Solutions of partial differential equations using the

Eur. Phys. J. Plus (2018) 133: 215 DOI 10.1140/epjp/i2018-12051-9

THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel Mehmet Yavuz1 , Necati Ozdemir2 , and Haci Mehmet Baskonus3,a 1 2 3

Necmettin Erbakan University, Faculty of Science, Konya, Turkey Balikesir University, Faculty of Art and Science, Balikesir, Turkey Munzur University, Faculty of Engineering, Tunceli, Turkey Received: 7 February 2018 / Revised: 22 March 2018 Published online: 6 June 2018 c Societ`  a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2018 Abstract. In this paper, time-fractional partial differential equations (FPDEs) involving singular and nonsingular kernel are considered. We have obtained the approximate analytical solution for linear and nonlinear FPDEs using the Laplace perturbation method (LPM) defined with the Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators. The AB fractional derivative is defined with the MittagLeffler function and has all the properties of a classical fractional derivative. In addition, the AB operator is crucial when utilizing the Laplace transform (LT) to get solutions of some illustrative problems with initial condition. We show that the mentioned method is a rather effective and powerful technique for solving FPDEs. Besides, we show the solution graphs for different values of fractional order α, distance term x and time value t. The classical integer-order features are fully recovered if α is equal to 1.

1 Introduction Researchers are increasingly working on modelling fractional accounts that draw attention in everyday life. Many studies on new fractional derivative operators have been done in the last 30 years. Fractional calculations (FC) and approximate analytical solution methods are used widely when solving real-life problems, especially in mathematical systems of problems. Among them, the fractional models have been used to calculate the performance of the electrical RLC circuit [1], to solve PDEs arising in fluid mechanics [2], to approximate for two-sided space-fractional PDEs [3], to apply the Shannon wavelets [4], to price the options of fractional order [5,6] and to model the population growth [7]. In addition, these fractional models have been used to demonstrate the applications of approximate-analytical solution methods; for instance, Adomian decomposition method [8,9], homotopy perturbation method [10,11], inverse Laplace homotopy technique [12], homotopy analysis method [13,14], He’s variational/(local) variational iteration method [15, 16], generalized differential transform method [17], finite difference method [18,19], Podlubny’s matrix approach [20], reproducing kernel method [21], multivariate Pad´e approximation method [22,23], etc. These mentioned methods are also used to get the solution of optimal control problem [24], constrained optimization problem [25], portfolio optimization problem [26], diffusion-wave problem [27], heat problem on radial symmetric plate [28], etc. Moreover, some researchers [29–33] have developed various illustrative solution methods for nonlinear PDEs and they solved partial differential equations. Certain novel definitions of fractional derivative operators were specified in order to model real-life problems by Gr¨ unwald-Letnikov [34], Liouville-Caputo [35], Caputo-Fabrizio [36] and Atangana-Baleanu [37]. Especially LiouvilleCaputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators have a number of applications, for example, Morales-Delgado [38] presented an analysis on Liouville-Caputo and Caputo-Fabrizio fractional derivatives. Sheikh et al. [39] used the AB and CF fractional derivatives to analyse a generalized Casson fluid model. Atangana and Alkahtani [40] modelled the groundwater flowing within a confine aquifer by using CF derivative. Koca and Atangana [41] solved the Cattaneo-Hristov model with CF and AB fractional derivative operators. Dokuyucu et al. [42] examined a model for cancer treatment by using the CF operator. In another study, Algahtani [43] compared AB and CF operators with the Allen Cahn model. a

e-mail: [email protected] (corresponding author)

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The main purpose of this study is to demonstrate the solution of the one-dimensional and non-homogeneous fractional Cauchy problem with initial condition by using the LC and AB fractional derivative operators. We have also aimed at determining the differences between the methods by comparing the numerical and analytical results.

2 Some preliminaries Definition 1. The Liouville-Caputo (LC) time-fractional derivative is given by [44] LC α 0 Dt f (t)

=

1 Γ (n − α)



t

(t − τ )n−α−1 f (n) (τ )dτ,

t > 0,

(1)

0

where f ∈ L1 (a, b) and n < α ≤ n + 1. Definition 2. The Laplace transform (LT) of the LC fractional derivative L

 α 0 Dt f (t) (s) =

LC

1 sn−α

LC α 0 Dt f (t)

has the form [40]



 sn L{f (x, t)}(s) − sn−1 f (x, 0) − · · · − f (n−1) (x, 0) .

