SQU Journal for Science, 2018, 23(2),137-146 DOI: http://dx.doi.org/10.24200/squjs.vol23iss2pp137-146 Sultan Qaboos University 137 Initial and Boundary Value Problems for Fractional Differential Equations Involving Atangana-Baleanu Derivative Fatma S. Al-Musalhi 1 , Nasser S. Al-Salti 1,2 *, and Erkinjon Karimov 2,3 1 Department of Mathematics, Sultan Qaboos University, P.O. Box 36, PC 123, Al Khodh, Muscat, Sultanate of Oman; 2 FracDiff Research Group, Sultan Qaboos University, Muscat, Sultanate of Oman; 3 Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent 170, Uzbekistan, *Email: nalslti@squ.edu.om. Abstract: In the present work, an initial value problem involving the Atangana-Baleanu derivative is considered. An explicit solution of the given problem in integral form is obtained by using the Laplace transform. 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 a series expansion using an orthogonal basis obtained by separation of variables. Some examples are also given to illustrate the obtained results. Keywords: Atangana-Baleanu derivative, Initial-Boundary value problem, Fractional differential equation. باليانومشتقة أتنجانا معادالت تفاضلية كسرية ذات شروط ابتدائية وحدودية والتي تحتوي على فاطمة المصلحي، ناصر السلطي وإيركن كريموف في هذه الورقة تم دراسة معادلة تفاضليه تحتوي على مشتقة كسريه ألتنجانا باليانو في ظل وجود شروط ابتدائية تارة وباستخدام الملخص: باستخدام تحويل البالس. ولقد اتضحت أهمية هذه شروط حدودية تارة أخرى، وقد تم أوال إيجاد حل المعادلة في ظل وجود شروط ابتدائية المعادلة في إيجاد حل المعادلة التفاضلية في ظل وجود شروط ابتدائية وحدودية معا، والتي تم التعبير عن حلها في صورة متسلسلة رياضية. وفي األخير تم عرض بعض األمثلة لتوضيح النتائج التي توصلت لها الدراسة. مشتقة أتنجانا باليانو ، مسألة ذات شروط إبتدائية وحدودية، معادالت تفاضلية كسرية. ية:الكلمات المفتاح 1. Introduction and Preliminaries ecently, two newly defined fractional derivatives without a singular kernel have been suggested, namely, the Caputo-Fabrizio fractional derivative [1] and the Atangana-Baleanu fractional derivative [2]. These two new derivatives have been applied to real-life problems in various areas, for example in the fields of thermal science, material sciences, groundwater modelling and mass-spring systems [2 -7], and have been considered in a number of other recent works, see for example [8-17]. The main difference between these two definitions is that the Caputo- Fabrizio derivative is based on an exponential kernel, while the Atangana-Baleanu definition uses a Mittag- leffler function as a non-local kernel. The non-locality of the kernel gives a better description of the memory within structures with different scales. These two new derivatives are defined as follows: R FATMA S. Al-MUSALHI ET AL 138 Σ ∈ Definition 1.1. [2] Let f ∈ H 1 (a, b), b > a, α ∈ [0, 1]. The Caputo-Fabrizio fractional derivative is defined as ( ) ( ) '( ) exp ( ) , 1 1 t CF a t a B D f t f s t s ds              (1) and the Atangana-Baleanu fractional derivative is given by ( ) ( ) '( ) ( ) , 1 1 t ABC a t a B D f t f s E t s ds                (2) where H 1 (a, b) is the usual Sobolev space, i.e., 𝐻1(𝑎, 𝑏) = {𝑓 ∈ 𝐿2(𝑎, 𝑏), 𝑓 ′ ∈ 𝐿2(𝑎, 𝑏) }, ( )B  denotes a normalization function such that B(0) = B(1) = 1, and 0 ( ) , Re( ) 0, , ( 1) k k z E z z k            C is the Mittag-Leffler function of one parameter [18]. For properties related to these derivatives, see [2], [11], [15]. 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 , 0 ( ) , Re( ) 0, Re( ) 0, . ( ) k k z E z z k                C Moreover, the Mittag-Leffler function ( )E t    is bounded (see [18]), i.e. ( ) ,E t M     (3) where M denotes a positive constant. In [2], Atangana and Baleanu considered the time fractional ordinary differential equation 0 ( ) ( ), ABC t D f t u t   and, on using the Laplace transform, they found the following solution 1 0 0 1 ( ) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) t AB t f t I u t u t u s t s ds B B              where they have defined 1 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) t AB t I f t f t f s t s ds B B             to be the fractional integral associated with the fractional derivative (2). In this paper, we consider the following initial value problem (IVP) 0 0 ( ) ( ) ( ), 0, (0) , ABC t D u t u t f t t u u      (4) INTIAL AND BOUNDARY VALUE PROBLEMS OF FDES WITH A-B DERIVAITVE 139 where 0 , .u R The solution of this IVP is obtained here by using the Laplace transform, which can also be obtained by using the successive approximation method as illustrated in the arXiv version of this paper [19]. The use of such an IVP is illustrated by considering a boundary value problem in which the solution is expressed in the form of a series expansion using an orthogonal basis obtained by separation of variables. It is worth mentioning here that the fractional differential equation in (4), with or without initial condition, has also been considered by other authors, see for example [7], [20-22]. However, most of previous studies have either considered special forms of (t)f or considered only one case of  , namely, ( ) 1      B or ( ) 1      B . Moreover, most of the solutions obtained previously fail to satisfy the initial condition, since the existence of a solution requires a necessary condition, namely, 0 (0)  f u , which was first addressed in the arXiv version of this paper [19]. The rest of the present paper is organized as follows. Section 2 is devoted to our main result, which is an explicit solution of the IVP (4) and to solving a boundary value problem utilizing the solution of the IVP (4). We conclude this paper by presenting some examples to illustrate the obtained results. 2. Main Result 2.1 Initial value problem Here, we consider the following problem: Find a solution 1 ( ) (0, )u t H  that satisfies the following equation 0 ( ) ( ) ( ), 0, ABC t D u t u t f t t     (5) and the initial condition 0, (0)u u (6) where 0 , .u R The solution of this initial value problem is formulated in the following theorem: Theorem 2.1.  If 0 ( ) , ( ) (0, ) , (0) , 1 B f t C f u           then the solution of the initial value problem (5) - (6) is given by 0 1 ,2 0 ( ) 1 ( ) ( ) ( ) (1 ) ( ) (1 ) ( ) (1 ) ( ) ( ) ( ) ( ) . ( ( ) (1 )) ( ) (1 ) t B u u t E t f t B B B B f t E t d B B                                                            (7)  If 1 0 ( ) , ( ) [0, ) , (0) 1 B f t AC f u           , then the solution of the given IVP is given by 2 0 (1 ) (1 ) ( ) ( ) ( ), ( ) ( ) C t u t D f t f t B B           (8) where the space 1 [0, )AC  is defined as 𝐴𝐶1 ≔ {     1 0 ( ) 0 , [0, ) t f t f g z dz g L    } and 0 ( ) C t D f t  represents the Caputo fractional derivative of order ( 0 1  ) and is defined as [15] 0 0 1 ( ) ( ) ( ) . (1 ) t C t D f t t y f y dy          FATMA S. Al-MUSALHI ET AL 140 Proof. Applying the Laplace transform to both sides of equation (5) and using (6), we have 1 0 ( )( ) ( ) ( ), 1 1 s U s s uB U s F s s               (9) where  ( ) ( ) ( )U s u t s and   1 0 0 ( )( ) ( ) ( ) . 1 1 ABC t s U s s uB D u t s s              Simplifying and solving for U (s), we get 1 0 ( ) (1 ) ( ) ( ), ( ( ) (1 )) ( ( ) (1 )) B s u s U s F s s B s B                           for ( ) , 1 B      which can be rewritten as 1 0 1 0 2 ( ) (1 ) ( ) ( ) ( ( ) (1 )) ( ( ) (1 )) ( ) (1 ) ( ) (1 ) ( ) (1 ) ( ) ( ( ) (1 )) ( ( ) (1 )) ( ) (1 ) ( ) ( ( ) (1 )) ( ) ( B s u s U s F s B s B s B B B s u F s B B s B B B s B                                                                                            ( ) 1 ) F s        Since the Laplace transform of the Mittag-Leffler function is given by  1 , ( ) ( ) , s t E t s s              applying the Laplace inverse will then give 0 ,1 1 ,2 ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) (1 ) ( ) (1 ) ( ) * ( ) , ( ( ) (1 )) ( ) (1 ) B u u t E t f t B B B B t E t f t B B                                                         which is equivalent to the desired result as given in Equation (7). Note that the condition 0 (0)f u  is needed to ensure that 0 (0)u u . For ( ) , 1 B      equation (9) gives, 2 1 0 (1 ) (1 ) ( ) ( ). ( ) 1 U s s u s F s B                    INTIAL AND BOUNDARY VALUE PROBLEMS OF FDES WITH A-B DERIVAITVE 141 Applying the Laplace inverse, we obtain 2 2 0 (1 ) (1 ) (1 ) (0) 1 ( ) * ( ) ( ). (1 ) ( ) (1 ) ( ) (1 ) ( ) t t f u t t u f t f t B B B                                    Using the condition 0 (0) ,f u  one can obtain the desired solution (8). Remark 2.2. For the case 0  and (0) 0,u  we get 1 0 1 ( ) ( ) ( ) ( ) , ( ) ( ) ( ) t u t f t f t d B B               which coincides with the result obtained in [2]. Note that the obtained results for the IVP (4) are analogous to the results obtained for the case of Caputo-Fabrizio fractional derivative [8]. In the next section, we consider a direct initial-boundary value problem including a partial fractional differential equation with the Atangana-Baleanu derivative. To obtain the solution of the direct problem, we have utilized the solution of the IVP (4). It is also worth mentioning here that the condition 0 (0)f u  makes it impossible to consider inverse source or inverse initial problems with the Atangana-Baleanu fractional derivative. 2.2 Initial-boundary value problem Now, we consider a direct problem of determining ( , )u x t in a domain  ( , ) : 0 1, 0 ,x t x t T      such that 2 1 (., ) (0,1), ( ,.) (0, )u t C u x H T  and satisfies the following initial-boundary value problem: 0 ( , ) ( , ) ( , ), ( , ) ABC t xx D u x t u x t f x t x t     (10) (0, ) 0, (1, ) 0, 0 ,u t u t t T    (11) ( , 0) 0, 0 1,u x x   (12) where ( , )f x t is a given function. We begin by using the separation of variables method to solve the homogeneous equation corresponding to equation (10) along with the boundary conditions (11). Thus, we obtain the following spectral problem: '' 0, (0) 0, (1) 0. X X X X      (13) which is self-adjoint and has the following eigenvalues 2 ( ) , 1, 2, 3, ... . k k k   The corresponding eigenfunctions are sin( ), 1, 2, 3, ... . k X k x k  (14) Since the system of eigenfunctions (14) forms an orthogonal basis in 2 (0,1)L [23], we can then write the solution ( , )u x t and the given function ( , )f x t in the form of series expansions as follows: 1 ( , ) ( ) sin( ), k k u x t u t k x     (15) 1 ( , ) ( ) sin( ), k k f x t f t k x     (16) FATMA S. Al-MUSALHI ET AL 142 where ( ) k u t are the unknown coefficients to be found, and the coefficients ( ) k f t are given by 1 0 ( ) 2 ( , ) sin( ) . k f t f x t k x dx  Substituting (15) and (16) into (10) and (12), we obtain the following fractional differential equation 2 2 0 ( ) ( ) ( ), ABC t k k k D u t k u t f t    along with the following condition (0) 0. k u  Whereupon using Theorem 2.1, the solution is given by 2 2 2 2 2 2 2 1 , 2 2 0 1 ( ) ( ) ( ) ( ) (1 ) ( ( ) (1 )) ( ) ( ) ( ) , ( ) (1 ) k k t k B u t f t B k B k k f t t E t d B k                                       with (0) 0, k f  which is achieved by assuming ( , 0) 0.f x  Thus, the solution ( , )u x t can now be written as 2 2 2 2 2 1 2 2 1 , 2 2 0 1 ( ) ( , ) ( ) ( ) (1 ) ( ( ) (1 )) ( ) ( ) ( ) sin( ). ( ) (1 ) k k t k B u x t f t B k B k k f t E t d k x B k                                               In order to complete the proof of the existence of solution, we need to show the uniform convergence of the series representations of 0 ( , ), ( , ), ( , ), ( , ). ABC x xx t u x t u x t u x t D u x t  Since the Mittag-Leffler function appearing in the expression of u(x, t) is bounded, the uniform convergence of the series representation of ( , )u x t is ensured by assuming ( ,.) (0, ).f x C T Now, the series representation of ( , ) xx u x t is given by 2 2 2 2 2 2 2 2 2 1 2 2 1 , 2 2 0 2 2 1 1 0 (1 ) ( ) ( , ) ( ) ( ) (1 ) ( ( ) (1 )) ( ) ( ) ( ) sin( ) ( ) (1 ) ( ) ( ) sin( ) sin( ) ( ) ( ) (1 ) xx k k t k t k k k k k k B u x t f t B k B k k f t E t d k x B k B f t k x k x f E B k                                                                     2 2 ,1 2 2 ( ) . ( ) (1 ) k t d B k               Assuming ( , ) t f x t is integrable, it is clear that the second term of the above series converges uniformly. For the convergence of the first term, we assume (0, ) (1, ) 0f t f t  and use integration by parts to get 1 1 1 1 1 1 1 ( ) sin( ) ( ) sin( ) ( ) , k k k k k kk k f t k x f t k x f t               INTIAL AND BOUNDARY VALUE PROBLEMS OF FDES WITH A-B DERIVAITVE 143 where 1 1 0 ( ) 2 ( , ) cos( ) . k x f t f x