 2 α (s − ξ) Eα,2α (s − ξ) ds, 1−α and end with K3 (t, ξ) =



B(α) B(α) − λ(1 − α)

3 

α 1−α

3

(t − ξ)

3α−1

3 Eα,3α



 −α α (t − ξ) . 1−α

Consequently, the general expression for the kernel is given by  i  i   B(α) α −α α iα−1 i Ki (t, ξ) = (t − ξ) , i = 1, 2, 3, · · · . (t − ξ) Eα,iα B(α) − λ(1 − α) 1−α 1−α As n → ∞, the approximations of un (t) converges to the solution u(t) u(t) = fb(t) +

Z

t 0

fb(ξ)

∞ X

Ki (t, ξ)dξ.

i=1

According to Lemma 1.2, we get the following  Z t    α αλ B(α) α α−1 b b (t − ξ) dξ, f (ξ)(t−ξ) Eα,α u(t) = f (t)+ B(α) − λ(1 − α) 1−α B(α) − λ(1 − α) 0 simplifying the above integral using formula (1.107) in [12] and properties of Mittag-Leffler

function, we obtain the desired solution given by (8). An alternative way of finding the solution of the initial value problem (6) − (7) is using Laplace transform method. So, by applying Laplace transform to both sides of equation (6), we have B(α) sα U(s) − sα−1 u(0) − λU(s) = F (s), α 1−α sα + 1−α where U(s) = L{u(t)}(s) and L{ ABC 0 Dtα u(t)}(s) =

B(α) sα U(s) − sα−1 u(0) . α 1−α sα + 1−α

Simplifying and solving for U(s), we get

U(s) =

B(α)sα−1 u0 (1 − α)sα + α + F (s), sα (B(α) − λ(1 − α)) − λα sα (B(α) − λ(1 − α)) − λα 7

which can be rewritten as B(α)sα−1 u0   λα α (B(α) − λ(1 − α)) s − B(α) − λ(1 − α) (1 − α)sα + α  F (s).  + λα (B(α) − λ(1 − α)) sα − B(α) − λ(1 − α)

U(s) =

Since the Laplace transform of Mittag-Leffler function is given by L{tβ−1 Eα,β (λtα )}(s) =

sα−β , sα − λ

then, applying Laplace inverse gives   B(α)u0 αλ α u(t) = Eα,1 t B(α) − λ(1 − α) B(α)    − λ(1 − α) d αλ 1−α α Eα,1 t ∗ f (t) + B(α) − λ(1 − α) dt B(α) − λ(1  − α)    αλ (1 − α)f (t) α α−1 α t Eα,α + + t ∗ f (t) . B(α) − λ(1 − α) B(α) − λ(1 − α) B(α) − λ(1 − α) Consequently,   (1 − α) αλ B(α)u0 α Eα,1 t + f (t) u(t) = B(α) − λ(1 − α)  B(α) − B(α)  λ(1 − α)  − λ(1− α) αλ αB(α) α−1 t E tα ∗ f (t) , + α,α B(α) − λ(1 − α))2 B(α) − λ(1 − α) which is the same as the solution obtained by successive iterations. Remark 2.2. For the case λ = 0 and u(0) = 0, we get Z t α 1−α f (t) + f (ξ)(t − ξ)α−1 dξ, u(t) = B(α) B(α)Γ(α) 0 which coincides with the result obtained in [6].

2.2

Boundary value problem

Now, we consider a direct problem of determining u(x, t) in a rectangular domain Ω = {(x, t) : 0 < x < 1, 0 < t < T } , such that u ∈ C 2 (0, 1)×H 1 (0, T ) and satisfies the following initial-boundary value problem: ABC

α 0 Dt u(x, t)

− uxx (x, t) = f (x, t),

u(0, t) = 0,

u(1, t) = 0,

u(x, 0) = 0,

8

(x, t) ∈ Ω

(10)

0 ≤ t ≤ T,

(11)

0 ≤ x ≤ 1,

(12)

where f (x, t) is a given function. We begin by using separation of variables method to solve the homogeneous equation corresponding to equation (10) aloong with the boundary conditions (11). Thus, we obtain the following spectral problem:   X ′′ + λX = 0,  X(0) = 0, X(1) = 0.

(13)

which is self adjoint and has the following eigenvalues λk = (kπ)2 ,

k = 1, 2, 3, · · · .

The corresponding eigenfunctions are Xk = sin(kπx) k = 1, 2, 3, · · · .

(14)

Since the system of eigenfunctions (14) forms an orthogonal basis in L2 (0, 1), we can then write the solution u(x, t) and the given function f (x, t) in the form of series expansions as follows: ∞ X

u(x, t) =

uk (t) sin(kπx),

(15)

fk (t) sin(kπx),

(16)

k=1

f (x, t) =

∞ X k=1

where uk (t) is the unknown to be found and fk (t) is given by fk (t) = 2

R1 0

f (x, t) sin(kπx)dx.

Substituting (15) and (16) into (10) and (12), we obtain the following fractional differential equation ABC

α 0 Dt uk (t)

+ k 2 π 2 uk (t) = fk (t),

(17)

along with the following condition uk (0) = 0.