Definition 3. The Atangana-Baleanu (AB) time-fractional derivative in the Caputo sense is given as [37]    α(t − τ )α B(α) t  ABC α dτ, f (τ )Eα − b Dt f (t) = 1−α b 1−α

(2)

(3)

where B(α) is a normalization function such that B(0) = B(1) = 1, f ∈ H 1 (a, b), b > a, α ∈ [0, 1]. Definition 4. The LT of the AB fractional derivative L

ABC 0

ABC α 0 Dt f (t)

is given by [37]

 B(α) sα L{f (t)}(s) − sα−1 f (0) Dtα f (t) (s) = . α 1−α sα + 1−α

(4)

3 Description of the proposed method using the new fractional operators Suppose the following fractional nonlinear partial differential equation [38,45]: ∗ (α+n) u(x, t) 0 Dt

+ γu(x, t) + ηu(x, t) = τ (x, t),

with initial conditions

∂mu (x, 0) = fm (x), ∂tm

(x, t) ∈ [0, 1] × [0, T ],

(5)

m = 0, 1, . . . , k − 1

(6)

t ≥ 0,

(7)

and the boundary conditions u(0, t) = g0 (t),

k − 1 < α + n ≤ k,

u(1, t) = g1 (t),

where fm , m = 0, 1, . . . , k − 1, τ , g0 , g1 , η and γ are known functions and T > 0 is a real number. In eq. (5), the (α+n) represents the LC or the AB fractional derivative operators. We define the method of solution for notation ∗0 Dt solving problems (5)–(7). Using the Laplace transform of the Liouville-Caputo (2) and Atangana-Baleanu (4) fractional derivative we define the L{u(x, t)}(s) = Ω(x, s) for eq. (5). Then we can write the homotopies for the LC operator

τ˜(x, s) 1  1 . (8) Ω(x, s) = − α L{γu(x, t) + ηu(x, t)} + n sn−1 u0 (x) + sn−2 u1 (x) + · · · + un−1 (x) + s s sα In addition, we can develop the homotopies for the AB fractional operator



1  α−1 (1 − α)sα + α (1 − α)sα + α L{γu(x, t) + ηu(x, t)} + α s τ˜(x, s), u0 (x) + Ω(x, s) = − sα s sα

(9)

where Ω(x, s) = L{u(x, t)} and τ˜(x, s) = L{τ (x, t)}. The LTs of the boundary conditions are given as Ω(0, s) = L{g0 (t)},

Ω(1, s) = L{g1 (t)},

s ≥ 0.

(10)

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Then we apply the LPM in order to get the solution of eqs. (8) and (9): Ω(x, s) =

∞

¯ m Ωm (x, s), h

m = 0, 1, 2, . . . ,

(11)

m=0

where Ωm (x, s) are known functions. The nonlinear term in eq. (5) can be obtained as ηu(x, t) =

∞

¯ m Nm (x, t). h

(12)

m=0

The polynomials Nm (x, t) are given, in [46], as  ∞ 

1 ∂m i η Nm (u0 , u1 , . . . , um ) = ¯h ui m! ∂¯hm i=0

,

m = 0, 1, 2, . . .

(13)

h ¯ =0

Substituting eqs. (11) and (12) into eq. (8), we make the solution series for the LC operator   ∞  ∞ ∞

1 m m m h Ωm (x, s) = −¯ ¯ h αL γ ¯h Ωm (x, t) + ¯h Nm (x, t) s m=0 m=0 m=0 +

τ˜(x, s) 1  n−1 s u0 (x) + sn−2 u1 (x) + · · · + un−1 (x) + n s sα

and, substituting eqs. (11) and (12) into eq. (9), we have the solution series for the AB operator: 

 ∞ ∞ ∞

(1 − α)sα + α m m m L γ h Ωm (x, s) = −¯ ¯ h ¯h Ωm (x, t) + ¯h Nm (x, t) sα m=0 m=0 m=0

(1 − α)sα + α 1  α−1 τ˜(x, s). u0 (x) + + α s s sα

(14)

(15)

By comparing the coefficients of powers of h ¯ , we have the homotopies of the LC fractional derivative as follows: τ˜(x, s) 1  n−1 s u0 (x) + sn−2 u1 (x) + · · · + un−1 (x) + , n s

sα 1 h1 : Ω1 (x, s) = − α L{γu0 (x, t) + ηu0 (x, t)}, ¯ s

1 h2 : Ω2 (x, s) = − α L{γu1 (x, t) + ηu1 (x, t)}, ¯ s .. .

