Next Article in Journal
Generalized Memory: Fractional Calculus Approach
Previous Article in Journal
Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs

1
E3MI, Faculty of Sciences and Techniques Errachidia, University Moulay Ismail, Meknes 50050, Morocco
2
Department of Mathematics, Faculty of Basic Education, Public Authority for Applied Education and Training, Al-Ardhiya 92400, Kuwait
*
Author to whom correspondence should be addressed.
Fractal Fract. 2018, 2(3), 22; https://doi.org/10.3390/fractalfract2030022
Submission received: 28 March 2018 / Revised: 30 July 2018 / Accepted: 31 July 2018 / Published: 7 September 2018

Abstract

:
In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives.

1. Introduction

In recent years, fractional calculus is considered to be a fascinating field of research, due to the wide applications of fractional integrals and derivatives in mathematical modeling of systems and processes in many fields of engineering and science [1,2,3,4,5,6,7,8,9,10,11,12,13]. Generally speaking, most of fractional differential equations are not solvable toward exact solutions. Therefore, numerous analytical and numerical methods were successfully utilized to treat this sort of problems. Among the aforementioned methods, we can refer to fractional Homotopy Perturbation [14], fractional Adomian decomposition [15], Yang-Laplace transform [16], Variational Iteration and function decomposition in local fractional sense [15,17].
Homotopy Perturbation coupled with Sumudu transform technique is an integration between two powerful methods: Sumudu transform, which was proposed by Watugala [18] in 1993, and Homotopy perturbation, which was introduced by He [19,20]. This combined method was applied to solve fractional nonlinear problems, arising in the field of nonlinear sciences such as engineering and mathematical physics. Several researchers used this simple tool to obtain solutions of nonlinear differential equations. For example, both Singh and Kumar [21] solved the nonlinear fractional gas dynamics equation. On the other hand, Sharma and Singh [22] derived the solutions for the fractional nonlinear partial differential equations. As far as Ait touchent and Belgacem [23] are concerned, they presented the solutions for the system of nonlinear fractional PDEs. For Eltayeb A. Youssif [24], he found the exact solution for nonlinear Schrodinger equation.
In recent years, not only was the local fractional analysis theory given much more consideration and concern, but also it was well employed for describing non-differentiable problems appearing in different sciences. For example, local fractional Laplace equation [15], diffusion equations on Cantor sets [25], Korteweg-ed Vries equation with local fraction operator [26] and fractal heat conduction equation [27] as well as fractal wave equation [28].
The aim of this paper is to implement the combination of local fractional Sumudu transform and homotopy perturbation in order to get analytical solutions of some local fractional problems in mathematical physics, for example dissipative and damped wave equations. Furthermore, we prove the convergence analysis and we show the advantages of this method for constructing solutions of local-fractional PDEs.
This article is divided into seven sections: In Section 2, we introduce some preliminaries about local fractional calculus. Section 3, is dedicated to present the local fractional Sumudu transform. In Section 4, the main steps of homotopy perturbation coupled with local fractional Sumudu transform are mentioned. Analysis on convergence with some examples are given in Section 5 and Section 6, are followed by the conclusion in Section 7.

