A NEW NUMERICAL TECHNIQUE FOR SOLVING FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
Omer Acan1*, Dumitru Baleanu2,3 1 Department of Mathematics, Faculty of Art and Science, Siirt University, 56100, Siirt, Turkey Department of Mathematics and Computer Sciences, Faculty of Art and Science, Balgat 06530, Ankara, Turkey 3 Institute of Space Science, Magurele-Bucharest, Romania
2
Abstract we propose conformable Adomian decomposition method (CADM) for fractional partial differential equations (FPDEs) at the first time. This method is a new Adomian decomposition method based on conformable derivative to solve FPDEs. At the same time, conformable reduced differential transform method (CRDTM) for FPDEs is briefly given and a numerical comparison is made between this method and the newly introduced CADM. In applied science, CADM can be used as an alternative method to obtain approximate and analytical solutions for FPDEs as CRDTM. In this study, linear and non-linear three problems are solved by these two methods. In these schemes, the obtained solutions take the form of a convergent series with easily computable algorithms. For the applications, the obtained results by these methods are compared to each other and with the exact solutions. When applied to FPDEs, it is seem that CADM approach produces easy, fast and reliable solutions as CFRDTM. Keywords: Numerical Solution, Adomian decomposition method, Differential transform method, Fractional derivative, Conformable derivative, partial differential equations, Fractional diffusion equation, Fractional gas dynamical equation.
1.
Introduction
Linear and non-linear both fractional and non-fractional problems of differential equations play a major role in various fields such as biology, physics, chemistry, mathematics, astronomy, fluids mechanics, optics, applied mathematics, and engineering. It is not always possible to find analytical solutions to these problems [1–10]. Therefore, it is very important to handle these problems appropriately and solve them or develop solutions. Therefore, it is very important to handle these problems appropriately and solve them or develop solutions. Adomian decomposition method (ADM), which is introduced [11–13] in the 1980’s, is one of the important mathematical methods used to solve many problems in real world. Since then, Since then, a number of studies have been conducted on ADM such as linear and nonlinear, homogeneous and nonhomogeneous operator equations, including fractional or nonfractional ordinary differential equations (ODEs), partial differential equations (PDEs), integral equations, integro-differential equations, etc. (see [14– 24] and references therein). Recently, a new derivative called CDO was introduced [25–27] In science, by the help of this new derivative, the behaviors of many problems has been studied and some solutions techniques have been developed [27–37]. This new subject gives academicians an opportunity to study further in many engineering, physical and applied mathematics problems. The aim of this study is to introduce CADM by using CDO and ADM for the first time in the literature. This method can be used to solve many linear and non-linear FPDEs. We will briefly mentioned CRDTM to compare our CADM with it. The problems will be solved both by the CRDTM and the first proposed CADM. The obtained solutions by these methods will be compared. For this, in section 2, we give some basic definitions and important properties of CDO. In section 3, we propose CADM. In sections 4, we introduce CRDTM to compare with our method. In section 5, we give applications of CADM and CRDTM. We give the conclusion in the final section. 2.
On The Conformable Derivative
Definition 2.1. Given a functio f : [0, ∞) → . Then the conformable derivative of f order 𝛼𝛼 is defined by [25,26]
(Tα f )( x) = lim ε→ 0
f ( x + εx1−α ) − f ( x) ε
for all x > 0 , α ∈ (0,1] .
*
Corresponding author. E-mail addresses:
[email protected] (O. Acan),
[email protected] (D. Baleanu)
Lemma 2.1. [25,26] Let α ∈ (0,1] and f , g be α -differentiable at a point x > 0 . Then
(i) Taaa (af + bg= ) a (T f ) + b(T g ) for a, b ∈ , p (ii) Tα ( x ) = px p −α , for all p ∈ , (iii) Tα ( f ( x)) = 0 , for all constant functions f ( x) = λ , (iv) T= f (Tα g ) + g (Tα f ) , α ( fg ) (v) Tα ( f / g ) =
g (Tα f ) − f (Tα g ) , g2
(vi) If f ( x) is differentiable, then Tα ( f ( x)) = x1−α
d f ( x) . dx
Definition 2.2. Given a function f :[a, ∞) → . Then the conformable derivative of f order 𝛼𝛼 is defined by [25,26]:
(Taa f )( x) = lim ε→ 0
f ( x + ε( x − a )1−a ) − f ( x) ε
for all x > 0 , α ∈ (0,1] . Definition 2.3. Let f be an n times differentiable at x. Then the conformable derivative of f order α is defined by [25,26]:
(Tα f )( x) = lim
f (
α −1)
( x + εx (
α −α )
α −1 ) − f ( ) ( x)
ε for all x > 0 , α ∈ (n, n + 1] , α is the smallest integer greater than or equal to α . ε→ 0
Lemma 2.2. [25,26] Let f be an n times differentiable at x. Then
Tα ( f ( x)) = x
α −α
α f ( ) ( x)
for all x > 0 , α ∈ (n, n + 1] . 3.