1 hn+1 : Ωn+1 (x, s) = − α L{γun (x, t) + ηun (x, t)}. ¯ s ¯ 0 : Ω0 (x, s) = h

In addition, the homotopies with respect to the AB fractional operator is given by

1  α−1 (1 − α)sα + α 0 τ˜(x, s), h : Ω0 (x, s) = α s ¯ u0 (x) + s sα

(1 − α)sα + α L{γu0 (x, t) + ηu0 (x, t)}, h1 : Ω1 (x, s) = − ¯ sα

(1 − α)sα + α L{γu1 (x, t) + ηu1 (x, t)}, h2 : Ω2 (x, s) = − ¯ sα .. .

(1 − α)sα + α hn+1 : Ωn+1 (x, s) = − ¯ L{γun (x, t) + ηun (x, t)}. sα

(16)

(17)

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When h ¯ → 1, we see that eqs. (16) and (17) give the approximate solution for problems (14), (15) and the solution is Hn (x, s) =

n

Ωj (x, s).

(18)

j=0

If we take the inverse Laplace transform of eq. (18), we get the approximate solution of eq. (5), uapprox (x, t) ∼ = un (x, t) = L−1 {Hn (x, s)}.

(19)

Moreover, we are going to determine the error rates of the LPM solution by applying this method to the linear nonhomogeneous fractional Cauchy problem. If we represent un (x, t) = L−1 {Hn (x, s)}, which is the n-th partial sum of the infinite series of approximate solution of eq. (19), the amount of absolute error AE is calculated as AE = |un (x, t) − uexact (x, t)|.

(20)

4 Applications We present some illustrative examples as the application of the suggested method with the LC and AB fractional operators.

4.1 Solution of the time-fractional non-homogeneous Cauchy problem The time-fractional one-dimensional non-homogeneous Cauchy problem is solved using the proposed method described with the Liouville-Caputo and Atangana-Baleanu fractional derivative operators in this section. In addition to this, we show the effectiveness and appropriateness of the LT of the LC and AB fractional derivatives by applying them to the fractional non-homogeneous Cauchy problem. Moreover, we analyze the stability and convergence of the mentioned method. Now we consider the following non-homogeneous time-fractional Cauchy problem [47]: ∗ α 0 Dt u

+ ux = x,

t > 0,

x ∈ R,

0 < α ≤ 1,

(21)

with the initial condition u(x, 0) = ex .

(22)

Firstly, we solve eq. (21) with the initial condition (22) by using the LPM defined with the Liouville-Caputo fractional derivative operator. We can create the homotopies as follows: 1 x ex x 1 u(x, 0) + α = + α+1 , s s s s s   x e ∂Ω 1 1 (x, s) 0 = − α+1 − 2α+1 , ¯ 1 : Ω1 (x, s) = α − h s ∂x s s   x e ∂Ω1 (x, s) 1 = 2α+1 , h2 : Ω2 (x, s) = α − ¯ s ∂x s ¯ 0 : Ω0 (x, s) = h

.. .

  ∂Ωn−1 (x, s) 1 ex ¯ : Ωn (x, s) = α − h = (−1)n nα+1 . s ∂x s n

(23)

By totalising the iteration term up to the n-th order, we have Hn (x, s) =

n

j=0

=

Ωj (x, s) =

x ex ex 1 ex ex ex + α+1 − α+1 − 2α+1 + 2α+1 − 3α+1 + · · · + (−1)n nα+1 s s s s s s s

n

x ex 1 (−1)m + α+1 − 2α+1 + ex . s s s smα+1 m=1

(24)

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Fig. 1. The solution functions of (26) via the LC fractional operator for α = 1.0 (left) and α = 0.45 (right).

Fig. 2. Numerical simulations of (26) in the LC fractional derivative sense for various values of α and t.

Getting the inverse LT of eq. (24), we have the approximate solution of eq. (21) with the initial condition (22) as u(x, t) ≈ un (x, t) = L−1 {Hn (x, s)} t2α xtα − − ex =e + Γ (α + 1) Γ (2α + 1) x



t2α tnα tα − + ··· + Γ (α + 1) Γ (2α + 1) Γ (nα + 1)

.

(25)

When n → ∞, we have u(x, t) = ex +

  tα t2α xtα − − ex 1 − e− Γ (α+1) Γ (α + 1) Γ (2α + 1)

(26)

and, for the special case α = 1, the exact solution of the Cauchy equation is presented by

uα=1 (x, t) =

lim

n→∞,α→1

x−t

Hn (x, t) = e

t +t x− 2

.