2. Local Fractional Calculus Preliminaries

The following section sheds light on some necessary definitions of local fractional calculus exploited in the present article.
Definition 1.
(see [29]) We say that the function g is continuous in local fractional sense at x 0 , if
g ( x ) g ( x 0 ) < ϵ r , 0 < r 1 ,
with x x 0 < δ , for ϵ > 0 and ϵ R .
Definition 2.
(see [29]) The derivative of g in local fractional sense at x 0 is presented as
D x r g ( x 0 ) = d r g ( x 0 ) d x r = lim x x 0 Δ r ( g ( x ) g ( x 0 ) ) ( x x 0 ) r ,
where Δ r ( g ( x ) g ( x 0 ) ) Γ ( r + 1 ) Δ ( g ( x ) g ( x 0 ) ) .
The formula of local fractional derivative of high order is given by
g ( k r ) ( x ) = D x r D x r D x r k t i m e s g ( x ) .
The expression of local fractional partial differential operator of order r ( 0 < r < 1 ) has the form:
r t r g ( x 0 , t ) = Δ r ( g ( x 0 , t ) g ( x 0 , t 0 ) ) ( t t 0 ) r ,
where Δ r ( g ( x 0 , t ) g ( x 0 , t 0 ) ) Γ ( r + 1 ) Δ ( g ( x 0 , t ) g ( x 0 , t 0 ) ) .
Moreover, the form of local fractional partial derivative of high order is presented below:
k r x k r g ( x , y ) = r x r r x r r x r k t i m e s g ( x , y ) .
The formula of local fractional derivative applied in this work, is in the following form [30]:
d r d x r x n r Γ ( 1 + n r ) = x ( n 1 ) r Γ ( 1 + ( n 1 ) r ) , n N .
Definition 3.
In fractal space, the Mittag–Leffler function is defined as
E r ( x r ) = m = 0 x m r Γ ( 1 + m r ) , 0 < r 1 .

3. Local Fractional Sumudu Transform

The integral transform method named Sumudu transform was proposed in 1993 by Watugula, who applied it to obtain solutions for problems in mathematical physics. Some fundamental properties of this transform were introduced and investigated by Belgacem [31]. Katatbeh and Belgacem [32] put into practice this method to get solutions for fractional differential equations. Also Ait touchent and Belgacem used the combination of Sumudu transform and homotopy perturbation for handling nonlinear fractional PDEs [23]. In this part, we define the Sumudu transform in local fractional sense and we give some of its important properties.
The definition of this transform of function g is given by Srivastava et al. [29] as follows:
L F S r [ g ( x ) ] = G r ( z ) = 1 Γ ( 1 + r ) 0 E r ( z r x r ) g ( x ) z r ( d x ) r , 0 < r 1 ,
where the integral of g in local fractional sense is given by [29]
a I b r g ( x ) = 1 Γ ( 1 + r ) a b g ( x ) ( d x ) r = 1 Γ ( 1 + r ) lim Δ x 0 i = 0 M 1 g ( x i ) ( Δ x i ) r ,
where the partitions of the interval a , b are ( x i , x i + 1 ) , i = 0 , , M 1 , Δ x = x i + 1 x i .
The inverse of local fractional Sumudu transform is provided by
L F S r 1 [ G r ( z ) ] = g ( x ) , 0 < r 1 .
The local fractional derivative is transformed by the local fractional Sumudu transform as is shown below [29]:
L F S r d r g ( x ) d x r = G r ( z ) g ( 0 ) z r ,
where G r ( z ) = L F S r [ g ( x ) ] . Also, we have the following results [29]:
L F S r d n r g ( x ) d x n r = 1 z n r G r ( z ) k = 0 n 1 z k r g ( k r ) ( 0 ) .
When n = 2 , from expression (12) we get
L F S r d 2 r g ( x ) d x 2 r = 1 z 2 r G r ( z ) g ( 0 ) z r g ( r ) ( 0 ) ,
with G r ( z ) = L F S r [ g ( x ) ] .