Conformable Adomian Decomposition Method
We will briefly introduce CADM for non-linear FPDEs in this section. For the purpose of illustration of the methodology to the proposed methods, we write the non-linear FPDEs in the standard operator form (3.1) Lα ( u ( x, t ) ) + R ( u ( x, t ) ) + N ( u ( x, t ) ) = g ( x, t ) where Lα = αT is a linear operator with conformable derivative of order α ( n < α ≤ n + 1 ), R is the other part of the
linear operator, N is a non-linear operator and g ( x, t ) is a non-homogeneous term. If the linear operator in eq. (3.1) is applied to Lemma 2.2, the following equation is obtained:
t Applying L−α1 =
t ξ1
∫ ∫ 0 0
ξn−1
∫ n
α −α
1 ξn
α −α
∂
α
∂t
α
u ( x, t ) + R ( u ( x, t ) ) N ( u ( x, t ) ) = g ( x, t )
(.) d ξn d ξn −1 d ξ1 , (n < α ≤ n + 1)
(3.2)
the inverse of operator, to both sides of (3.2), it is
obtained as
L−α1 Lα ( u ( x, t ) ) = L−α1 g ( x, t ) − L−α1 R ( u ( x, t ) ) − L−α1 N ( u ( x, t ) ) .
(3.3)
The general solution of the given equation is decomposed into the sum ∞
u ( x, t ) = ∑ u n ( x, t ) .
(3.4
n=0
The nonlinear term N (u ) can be decomposed into the infinite series of polynomial obtained by ∞
N (u ) = ∑ An , (u0 , u1 , , un ) , n=0
(3.5)
where An is the so-called Adomian polynomials. These polynomials can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian [12,17]. u and N ( u ) , respectively, is obtained as
= u
∞
∑λ u i
i
i =0
∞ ∞ , N ( u ) =N ∑ λ i ui =∑ λ i Ai = i 0= i 0
(3.6)
where λ is the convenience parameter. From (3.6), Adomian polynomials An are obtained as
= n ! An
dn d λn
∞ n N ∑ λ un . λ=0 n=0
These polynomials can be calculated easily with the following Maple code: restart: ord:=m: #Order (Enter a number for m.) NF:=N(u(x,t)) #Nonlinear Function u[t]:=sum(u[b]*t^b,b=0..ord): NF[t]:=subs(u(x,t)=u[t],NF): s:=expand(NF[t],t): dt:=unapply(s,t): for i from 0 to ord do A[i]:=((D@@i)(dt)(0)/i!): print(AA[i],A[i]): #Adomian Polinoms od: Substituting (3.4) and (3.5) into (3.3), it is obtained ∞
∑u
n 0 =
n
∞ ∞ = θ + L−α1 g − L−α1 R ∑ un − L−α1 ∑ An . = n 0= n 0
(3.7)
where θ =u ( x, 0 ) is initial condition. From (3.7), the iterates are defined by the following recursive way:
u0 = θ + L−α1 g , − L−α1 R u0 − L−α1 A0 , u1 =
(3.8)
− L−α1 R un − L−α1 An , n ≥ 0 . un +1 = Therefore, the approximation solution of (3.1) is obtained by m
um ( x, t ) = ∑ un ( x, t ) . n=0
Hence the exact solution of (3.1) given as
u ( x, t ) = lim um ( x, t ). m →∞
4. Conformable Reduced Differential Transform Method In this section, it is given basic definitions and properties of CFRDTM for fractional PDEs. The lowercase u ( x, t ) represent the original function while the uppercase U kα ( x ) stand for the conformable fractional reduced differential transformed (CRDT) function. Definition 4.1. [36] Assume u ( x, t ) is analytic and differentiated continuously with respect to time t and space x in the its domain. CFRDT of u ( x, t ) is defined as U kα ( x ) =
where some 0 < α ≤ 1 , α (k )
tTα
(
)
1 (k ) tTα u t = t0
αkk!