(18)

Whereupon using Theorem 2.1, the solution is given by 1−α α B(α) uk (t) = fk (t) + 2 2 (B(α) + k 2 π 2 (1 − α))2 Z tB(α) + k π (1 − α)  −αk 2 π 2 (t − ξ)α dξ, fk (ξ)(t − ξ)α−1 Eα,α 2 π 2 (1 − α) B(α) + k 0 with fk (0) = 0, which is acheived by assuming f (x, 0) = 0. Thus, the solution u(x, t) can now be written as ∞  X

α B(α) 1−α fk (t) + 2 2 B(α) + k π (1 − α) (B(α) + k 2 π 2 (1 − α))2 k=1    t −αk 2 π 2 α α−1 (t − ξ) dξ sin(kπx). fk (ξ)(t − ξ) Eα,α B(α) + k 2 π 2 (1 − α) 0

u(x, t) = Z

9

In order to complete the proof of existence, we need to show the uniform convergence of the series representations of u(x, t), ux (x, t), uxx (x, t),

ABC

α 0 Dt u(x, t).

Since Mittag-Leffler functions is bounded, then the uniform convergence of the series representation of u(x, t) is ensured by assuming f (x, ·) ∈ C(0, T ). Now, the series representation of uxx (x, t) is given by ∞  X

k 2 π 2 (1 − α) k 2 π 2 α B(α) uxx (x, t) = − f (t) + k B(α) + k 2 π 2 (1 − α) (B(α) + k 2 π 2 (1 − α))2    Z t k=1 −αk 2 π 2 α α−1 (t − ξ) dξ sin(kπx) fk (ξ)(t − ξ) Eα,α B(α) + k 2 π 2 (1 − α) 0 ∞ ∞ X X B(α) sin(kπx) = fk (t) sin(kπx) + B(α) + k 2 π 2 (1 − α) k=1 k=1   Z t −αk 2 π 2 ′ α fk (ξ)Eα,1 (t − ξ) dξ. B(α) + k 2 π 2 (1 − α) 0 Assuming ft (x, t) is integrable, it is clear that the second term of the above series converges uniformly. For convergence of the first term, we assume f (0, t) = f (1, t) = 0 and use integration by parts to get X X X ∞ ∞ 1 ∞ 1 fk (t) sin(kπx) = f1k (t) sin(kπx) ≤ |f1k (t)|, kπ kπ k=1

k=1

where

f1k (t) = 2

Z

k=1

1

fx (x, t) cos(kπx) dx.

0

1 2 (a + b2 ) and the Bessel’s inequality, we then have the following 2 ∞   ∞ X X 1 1 2 fk (t) sin(kπx) ≤ + |f1k | 2 k2 π2 k=1 k=1 ∞ X 1 1 ≤ + kfx (x, t)k2L2 (0,1) . 2 2 2k π 2 k=1

Using the inequality ab ≤ estimate

Therefore, the expression of uxx (x, t) is uniformly convergent. Finally, the uniform convergence of

ABC

α 0 Dt u(x, t),

follows from equation (10).

The uniqueness of solution can be shown using the completeness properties of the system {sin(kπx)}. The main result for the direct problem can be summarized in the following theorem: Theorem 2.3. Assume f (x, t) ∈ C[0, 1] × C[0, T ] such that f (x, 0) = 0, f (0, t) = f (1, t) = 0, ft (x, t) ∈ L[0, T ] and fx (x, t) ∈ L2 [0, 1], then the problem (10) - (12) has a unique solution 10

u(x, t) given by ∞  X

α B(α) 1−α fk (t) + 2 2 B(α) + k π (1 − α) (B(α) + k 2 π 2 (1 − α))2 k=1    t −αk 2 π 2 α α−1 (t − ξ) dξ sin(kπx). fk (ξ)(t − ξ) Eα,α B(α) + k 2 π 2 (1 − α) 0

u(x, t) = Z where,

fk (t) = 2

Z

1

f (x, t) sin(kπx) dx.

0

Acknowledgements. The first two authors acknowledge financial support from The Research Council (TRC), Oman. This work is funded by TRC under the research agreement no. ORG/SQU/CBS/13/030.

References [1] B.S.T. Alkahtani, Chuas circuit model with AtanganaBaleanu derivative with fractional order, Chaos, Solitons and Fractals, 89(2016), pp.547-551. [2] B.S.T. Alkahtani and A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos, Solitons and Fractals, 89(2016), pp.539-546. [3] R.T. Algahtani, Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, J. Nonlinear Sci. App., 9(2016), pp.3647-3654. [4] N. Al-Salti, E. Karimov and S. Kerbal, Boundary-value problem for fractional heat equation involving Caputo-Fabrizio derivative, New Trend in Mathematical Sciences, 4 (2016), pp. 79-89. [5] N. Al-Salti, E. Karimov and K. Sadarangani , On a Differential Equation with CaputoFabrizio Fractional Derivative of Order 1 < β ≤ 2 and Application to Mass-Spring-Damper System, Progr. Fract. Differ. Appl., 4 (2016), pp. 257-263. [6] A. Atangana and D. Baleanu, New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model, Thermal Science, 20(2016), pp.763-769. [7] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89(2016), pp.446-454. 11

[8] M. Caputo and and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1, 2 (2015), pp. 73-85. [9] M. Caputo and and M. Fabrizio, Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progr. Fract. Differ. Appl. 2, 1 (2016), pp. 1-11. [10] O. Jefain and J. Algahtani, Comparing the Atangana -Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons and Fractals, 89(2016), pp.552-559. [11] J. Losada and J. Nieto , Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 2 (2015), pp. 87-92. [12] I. Podlubny, Fractional differential equations. Academic Press Inc., San Diego, CA, 1999. [13] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernal, Yokohama Math. J., 19 (1971), pp. 7-15. [14] N.A. Sheikh, F. Ali, M. Saqib, I. Khan, S.A.A. Jan, A.S. Alshomrani, M.S. Alghamdi, Comparison and analysis of the AtanganaBaleanu and CaputoFabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), pp. 789-800.

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