4. Local Fractional Sumudu Transform Coupled with Homotopy Perturbation (LHPSTM)

The following section is dedicated to the introduction of the main steps of the exploited method (LHPSTM) [33]. To illustrate this, we consider the next local fractional differential problem:
L w ( x , t ) + R w ( x , t ) = h ( x , t ) , 0 < x < 1 , 0 < t < 1 ,
where L and R represent respectively local fractional differential linear operator and nonlinear operator and the function h is a source term.
Applying the technique of local fractional Sumudu transform on both sides of Equation (14) we obtain
w r ( x , z ) = w ( x , 0 ) z r L F S r [ R w ( x , t ) ] + z r L F S r [ h ( x , t ) ] .
The inverse local fractional Sumudu transform method implies that
w ( x , t ) = w ( x , 0 ) L F S r 1 z r L F S r R w ( x , t ) + L F S r 1 z r L F S r h ( x , t ) .
By the homotopy perturbation for (16), we get:
H ( w , p ) = w ( x , t ) w ( x , 0 ) + p × L F S r 1 z r L F S r R w ( x , t ) L F S r 1 z r L F S r h ( x , t ) = 0 ,
where p 0 , 1 is a homotopy parameter, which gives
w ( x , t ) = w ( x , 0 ) p × L F S r 1 z r L F S r R w ( x , t ) + L F S r 1 z r L F S r h ( x , t ) ,
and
w ( x , t ) = n = 0 p n w n ( x , t ) ,
and the nonlinear term is decomposed as:
R w ( x , t ) = n = 0 p n H n ( w ) ,
for some He’s polynomials H n ( w ) that are given by:
H n ( w 0 , w 1 , , w n ) = 1 n ! n p n R i = 0 n p i w i p = 0 , n = 0 , 1 , 2 ,
Substituting Equations (19) and (20) in Equation (18), we obtain:
n = 0 p n w n ( x , t ) = w ( x , 0 ) p × L F S r 1 z r L F S r n = 0 p n H n ( w ) + L F S r 1 z r L F S r h ( x , t ) ,
which is the combination of local fractional Sumudu transform and HPM with He’s polynomials.
Making a comparison of the terms having same power of p, we achieve:
p 0 : w 0 ( x , t ) = w ( x , 0 ) + L F S r 1 [ z r L F S r [ h ( x , t ) ] ] , p 1 : w 1 ( x , t ) = L F S r 1 [ z r L F S r [ H 0 ( w ) ] ] , p 2 : w 2 ( x , t ) = L F S r 1 [ z r L F S r [ H 1 ( w ) ] ] , . . .
Finally, the analytical solution of Equation (14) has this structure:
w ( x , t ) = lim N n = 0 N w n ( x , t ) .