is a parameter describing the order of conformable fractional derivative,
u = ( tTα tTα tTα ) u ( x, t ) and the t dimensional spectrum function U kα ( x ) is the CFRDT function. (((( k times
Definition 4.2. [36] Let U k ( x ) be the CFRDT of u ( x, t ) . Inverse CFRDT of U kα ( x ) is defined as α
∞ αk 0 k = k 0= k 0
u= ( x, t )
∞
t ) ∑U α ( x )( t − = ∑α
1 (k ) αk ( t − t0 ) . tTα u k t = t0 k!
CFRDT of initial conditions for integer order derivatives are defined as
1 ∂α k α k u ( x, t ) α U k ( x ) (α k )! ∂t = t =t0 0
if α k ∈ + n for k 0,1, 2,..., − 1 = α if α k ∉ +
where n is the order of conformable fractional PDE. α
By consideration of U 0 ( x ) = f ( x ) as transformation of initial condition u ( x, 0) = f ( x ). α
A straightforward iterative calculations gives the U k ( x ) values for k = 1, 2, 3, ..., n . Then the inverse transformation of
{
α
}
the U k ( x )
n k =0
gives the approximation solution as: u n ( x, t ) =
n
∑U ( x) t α
kα
k
,
k =0
where n is order of approximation solution. The exact solution is given by: u ( x, t ) = lim u n ( x, t ) n →∞
The fundamental operations of CFRDT that can be deduced from Definition 4.1 and Definition 4.2 are listed Table 4.1.
Table 4.1. Basic operations of CFRDTM [36]. Original function
Transformed function
u ( x, t )
U kα ( x ) =
= u ( x, t ) av ( x, t ) ± bw ( x, t )
= U kaaa ( x ) aVk ( x ) ± bWk ( x )
u ( x, t ) = v ( x, t ) w ( x, t )
U kα ( x ) = ∑ Vsα ( x ) Wkα− s ( x )
(
)
1 (k ) tTα u t = t0
αkk!
k
s =0
u ( x, t ) = tTα v ( x, t )
u(= x, t ) x m ( t − t0 )
U kα = ( x ) α ( k + 1)Vkα+1 ( x ) n
n α U= x mδ k − k ( x) α
5.
Applications
To illustrate the effectiveness of the given CADM and CRDTM, three examples are considered in this section. All the results are calculated by software MAPLE. Example 4.1: Firstly, Consider the linear time and space fractional diffusion equation: ∂α ∂ 2β = u ( x, t ) u ( x, t ) t > 0, x ∈ R, 0 < α, β ≤ 1 α ∂t ∂x 2β subject to the initial condition xβ u ( x, 0) = sin . β
(5.1)
(5.2)
Exact solution of the problem) in conformable sense is
xβ u ( x, t ) = sin β
α
− tα e .
Solution by CADM:
∂α be a linear operator, then the operator form of (5.1) is as follows ∂t α ∂ 2 β u ( x, t ) (5.3) = Tα u ( x, t ) t > 0, x ∈ R, 0 < α, β ≤ 1 ∂x 2β By the help of Lemma 2.2, eq. (4.4) can be written as ∂u ( x, t ) ∂ 2β u ( x, t ) (5.4) t1−β t > 0, x ∈ R, 0 < α, β ≤ 1 . = ∂t ∂x 2β t 1 If L−α1 ∫ 1−α (.) d ξ ,which is the inverse of Lα , is applied to both sides of eq. (5.4), we get = 0 ξ Solve this problem by using CADM. Let L= T= α α
∂ 2β u= ( x, t ) u ( x, 0) − L−α1 2β u ( x, t ) . ∂x According to (3.8) and the initial condition (5.2), we can write xβ u0 = sin , β ∂2 u1 = − L−α1 2 u0 ∂x
(5.5)
∂2 un +1 = − L−α1 2 un , n ≥ 0 . x ∂ From (5.5), we conclude the terms of decomposition series as: xβ xβ t α xβ t 2 α u0 = sin , u1 = sin 2 , − sin , u2 = β β α β 2α
xβ t 3α x β t nα , , un = (−1) n sin , u3 = − sin 3 n β 3!α β n !α Thus, by using (5.6) the approximate solution of (5.1) obtained by CADM is m m x β t nα . u!m ( x,= t ) ∑ un ( x,= t ) ∑ (−1) n sin n = n 0= n 0 β n !α
(5.6)
(5.7)
From (5.7) we obtain α
xβ − t = u ( x, t ) lim = um ( x, t ) sin e α . m →∞ β This analytical approximate solution (5.8) is the exact solution.