t k x dx  Using the inequality 2 21 ( 2 )aa b b  and the Bessel’s inequality for trigonometric series, we then have the following estimate 2 2 2 2 2 12 2 (0,1) 1 1 1 1 1 1 2 2 2 1 ( ) sin( ) ( , ) . k k x L k k k k f t k x f f x t k                      Therefore, the expression of ( , ) xx u x t is uniformly convergent. The uniform convergence of 0 ( , ), ABC t D u x t  can be proved in a similar way. The uniqueness of the solution can be obtained using the completeness properties of the system sin( ) .k x The main result for the direct problem can be summarized in the following theorem: Theorem 2.3. Assume (., ) (0,1), ( ,.) (0, )f t C f x C T  such that ( , 0) 0, (0, ) (1, ) 0,f x f t f t   1 ( , ) (0, ), t f x t L T and 2 ( , ) (0,1), x f x t L then the problem (10) - (12) has a unique solution ( , )u x t given by 2 2 2 2 2 1 2 2 1 , 2 2 0 1 ( ) ( , ) ( ) ( ) (1 ) ( ( ) (1 )) ( ) ( ) ( ) sin( ). ( ) (1 ) k k t k B u x t f t B k B k k f t E t d k x B k                                               where, 1 0 ( ) 2 ( , ) sin( ) . k f t f x t k x dx  3. Illustrative Examples In this section, we give some examples to illustrate our results presented in Theorems (2.1) and (2.3), choosing ( ) 1.B   Example 1. Here, we present solutions of three different initial value problems by considering different choices of ( ),f t  and 0 u .  The constant function ( ) 1u t  solves the following IVP: 0 ( ) ( ) 1, (0) 1. A BC t D u t u t u       For 1 ( ) , ( 1)f t t      , the solution of IVP (5)-(6) with 1  and 0 0u  is given by 1 , 1 ( ) ( ) ( );u t t t E t                    The solution of the following IVP: 0 ( ) ( ) ( ), (0) 1. A BC t D u t u t E t u            FATMA S. Al-MUSALHI ET AL 144 is given by:  (1 ) ( ) ( ) . (2 ) (2 ) 2 u t E t E t                             Example 2. Consider the boundary value problem (10) − (12) with ( , ) sin( ).f x t t x Using Theorem 2.3, the solution ( , )u x t is given by 2 1 , 22 2 2 2 1 ( , ) sin( ). 1 (1 ) (1 (1 )) 1 (1 ) u x t t t E t x                                 The obtained solution is illustrated in Figures (1 − 2) for T = 1. Figure 1 demonstrates the solution profile at different times for a fixed value of the order α of the fractional derivative. The effect of the nonhomogeneous term, particularly its x-dependence, is clearly seen in the solution profile. Moreover, the solution increases with time and reaches its maximum when t = T = 1. The effect of the order α of the fractional derivative is shown in Figure 2 at a fixed time. It shows that as α increases, the solution decreases and reaches its minimum when α = 1, which represents the classical derivative case. Figure 1. Graphs of u(x, t) at different times for 0.5.  Figure 2. Graphs of u(x, t) at t = 0.5 for different values of . INTIAL AND BOUNDARY VALUE PROBLEMS OF FDES WITH A-B DERIVAITVE 145 4. Conclusion In this paper, we have considered an initial value problem (IVP) for a nonhomogeneous linear fractional differential equation with the Atangana-Baleanu derivative. An explicit solution of the IVP in integral form has been obtained using the Laplace transform. The solution can be also obtained using the successive approximation method as shown in the arXiv version of this paper. A condition relating the initial data and the initial value of the nonhomogeneous term, namely, 0 (0)f u  , has been found to be necessary for the existence of a solution of the IVP. On the other hand, this condition makes it impossible to consider inverse source or inverse initial problems with the Atangana-Baleanu fractional derivative. We have also considered an initial-boundary value problem for a nonhomogeneous linear partial fractional differential equation with the Atangana-Baleanu derivative and a given time and space dependent nonhomogeneous term. A regular classical solution of this direct problem was obtained in the form of a series expansion using an orthogonal basis, which is a set of eigenfunctions of a self-adjoint spectral problem obtained by considering the corresponding homogeneous equation and using the method of separation of variables. 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