5. Analysis on Convergence

The goal of this section is to demonstrate the convergence theorem which is used further in this work.
Lemma 1.
C r [ 0 , 1 ] , R is a Banach space of local fractional continuous functions on the interval [ 0 , 1 ] with the norm
w = max t [ 0 , 1 ] | w ( x , t ) | .
Proof. 
Let ( f n ) n N a Cauchy sequence in C r [ 0 , 1 ] , R . For t fixed in [ 0 , 1 ] .
p , q N , we have | f p ( t ) f q ( t ) | f p f q < ϵ r ,
which implies that f n ( t ) is a Cauchy sequence in R . While R is complete, then f n ( t ) is convergent.
Let f ( t ) its limit, where the function f : [ 0 , 1 ] R verify:
f ( t ) = lim n f n ( t ) t [ 0 , 1 ] .
Now, we prove that f is local fractional continuous. The sequence ( f n ) n N is a sequence of local fractional continuous functions. Therefore
| f n ( t ) f n ( t 0 ) | < ϵ r t [ 0 , 1 ] , with | t t 0 | < δ .
Using limit we get
| f ( t ) f ( t 0 ) | < ϵ r t [ 0 , 1 ] .
Hence the function f C r [ 0 , 1 ] , R .
Let’s prove that ( f n ) n N converges to f in C r [ 0 , 1 ] , R . There exist N such that:
p , q N f p f q < ϵ r .
Therefore, for fixed t in [ 0 , t ] , we have:
p N , q N | f p ( t ) f q ( t ) | f p f q < ϵ r .
Now, we fixe p and we let q to infinity in the last expression, we get | f p ( t ) f ( t ) | < ϵ r which is true for every t [ 0 , 1 ] , we have f p f < ϵ r for every p N , then f p converges to f.
Finally, every Cauchy sequence ( f n ) n N converges in C r [ 0 , 1 ] , R , then C r [ 0 , 1 ] , R is complete. Therefore C r [ 0 , 1 ] , R is a Banach space. ☐
Theorem 1.
Let ( w n ) n N in Banach space C r [ 0 , 1 ] , R . If w k + 1 q w k for 0 < q < 1 then, the series n = 0 w n is convergent.
Theorem 2.
If the series solution n = 0 w n given in (24) is convergent then, it is an exact solution of the local fractional problem (14).
Proof. 
We determine the sequence
A n = w 1 + w 2 + + w n ,
by using the iterative scheme
A 0 = 0 ,
A n + 1 = w ( x , 0 ) L F S 1 z r L F S R n ( A n + w 0 ) + L F S 1 z r L F S h ,
where
R n i = 0 n w i = i = 0 n H i , n = 0 , 1 , 2 ,
Suppose that the series solution (24) converges, and ψ = n = 0 w n , then we have,
lim n A n + 1 = w ( x , 0 ) L F S 1 z r L F S lim n R n ( A n + w 0 ) + L F S 1 z r L F S h ,
then we obtain
ψ = w ( x , 0 ) L F S 1 z r L F S lim n R n i = 0 n w i + L F S 1 z r L F S h , = w ( x , 0 ) L F S 1 z r L F S lim n i = 0 n H i + L F S 1 z r L F S h , = w ( x , 0 ) L F S 1 z r L F S i = 0 H i + L F S 1 z r L F S h ,
using (20), for p = 1 , we get
ψ = w ( x , 0 ) L F S 1 z r L F S R i = 0 w i + L F S 1 z r L F S h ,
then
ψ = w ( x , 0 ) L F S 1 z r L F S R ψ + L F S 1 z r L F S h ,
applying the LFHST on both sides of Equation (37) then, we obtain,
L F S ψ w ( x , 0 ) z r = L F S R ψ + L F S h ,
L F S L ψ ( x , t ) = L F S R ψ ( x , t ) + L F S h ( x , t ) ,
Now, by applying the inverse of LFHST we obtain,
L ψ ( x , t ) + R ψ ( x , t ) = h ( x , t ) ,
therefore, ψ = n = 0 w n is an exact solution of Equation (14). ☐

6. Application

6.1. Exemple 1

Considering the following diffusion equation involving local fractional derivative
D t r w ( x , t ) = 1 2 x 2 w x x ( x , t ) , 0 < x < 1 , 0 < r 1 ,
with the next conditions:
w ( x , 0 ) = x 2 , w ( 0 , t ) = 0 , w ( 1 , t ) = E r ( t r ) ,
where the expression of exact solution is w ( x , t ) = x 2 E r ( t r ) .
Employing the expression of local fractional Sumudu transform for Equation (42) we have
L F S r [ D t r w ( x , t ) ] = L F S r 1 2 x 2 w x x ( x , t ) ,
then, we obtain
w r ( x , z ) = w ( x , 0 ) + z r L F S r 1 2 x 2 w x x ( x , t ) ,
which gives
w r ( x , z ) = x 2 + z r L F S r 1 2 x 2 w x x ( x , t ) .
The inverse of local fractional Sumudu transform method implies that
w ( x , t ) = x 2 + L F S r 1 z r L F S r 1 2 x 2 w x x ( x , t ) .
The homotopy perturbation gives
n = 0 p n w n ( x , t ) = x 2 + p × L F S r 1 z r L F S r 1 2 x 2 n = 0 p n ( w n ) x x ( x , t ) .
By comparison of the terms owning similar power of p, we get:
p 0 : w 0 ( x , t ) = x 2 , p 1 : w 1 ( x , t ) = L F S r 1 z r L F S r ( 1 2 x 2 ( w 0 ) x x = x 2 t r Γ ( r + 1 ) , p 2 : w 2 ( x , t ) = L F S r 1 z r L F S r ( 1 2 x 2 ( w 1 ) x x = x 2 t 2 r Γ ( 2 r + 1 ) , . . . p n : w n ( x , t ) = L F S r 1 z r L F S r ( 1 2 x 2 ( w n 1 ) x x = x 2 t n r Γ ( n r + 1 ) .
Therefore, the solution is formed as shown below:
w ( x , t ) = x 2 1 + t r Γ ( r + 1 ) + t 2 r Γ ( 2 r + 1 ) + t 3 r Γ ( 3 r + 1 ) + + t n r Γ ( n r + 1 ) + .
Hence
w ( x , t ) = x 2 m = 0 ( t r ) m Γ ( m r + 1 ) .
Finally, we come up with the following:
w ( x , t ) = x 2 E r ( t r ) ,
which is the exact solution of Equation (42).