(5.8)
Solution by CRDTM: Now solve this problem by using CRDTM. By taking the conformable fractional reduced differential transform (CRDT) of (4.1), it can be obtained that
∂2β ∂x
α ( k + 1)U kα+1 ( x ) = U kα ( x ) 2β where the
(5.9)
t -dimensional spectrum function U kα ( x ) is the conformable fractional reduced differential transform
function. From the initial condition (5.2) we write
xβ U 0α ( x ) = sin β
(5.10)
Substituting (5.10) into (5.9), we obtain the following U kα ( x ) values successively
xβ 1 xβ 1 x β ( −1) α sin , , sin , U1α ( x ) = − sin , U 2α ( x ) = U x = ( ) n 2 3 β α β 2!α β n !α n
Then, the inverse transformation of the set of values {U kα ( x )}
= u!n ( x, t )
n
k =0
gives the following approximation solution
x β ( −1) kα sin t . ∑ k 0= k 0 β k !α
= U kα ( x ) t kα ∑
= k
n
k
n
(5.11)
From (5.11) we obtain α
xβ − t u ( x, t ) lim um ( x, t ) sin e α . = = m →∞ β This analytical approximate solution (5.12) is the exact solution. Remark 4.1. If take α = β = 1 in the problem, then Example 4.1 is reduced to standard diffusion equation
(5.12)
∂ ∂2 = u ( x, t ) u ( x, t ) t > 0, x ∈ R ∂t ∂x 2 with initial condition
u ( x, 0) = sin( x) and our analytical approximate solution (5.8) and (5.12) imply u ( x, t ) = sin( x)e − t and this result is the exact solution of the standard problem in the literature. The Aproximate solutions obtained with both CADM and CRDTM give us the exact solution. Example 4.2: Now let us consider the nonlinear time- and space fractional gas dynamics equation: 1 ∂β 2 ∂α u ( x, t ) + u ( x, t ) − u ( x, t ) (1 − u ( x, t ) ) =0 , 0< α, β ≤ 1 α 2 ∂xβ ∂t subject to initial condition
u ( x, 0) = e
−
xβ β
.
The exact solutions of (5.13) in conformable sense is
u ( x, t ) = e
t α xβ − α β
.
(5.13)
(5.14)
Solution by CADM: Solve the problem by using CADM. Let L= T= α α
∂α be a linear operator, then the operator form of (5.13) is as ∂t α
follows
1 ∂β 2 u ( x, t ) + u ( x, t ) (1 − u ( x, t ) ) , 0< α, β ≤ 1. 2 ∂xβ By the help of Lemma 2.2, eq. (5.15) can be written as ∂u ( x, t ) ∂β t1−α = u ( x, t ) − u ( x, t ) β u ( x, t ) − u 2 ( x, t ), 0< α, β ≤ 1 . ∂t ∂x t 1 If L−α1 ∫ 1−α (.) d ξ ,which is the inverse of Lα , is applied to both sides of eq. (5.16), we get = ξ 0 Tα u ( x, t ) = −
(5.15)
(5.16)
∂β u ( x, t ) = u ( x, 0) + L−α1 ( u ( x, t ) ) − L−α1 u ( x, t ) β u ( x, t ) + u 2 ( x, t ) . ∂x According to (3.8) and initial condition (5.14), we can write the following recursive relations: u0 = e
−
xβ β
= u1 L−α1 ( u0 ) − L−α1 ( A0 )
(5.17)
un +1 = L ( un ) − L−α1 ( An ) , n ≥ 0. −1 α
where An ’s are
Adomian polynomials. By using the
Maple code above, for the nonlinear term
∂ u ( x, t ) + u 2 ( x, t ) , the Adomian polynomials can be obtain as: ∂x ∂β 2 A= u + u u0 0 0 0 ∂xβ ∂β ∂β 2u0 u1 + u0 β u1 + u1 β u0 A1 = ∂x ∂x β β ∂ ∂ ∂β A2 = u12 + 2u0 u2 + u0 β u2 + u1 β u1 + u2 β u0 ∂x ∂x ∂x β β β ∂ ∂ ∂ ∂β 2u1u2 + 2u0 u3 + u0 β u3 + u1 β u2 + u2 β u1 + u3 β u0 A3 = ∂x ∂x ∂x ∂x From (5.17) and (5.18), we conclude the terms of decomposition series as:
N ( u ( x ) ) u ( x, t ) =
xβ
xβ β
xβ
xβ
xβ
− − − tα t 2α t 3α t nα , u2 e β = , u3 e β , , un e β , = = 2 3 α 2α 3!α n !α n Thus, from (5.19), the approximate solution of (5.13) obtained by CADM is −