The numerical computations of eq. (26) for α = 1 and α = 0.45 with respect to the Liouville-Caputo fractional derivative operator are presented in fig. 1. The numerical simulations of eq. (26) for x = 0.75 and x = 0.3 considering the Liouville-Caputo fractional derivative operator for various values of α and t are represented in fig. 2. On the other hand, we consider the AB fractional operator. Now we solve problems (21)–(22) by using the LPM defined with the AB derivative. First of all, if we consider the LT of the non-homogeneous term, we get τ˜(x, s) =

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L{τ (x, t)} = L{x} = xs . Now we get the homotopies with respect to the AB derivative as



(1 − α)sα + α  x  ex x (1 − α)sα + α 1 = + , h0 : Ω0 (x, s) = u(x, 0) + ¯ s sα s s s sα 



x

(1 − α)sα + α (1 − α)sα + α ∂Ω0 e (1 − α)sα + α = − − − , h1 : Ω1 (x, s) = ¯ sα ∂x sα s sα+1 



∂Ω1 ((1 − α)sα + α)2 (1 − α)sα + α x = e h2 : Ω2 (x, s) = ¯ − , sα ∂x s2α+1 .. . 

 α n ∂Ωn−1 (1 − α)sα + α n x n ((1 − α)s + α) h : Ωn (x, s) = ¯ = e (−1) . sα ∂x snα+1

(27)

By considering the n-th–order approximate solution, we have Hn (x, s) =

n

Ωj (x, s)

j=0



ex x (1 − α)sα + α ((1 − α)sα + α)2 = + − s s sα s2α+1   α α α n ((1 − α)s + α)2 ((1 − α)sα + α)3 x (1 − α)s + α n+1 ((1 − α)s + α) . −e − + + · · · + (−1) sα+1 s2α+1 s3α+1 snα+1

(28)

Under the inverse LT of eq. (28), we acquire the approximate solution of (21), (22) as follows: u(x, t) ≈ un (x, t) = L−1 {Hn (x, s)}

(αtα )2 (αtα )2 αtα xαtα (αtα )n − + ex − + + · · · + (−1)n Γ (α + 1) Γ (2α + 1) Γ (α + 1) Γ (2α + 1) Γ (nα + 1)

α 2 α 3 2αtα αtα x x x (αt ) x (αt ) − 2e + 3e − 4e + ··· + (α − 1) e − x + Γ (α + 1) Γ (α + 1) Γ (2α + 1) Γ (3α + 1)

αtα (αtα )2 (αtα )3 + 6ex − 10ex + ··· + (α − 1)2 ex − 1 − 3ex Γ (α + 1) Γ (2α + 1) Γ (3α + 1)

αtα (αtα )2 (αtα )3 + 10ex − 20ex + ··· + (α − 1)3 ex − 4ex Γ (α + 1) Γ (2α + 1) Γ (3α + 1)

αtα (αtα )2 + 15ex + ··· + ··· . (29) + (α − 1)4 −5ex Γ (α + 1) Γ (2α + 1)

= ex +

If we take n → ∞ and, for the special case α = 1, the exact solution of the mentioned problem is presented by uα=1 (x, t) = limn→∞,α→1 Hn (x, t) = ex−t + t(x − 2t ), which is the same solution as that with the LC derivative sense. The numerical computations of Cauchy problem in eq. (21) and its solution functions equation (29) for different values of α, x and t are represented in figs. 3 and 4. 4.2 Solution of nonlinear time-fractional advection equation In this subsection, we take into account the following nonlinear time-fractional advection equation [48]: ∗ α 0 Dt u

+ uux = x + xt2 ,

t > 0,

0 < α ≤ 1,

(30)

subject to the initial condition u(x, 0) = 0.

(31)

In order to solve this problem with LPM in the Liouville-Caputo sense, we apply the Laplace transform to eq. (30) with its condition (31). Then we get    1  1 ∂u 1 2 . (32) Ω(x, s) = [u(x, 0)] + α L x + xt − α L u s s s ∂x

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Fig. 3. The solution function of (29) in the AB fractional derivative operator for α = 0.25 (left) and α = 0.75 (right) for various values of x and t.

Fig. 4. Numerical results of eq. (29) for x = 0.25 (left) and x = 0.50 (right).