6.2. Example 2: Local Fractional Dissipative Wave Equation

Considering the next dissipative wave equation including local fractional derivative [34]:
D t t 2 r w ( x , t ) D t r w ( x , t ) D x x 2 r w ( x , t ) D x r w ( x , t ) t r Γ ( r + 1 ) = 0 0 x l , t > 0 ,
with the initial conditions
w ( x , 0 ) = x r Γ ( 1 + r ) D t r w ( x , 0 ) = 0 .
Using the Sumudu transform in local fractional sense for Equation (53) we obtain
L F S r D t t 2 r w ( x , t ) = L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + D x r w ( x , t ) + t r Γ ( r + 1 ) .
Then, we get
w r ( x , z ) = w ( x , 0 ) + z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + D x r w ( x , t ) + t r Γ ( r + 1 ) ,
which implies
w r ( x , z ) = x r Γ ( 1 + r ) + z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + D x r w ( x , t ) + t r Γ ( r + 1 ) .
By the inverse local fractional Sumudu transform, it yields
w ( x , t ) = x r Γ ( 1 + r ) + L F S r 1 z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + D x r w ( x , t ) + t r Γ ( r + 1 ) .
Now, the use of homotopy perturbation gives
n = 0 p n w n ( x , t ) = x r Γ ( 1 + r ) + p × L F S r 1 z 2 r L F S r D t r n = 0 p n w n ( x , t ) + D x x 2 r n = 0 p n w n ( x , t ) D x r n = 0 p n w n ( x , t ) + t r Γ ( r + 1 ) .
Taking the terms of alike power of p, we obtain
p 0 : w 0 ( x , t ) = x r Γ ( 1 + r ) p 1 : w 1 ( x , t ) = L F S r 1 z 2 r L F S r D t r w 0 ( x , t ) + D x x 2 r w 0 ( x , t ) + D x r w 0 ( x , t ) + t r Γ ( r + 1 ) . . . p n : w n ( x , t ) = L F S r 1 z 2 r L F S r D t r w n 1 ( x , t ) + D x x 2 r w n 1 ( x , t ) + D x r w n 1 ( x , t ) .
We find
w 0 ( x , t ) = x r Γ ( 1 + r ) w 1 ( x , t ) = t 2 r Γ ( 1 + 2 r ) + t 3 r Γ ( 1 + 3 r ) . . . w n ( x , t ) = t ( n + 1 ) r Γ ( 1 + ( n + 1 ) r ) + t ( n + 2 ) r Γ ( 1 + ( n + 2 ) r ) .
Therefore, the solution is given as
w ( x , t ) = x r Γ ( 1 + r ) j = 0 1 t j r Γ ( 1 + j r ) j = 0 2 t j r Γ ( 1 + j r ) + 2 E r ( t r ) .
For the convergence of the above series, we have to prove that
lim n w n w n 1 < 1 .
Proof. 
For 0 < t < 1 , 0 < r < 1 , we have
w n w n 1 = t n + 1 r Γ 1 + n + 1 r + t n + 2 r Γ 1 + n + 2 r t n r Γ n r + 1 + t n + 1 r Γ 1 + n + 1 r 1 = t ( n + 1 ) r t n r 1 Γ 1 + n + 1 r + t r Γ 1 + n + 2 r 1 Γ n r + 1 + t r Γ 1 + n + 1 r 1 = t r 1 Γ 1 + n + 1 r + t r Γ 1 + n + 2 r 1 Γ n r + 1 + t r Γ 1 + n + 1 r 1 .
While the function Gamma is increasing, we find
1 Γ 1 + n + 1 r + t r Γ 1 + n + 2 r < 1 Γ n r + 1 + t r Γ 1 + n + 1 r .
Then, we get
1 Γ 1 + n + 1 r + t r Γ 1 + n + 2 r 1 Γ n r + 1 + t r Γ 1 + n + 1 r 1 < 1 .
Hence, we have this inequality
t r 1 Γ 1 + n + 1 r + t r Γ 1 + n + 2 r 1 Γ n r + 1 + t r Γ 1 + n + 1 r 1 < t r < 1 .
Then,
w n w n 1 < t r < 1 .
Therefore,
n N w n w n 1 < t r < 1 .
Finally,
lim n w n w n 1 t r < 1 .
Consequently, Theorem 1 ensures the convergence of the corresponding series for all values of t satisfying 0 < t < 1 .