β , u1 e u0 e= =
−
u!m ( x, t ) =
m
u n ( x, t ) = ∑
m
∑e
−
xβ β
n 0= n 0 =
(5.18)
t nα . n !α n
(5.19)
(5.20)
From (5.20) we obtain xβ − α β
tα
= = u ( x, t ) lim um ( x, t ) e m →∞
.
(5.21)
This analytical approximate solution (5.21) is the exact solution. Solution by CRDTM: Now solve this problem by using CFDTM. By taking the conformable fractional reduced differential transform (CRDTM) of (5.13), it can be obtained that k k ∂β (5.22) α ( k + 1)U kα+1 ( x) = −∑ U kα− r ( x) β U rα ( x) + U kα ( x) − ∑ U kα− r ( x)U rα ( x) ∂x = r 0= r 0 where the
t -dimensional spectrum function U kα ( x ) is the conformable fractional reduced differential transform
function. From the initial condition (5.14) we write
U 0α ( x ) = e
−
xβ
β
(5.23)
Substituting (5.23) into (5.22), we obtain the following U kα ( x ) values successively xβ
xβ
− − β 1 = U1α ( x ) e= , U 2α ( x ) e β
α
Then, the inverse transformation of the set of values {U kα ( x )}
= u!n ( x, t )
n
xβ
− 1 1 = , , U nα ( x ) e β , 2 n !α 3 2!α n k =0
n
= U ( x)t ∑ ∑e α
kα
gives the following approximation solution −
xβ
β
k = k 0= k 0
1 kα t . k !α k
(5.24)
From (5.24) we obtain tα
= = u ( x, t ) lim un ( x, t ) e
α
−
n →∞
xβ β
.
(5.25)
This analytical approximate solution (5.25) is the exact solution. Remark 4.2. If take α = β = 1 in the problem, then Example 4.2 is reduced to standard gas dynamics equation
1 ∂ 2 ∂ u ( x, t ) + u ( x, t ) − u ( x, t ) (1 − u ( x, t ) ) =0 2 ∂x ∂t with initial condition
u ( x, 0) = e − x our analytical approximate solution (5.21) and (5.25) imply u ( x, t ) = e t − x and this result is the exact solution of the standard problem in the literature. The approximate solutions obtained with both CADM and CRDTM give us the existed exact solution. Example 4.3: Now let us consider the nonlinear time- and space fractional partial differential equation ∂α ∂α (5.26) u ( x, t ) + (1 + u ( x, t ) ) α u ( x, t )=0 , 0< α ≤ 1 α ∂t ∂x subject to initial condition xα − α . (5.27) u ( x, 0) = 2α The exact solutions of (5.26) in conformable sense is xα − t α − α . u ( x, t ) = α t − 2α Solution by CADM: ∂α Solve the problem by using CADM. Let L= be a linear operator, then the operator form of (5.26) is as T= α α ∂t α follows ∂α (5.28) Tα u ( x, t ) = − (1 + u ( x, t ) ) α u ( x, t ) , 0< α ≤ 1. ∂x By the help of Lemma 2.2, eq. (5.28) can be written as ∂u ( x, t ) ∂α ∂α (5.29) = − α u ( x, t ) − u ( x, t ) α u ( x, t ), 0< α ≤ 1 . t1−α ∂t ∂x ∂x t 1 If L−α1 ∫ 1−α (.) d ξ ,which is the inverse of Lα , is applied to both sides of eq. (5.29), we get = ξ 0
∂α ∂α u ( x, t ) = u ( x, 0) − L−α1 α u ( x, t ) − L−α1 u ( x, t ) α u ( x, t ) . ∂ ∂ x x According to (3.8) and initial condition (5.27), we can write the following recursive relations:
xα − α 2α −1 = u1 Lα ( u0 ) − L−α1 ( A0 ) u0 =
(5.30)
un +1 = L ( un ) − L−α1 ( An ) , n ≥ 0. −1 α
where An ’s are
Adomian polynomials. By using the
Maple code above, for the nonlinear term
∂ u ( x, t ) + u 2 ( x, t ) , the Adomian polynomials can be obtain as: ∂x ∂α A0 = u0 α u0 ∂x α ∂ ∂α = A1 u0 α u1 + u1 α u0 ∂x ∂x ∂α ∂α ∂α A2 = u0 α u2 + u1 α u1 + u2 α u0 ∂x ∂x ∂x ∂α ∂α ∂α ∂α A3 = u0 α u3 + u1 α u2 + u2 α u1 + u3 α u0 ∂x ∂x ∂x ∂x From (5.30) and (5.31), we conclude the terms of decomposition series as: xα − α xα + α α xα + α 2α − = u0 = t u t , , u1 = , 2 2 3 2α ( 2α ) ( 2α )