Considering the inverse LT of eq. (32) we obtain

−1

u(x, t) = u(x, 0) + L



x sα+1

+

2x sα+3



−1



−L

  1 ∂u . L u sα ∂x

(33)

Now, if we apply the LPM, we have ∞

m

h Ωm (x, s) = x ¯

m=0

2tα+2 tα + Γ (α + 1) Γ (α + 3)



− ¯hL

−1

1 L sα



∞

 m

¯h Nm (u)

.

(34)

m=0

In the last equation, Nm (u) are the polynomials that show the nonlinear terms defined in (13). These polynomials are evaluated in the following way: N0 (u) = u0 (u0 )x ,

 N1 (u) = [(u0 + h ¯ u1 )(u0 + h ¯ u1 )x ]h¯ h¯ =0 = u0 (u1 )x + u1 (u0 )x ,    1  u0 + h N2 (u) = ¯ u1 + h ¯ 2 u2 u0 + h ¯ u1 + h ¯ 2 u2 x h¯ h¯ h¯ =0 = u0 (u2 )x + u1 (u1 )x + u2 (u0 )x , 2 .. .

(35)

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Fig. 5. Numerical computation of eq. (37) for α = 0.8 (left) and α = 1 (right).

α

α+2

t According to the above equation system, we can obtain N0 (u) = x( Γ (α+1) + Γ2t(α+3) )2 . The other parts of Nm (u) can be calculated in this manner: Comparing the coefficients of h ¯ in eq. (34), we get

2tα+2 tα + , h : u0 (x, t) = x ¯ Γ (α + 1) Γ (α + 3)   1 L {N0 (u)} , h1 : u1 (x, t) = −L−1 ¯ sα   4Γ (2α + 3)t3α+2 Γ (2α + 1)t3α 4Γ (2α + 5)t3α+4 + + , = −x Γ (α + 3)2 Γ (3α + 5) Γ (α + 1)Γ (α + 3)Γ (3α + 3) Γ (α + 1)2 Γ (3α + 1)   1 L{N1 (u)} , h2 : u2 (x, t) = −L−1 ¯ sα  8Γ (2α + 3)Γ (4α + 3)t5α+2 8Γ (2α + 5)Γ (4α + 5)t5α+4 + =x Γ (α + 1)Γ (α + 3)2 Γ (3α + 5)Γ (5α + 5) Γ (α + 1)2 Γ (α + 3)Γ (3α + 3)Γ (5α + 3) 16Γ (2α + 5)Γ (4α + 7)t5α+6 2Γ (2α + 1)Γ (4α + 1)t5α + + 3 Γ (α + 1) Γ (3α + 1)Γ (5α + 1) Γ (α + 3)3 Γ (3α + 5)Γ (5α + 7)  4Γ (2α + 3)Γ (4α + 3)t5α+2 16Γ (2α + 3)Γ (4α + 5)t5α+4 + , + Γ (α + 1)Γ (α + 3)2 Γ (3α + 3)Γ (5α + 5) Γ (α + 1)2 Γ (α + 3)Γ (3α + 1)Γ (5α + 3) .. .

0

(36)

In this way, the other components of the series can be obtained by using the Mathematica package program. Therefore the approximate solution of eq. (30) is given in the following series: u(x, t) =

∞

um (x, t).

(37)

m=0

Moreover, the analytical solution of the problem when α → 1 is given as u(x, t) = xt.

(38)

Figure 5 shows the numerical computations obtained LC derivative operator with respect to various values of α. On the other hand, let us consider the AB derivative operator in order to solve the suggested nonlinear equation. Firstly, applying the LT to eqs. (30)–(31), we get 1 Ω(x, s) = [u(x, 0)] + s



(1 − α)sα + α sα





L x + xt

2



−

(1 − α)sα + α sα





∂u L u ∂x

 .

(39)

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Then, taking the inverse LT of the last equation, we get  



 

 (1 − α)sα + α x 2x (1 − α)sα + α ∂u −1 −1 u(x, t) = u(x, 0) + L + 3 −L L u sα s s sα ∂x =x

 



2  αtα 2αtα+2 (1 − α)sα + α ∂u −1 + − (α − 1) t + 1 − L . L u Γ (α + 3) Γ (α + 1) sα ∂x

(40)