6.3. Example 3: Local Fractional Damped Wave Equation

Consider the damped wave equation with local fractional derivative as follows [34]
D t t 2 r w ( x , t ) D t r w ( x , t ) D x x 2 r w ( x , t ) x r Γ ( r + 1 ) = 0 0 x l , t > 0 ,
subject to the following conditions
w ( x , 0 ) = 0 , D t r w ( x , 0 ) = x r Γ ( 1 + r ) .
Using the expression of local fractional Sumudu transform for Equation (71) we find
L F S r D t t 2 r w ( x , t ) = L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + x r Γ ( r + 1 ) ,
then, we get
w r ( x , z ) = z r w r ( x , 0 ) + z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + x r Γ ( r + 1 ) ,
it yields
w r ( x , z ) = z r x r Γ ( 1 + r ) + z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + x r Γ ( r + 1 ) .
By applying the inverse local fractional Sumudu transform, we find
w ( x , t ) = t r Γ ( 1 + r ) x r Γ ( 1 + r ) + L F S r 1 z 2 r L F S r D t r w ( x , t ) + D x x 2 r w ( x , t ) + x r Γ ( r + 1 ) .
Using the homotopy perturbation, we obtain
n = 0 p n w n ( x , t ) = t r Γ ( 1 + r ) x r Γ ( 1 + r ) + p × L F S r 1 z 2 r L F S r D t r n = 0 p n w n ( x , t ) + D x x 2 r n = 0 p n w n ( x , t ) + x r Γ ( r + 1 ) .
Collecting the terms of similar powers of p, we get:
p 0 : w 0 ( x , t ) = t r Γ ( 1 + r ) x r Γ ( 1 + r ) , p 1 : w 1 ( x , t ) = L F S r 1 z 2 r L F S r D t r w 0 ( x , t ) + D x x 2 r w 0 ( x , t ) + x r Γ ( r + 1 ) , . . . p n : w n ( x , t ) = L F S r 1 z 2 r L F S r D t r w n 1 ( x , t ) + D x x 2 r w n 1 ( x , t ) .
we get
w 0 ( x , t ) = t r Γ ( 1 + r ) x r Γ ( 1 + r ) , w 1 ( x , t ) = 0 , . . . w n ( x , t ) = 0 .
Hence the solution is
w ( x , t ) = t r Γ ( 1 + r ) x r Γ ( 1 + r ) ,
which is an exact solution of Equation (71).

7. Conclusions

In this work, the technique that combines homotopy perturbation and Sumudu transform in local fractional sense was applied, to obtain solutions of local fractional PDEs. Further, we presented the sufficient condition for the convergence of this method. The obtained solutions show that this technique is a powerful tool to solve different types of local fractional PDEs arising in mathematical physics.