= N ( u ( x ) ) u ( x, t )
α x α + α 3α n x + α nα − u3 = t , , un = t , ( −1) n +1 4 ( 2α ) ( 2α )
Thus,from (5.32) the approximate solution of (5.26) obtained by CADM is α m xα − α m m x + α mα um ( x, t= ) ∑ un ( x, t= ) + ∑ ( −1) t . m +1 2α = n 0= n 1 ( 2α )
(5.31)
(5.32)
(5.33)
Solution by CRDTM: Now solve this problem by using CFDTM. By taking the conformable fractional reduced differential transform (CRDTM) of (5.26), it can be obtained that k
α ( k + 1)U k +1 ( x) = −U k ( x) − ∑ U k − r ( x) r =0
where the
∂ U r ( x) ∂x
(5.34)
t -dimensional spectrum function U kα ( x ) is the conformable fractional reduced differential transform
function. From the initial condition (5.27) we write
U 0α ( x ) =
xα − α 2α
(5.35)
Substituting (5.35) into (5.34), we obtain the following U kα ( x ) values successively α xα + α xα + α n x +α U1α ( x ) = − , U 2α ( x ) = , , U nα ( x ) = , ( −1) n +1 2 3 ( 2α ) ( 2α ) ( 2α )
Then, the inverse transformation of the set of values {U kα ( x )}
n k =0
α m k k = k 0= n 1
gives the following approximation solution
α x −α m x +α (5.36) + ∑ ( −1) t mα . m +1 2α ( 2α ) Compare the seventh iteration CADM and CFDTM solutions with the exact solution on the graph for some α values.
um ( x, t= )
m
α ∑U α ( x ) t =
Figure 6.1. Comparison of the seventh iteration approximate solutions of CADM (CRDTM) with the exact solutions for eq. (5.26).
Figure 6.2. Comparison of the seventh iteration approximate solutions of CADM (CRDTM) with the exact solutions for eq. (5.26) 6.
Conclusion
The fundamental goal of this article is to construct the approximate solutions of FPDEs. The goal has been achieved by using CADM for the first time and it is compared with CRDTM. CADM and CRDTM are applied to different linear and non-linear conformable time and space FPDEs. And also the approximate analytical solutions obtained by CADM and CRDTM are compared to each other and with the exact solutions. CADM and CRDTM offer solutions with easily computable components as convergent series. Approximate solutions obtained by CADM are exactly same as the solutions obtained by CRDTM for time and space FPDEs. The CADM gives quantitatively reliable results as CRDTM, and also it requires less computational work than existing other methods. As a result, in recent years, FDEs emerging as models in fields such as mathematics, physics, chemistry, biology and engineering makes it necessary to investigate the methods of solutions and we hope that this study is an improvement in this direction. References [1] [2] [3] [4] [5] [6] [7]
K. Oldham, J. Spanier, B. Ross, The Fractional Calculus: Theory and applications of Differentiation and Integration to Arbitary Order, Academic Press, 1974. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, 1998. X.-J. Yang, Advanced local fractional calculus and its applications, World Science, New York, NY, USA, 2012. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: models and numerical methods, World Scientific, 2016. X.-J. Yang, Local Fractional Functional Analysis & Its Applications, Asian Academic Publisher Limited, Hong Kong, 2011.