Furthermore, if we apply the proposed LPM, we have  



∞ ∞

2  αtα (1 − α)sα + α 2αtα+2 m m −1 + − (α − 1) t + 1 − ¯hL L h Ωm (x, s) = x ¯ ¯h Nm (u) . Γ (α + 3) Γ (α + 1) sα m=0 m=0 (41) The polynomials for the nonlinear terms Nm (u) in eq. (41) are calculated as in eq. (35). Therefore, the first component α+2 αtα 2 2 of Nm (u) is given as N0 (u) = x( Γ2αt (α+3) + Γ (α+1) − (α − 1)(t + 1)) . The other parts of Nm (u) can be obtained in the same way. Then, comparing the coefficients of ¯h in eq. (41), we get first few parts of the series as

  αtα 2αtα+2 h0 : u0 (x, t) = x ¯ + − (α − 1) t2 + 1 , Γ (α + 3) Γ (α + 1) 

 (1 − α)sα + α L {N h1 : u1 (x, t) = −L−1 ¯ (u)} , 0 sα  = −x 1 − 3α + 3α2 − α3 + 2t2 − 6αt2 + 6α2 t2 − 2α3 t2 + t4 − 3αt4 + 3α2 t4 − α3 t4 2α2 t2α − 2α3 t2α 4α2 t2α+2 − 4α3 t2α+2 24αtα+4 − 48α2 tα+4 + 24α3 tα+4 + + Γ (α + 5) Γ (2α + 1) Γ (2α + 3)     4(α − 1)αtα+2 αtα+2 Γ (α + 5) − (α − 1) 2 + t2 Γ (2α + 5) − Γ (α + 3)Γ (2α + 5) +

+

α2 t2α (αtα Γ (2α + 1) − (α − 1)Γ (3α + 1)) Γ (α + 1)2 Γ (3α + 1)

4α2 t2α+4 (αtα Γ (2α + 5) − (α − 1)Γ (3α + 5)) Γ (α + 3)2 Γ (3α + 5)   αtα 4α2 t2α+4 Γ (2α + 3)2 − 2(α − 1)αt2+α Γ (α + 3)2 Γ (3α + 3) + Γ (2α + 3)Γ (3α + 3)Γ (α + 1)Γ (α + 3)       αtα (α − 1) 4αt2+α − (α − 1) 3 + 2t2 Γ (α + 3) , − Γ (α + 1)Γ (α + 3) +

.. .

(42)

Taken the above into consideration, the other parts of the series can be simply obtained. Finally, the approximate solution and the analytical solution when α → 1 for eq. (30) are given in the following form, respectively: u(x, t) =

∞

um (x, t),

u(x, t) = xt,

(43)

m=0

which is the same result as that obtained by using the Liouville-Caputo operator. Figure 6 shows the numerical computations obtained via the AB derivative operator with respect to various values of α.

5 Error analysis and stability of the method It is very important to examine the behaviour of the error and to make the error stability of the method used for numerical solutions. Therefore, in this section, we investigate the convergence and the stability of the method.

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Fig. 6. Numerical computation of eq. (43) for α = 0.7 (left) and α = 1 (right).

Fig. 7. Absolute error values AE for some values of x and t.

If the series (18) converges such that Ω(x, s) is obtained by eq. (11), this series must be the solution of eq. (5). We can say that the mentioned solution technique is stable and convergent. Moreover, the method we suggested in this work generates a meaningful convergence region of the solution by generator functions (14) and (15). The numerical solutions we obtained are identical with the exact solution. In order to understand the convergent and stabilities of the LPM defined by using two different fractional operators in sect. 3, the amount of absolute error AE for some values of distance term x and time variable t were presented. In fig. 7, we show the absolute error rates based on the numerical solution in (29) and exact solution in (26) for the special case (x, t) = [0, 1] × [0, 1]. In this study, all computations were made by using the Maple and Mathematica package programs.

6 Conclusions In this paper, linear and nonlinear fractional problems have been solved approximate-analytically. The Laplace perturbation method defined with the Liouville-Caputo and Atangana-Baleanu fractional derivative operators have been used. The results have demonstrated that these fractional operators are very simple, clear and powerful when using the LT in order to solve the fractional linear/nonlinear problems with the initial conditions. The error analysis and convergence of the method have declared that both of the fractional derivative operators have given very close exact results. In addition, the approximate-analytical solutions have been presented with figures for different values of fractional operator α, distance term x and the time variable t. Finally, the results obtained in this study have shown that the suggested method verified the validity and effectiveness of the LPM in the LC and AB fractional derivative sense.

Eur. Phys. J. Plus (2018) 133: 215

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This research was supported by Balikesir University Scientific Research Projects Unit, BAP:2018/065.

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