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

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.


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], [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 and the Atangana-Baleanu fractional derivative is given by where B(α) denotes a normalization function such that B(0) = B(1) = 1 and is the Mittag-leffler function of one parameter [12].
Moreover, Mittag-Leffler function E α (λt α ) is bounded(see [13]), i.e, where M denotes a positive constant. In [6], Atangana and Baleanu considered the time fractional ordinary differential equation and on using Laplace transform they found the following solution to be the fractional integral associated with the fractional derivative (2). In this paper, we consider the following initial value problem (IVP) where λ, u 0 ∈ 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).

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: Proof. We begin by expanding the following series: Using the definition of Mittag-Leffler functions, it can be written as follows: and expanding the series representations of Mittag-Leffler functions, gives Now, combining like terms, we have and simplifying further gives 2 Main Result

Initial value problem
Here, we consider the following problem: Find a solution u(t) ∈ H 1 (0, T ) that satisfies the following equation and the initial condition where λ, u 0 ∈ R. The solution of this initial value problem is formulated in the following theorem: , then the solution of the initial value problem (6) − (7) is given by Proof. Using the definition of Atangana-Baleanu fractional derivative, we have which on integrating by parts leads to where and To solve equation (9), we use successive approximation method starting with u 0 (t) = f (t). Then, and similarly we obtain u 2 (t) Continuing the same process, the n th term will have the following form which can be derived using mathematical induction.
To obtain the general expression for the kernel K i (t, ξ), we substitute for K(t, s) and start with whereupon using Theorem 5. in [13], K 2 (t, ξ) reduces to Repeating the same procedure for K 3 (t, ξ), we have and end with Consequently, the general expression for the kernel is given by As n → ∞, the approximations of u n (t) converges to the solution u(t) According to Lemma 1.2, we get the following 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 where U(s) = L{u(t)}(s) and Simplifying and solving for U(s), we get which can be rewritten as Since the Laplace transform of Mittag-Leffler function is given by then, applying Laplace inverse gives Consequently, 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 which coincides with the result obtained in [6].

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: u(x, 0) = 0, 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, which is self adjoint and has the following eigenvalues The corresponding eigenfunctions are Since the system of eigenfunctions (14) forms an orthogonal basis in L 2 (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: where u k (t) is the unknown to be found and f k (t) is given by f k (t) = 2 1 0 f (x, t) sin(kπx)dx. Substituting (15) and (16) into (10) and (12), we obtain the following fractional differential along with the following condition Whereupon using Theorem 2.1, the solution is given by with f k (0) = 0, which is acheived by assuming f (x, 0) = 0. Thus, the solution u(x, t) can now be written as In order to complete the proof of existence, we need to show the uniform convergence of the series representations of 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 u xx (x, t) is given by Assuming f t (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 Using the inequality ab ≤ 1 2 (a 2 + b 2 ) and the Bessel's inequality, we then have the following Therefore, the expression of u xx (x, t) is uniformly convergent. Finally, the uniform convergence of ABC 0 D α t u(x, t), follows from equation (10). The uniqueness of solution can be shown using the completeness properties of the system {sin(kπx)}.