[8] [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
[33] [34] [35]
D. Baleanu, Z.B. Guvenc, J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer Netherlands, Dordrecht, 2010. H. Gray, N. Zhang, On a new definition of the fractional difference, Math. Comput. 50 (1988) 513–529. F. Atici, P. Eloe, Initial value problems in discrete fractional calculus, Proc. Am. Math. Soc. 133 (2009) 981– 989. G. Adomian, Stochastic Systems, Academic Press, New York, NY, USA, 1983. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501–544. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Springer Science & Business Media, 2013. S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488–494. Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182 (2006) 1048–1055. N. Bildik, A. Konuralp, The Use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for Solving Different Types of Nonlinear Partial Differential Equations, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) 65–70. J. Duan, R. Rach, D. Baleanu, A.-M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc. 3 (2012) 73–99. A. Wazwaz, R. Rach, L. Bougoffa, Solving the Lane–Emden–Fowler type equations of higher orders by the Adomian decomposition method, Comput. Model. Eng. Sci. 100 (2014) 507–529. J.-S. Duan, R. Rach, A.-M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method, J. Math. Chem. 53 (2015) 1054–1067. M. Turkyilmazoglu, Determination of the Correct Range of Physical Parameters in the Approximate Analytical Solutions of Nonlinear Equations Using the Adomian Decomposition Method, Mediterr. J. Math. 13 (2016) 4019–4037. B. Boutarfa, A. Akgül, M. Inc, New approach for the Fornberg-Whitham type equations, J. Comput. Appl. Math. 312 (2017) 13–26. A. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden-Fowler equation, Appl. Math. Comput. 161 (2005) 543–560. A. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166 (2005) 652–663. A.-M. Wazwaz, R. Rach, L. Bougoffa, Dual solutions for nonlinear boundary value problems by the Adomian decomposition method, Int. J. Numer. Methods Heat Fluid Flow. 26 (2016) 2393–2409. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65–70. T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57–66. D.R. Anderson, D.J. Ulness, Newly Defined Conformable Derivatives, Adv. Dyn. Syst. Appl. 10 (2015) 109– 137. A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math. 13 (2015) 889– 898. Y. Cenesiz, A. Kurt, The solutions of time and space conformable fractional heat equations with conformable Fourier transform, Acta Univ. Sapientiae, Math. 7 (2015) 130–140. A. Kurt, Y. Çenesiz, O. Tasbozan, On the Solution of Burgers’ Equation with the New Fractional Derivative, Open Phys. 13 (2015) 355–360. W.S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math. 290 (2015) 150–158. M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, A. Biswas, et al., Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Opt. - Int. J. Light Electron Opt. 127 (2016) 10659–10669. H. Karayer, D. Demirhan, F. Büyükkılıç, Conformable Fractional Nikiforov-Uvarov Method, Commun. Theor. Phys. 66 (2016) 12–18. A. Gökdogan, E. Ünal, E. Celik, Existence and uniqueness theorems for sequential linear conformable fractional differential equations, Miskolc Math. Notes. 17 (2016) 267–279. E. Ünal, A. Gökdoğan, Solution of conformable fractional ordinary differential equations via differential
[36] [37]
transform method, Opt. - Int. J. Light Electron Opt. 128 (2017) 264–273. O. Acan, O. Firat, Y. Keskin, G. Oturanc, Solution of Conformable Fractional Partial Differential Equations by Reduced Differential Transform Method, Selcuk J. Appl. Math. (2016) (acceped). O. Acan, O. Firat, Y. Keskin, G. Oturanc, Conformable Variational Iteration Method, New Trends Math. Sci. 5 (2017) 172–178.