Author Contributions

All authors contributed extensively to the work presented in this paper. Zakia Hammouch suggested the subject of the research and wrote the main paper, Toufik Mekkaoui and Kamal Ait Touchent obtained the solutions, Fethi B. M. Belgacem discussed and commented on the manuscript.

Funding

This research received the fund UMI2016 by the Moulay Ismail University.

Acknowledgments

The authors thank both referees and managing editor of Fractal and Fractional journal, for their careful reading of the manuscript, their useful comments and their corrections.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Caponetto, R.; Dongola, G.; Fortuna, L. Fractional Order Systems: Modeling and Control Application; World Scientific: Singapore, 2010. [Google Scholar]
  2. Miller, K.S.; Rosso, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. [Google Scholar]
  3. Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974. [Google Scholar]
  4. Petras, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation; Springe: Berlin, Germany, 2011. [Google Scholar]
  5. Podlubny, I. Fractional Differential Equations: Mathematics in Science and Engineering; Academic Press: New York, NY, USA, 1999. [Google Scholar]
  6. Belgacem, F.B.M.; Silambarasan, R.; Zakia, H.; Mekkaoui, T. New and Extended Applications of the Natural and Sumudu Transforms: Fractional Diffusion and Stokes Fluid Flow Realms. In Advances in Real and Complex Analysis with Applications; Chapter No. 6; Springer: Berlin, Germany, 2017; pp. 107–120. [Google Scholar]
  7. Zakia, H.; Mekkaoui, T.; Belgacem, F.B. Numerical simulations for a variable order fractional Schnakenberg model. In AIP Conference Proceedings; American Institute of Physics: College Park, ML, USA, 2014; Volume 1637. [Google Scholar]
  8. Singh, J.; Kumar, D.; Hammouch, Z.; Atangana, A. A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl. Math. Comput. 2018, 316, 504–515. [Google Scholar] [CrossRef]
  9. Zakia, H.; Mekkaoui, T. Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions. Int. J. Appl. Math. Res. 2012, 1, 206–212. [Google Scholar]
  10. Zakia, H.; Mekkaoui, T. Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives. Math. Eng. Sci. Aerosp. 2014, 5, 1–11. [Google Scholar]
  11. Wang, K.-L.; Liu, S.-Y. He’s Fractional Derivative for Nonlinear Fractional Heat Transfer Equation. Ther. Sci. 2016, 20, 793–796. [Google Scholar] [CrossRef]
  12. Liu, F.-J.; Li, Z.-B.; Zhang, S.; Liu, H.-Y. He’s Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silk-worm Cocoon Hierarchy. Ther. Sci. 2015, 9, 1155–1159. [Google Scholar] [CrossRef]
  13. Hammouch, Z.; Mekkaoui, T. Control of a new chaotic fractional-order system using Mittag-Leffler stability. Nonlinear Stud. 2015, 22, 565–577. [Google Scholar]
  14. Yang, X.-J.; Srivastava, H.M.; Cattani, C. Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics. Rom. Rep. Phys. 2015, 67, 752–761. [Google Scholar]
  15. Yan, S.-P.; Jafari, H.; Jassim, H.K. Local fractional Adomian decomposition and function decomposition methods for Laplace equation within local fractional operators. Adv. Math. Phys. 2014, 2014, 161580. [Google Scholar] [CrossRef]
  16. Zhang, Y.-Z.; Yang, A.-M.; Long, Y. Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform. Therm. Sci. 2014, 18, 677–681. [Google Scholar] [CrossRef]
  17. Yang, X.-J.; Baleanu, D. Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 2013, 17, 625–628. [Google Scholar] [CrossRef] [Green Version]
  18. Watugala, G.K. Sumudu transform: A new integral transform to solve differential equations and control engineering problems. Integr. Educ. 1993, 24, 35–43. [Google Scholar] [CrossRef]
  19. He, J.-H. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng. 1999, 178, 257–262. [Google Scholar] [CrossRef]
  20. He, J.-H. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non Linear Mech. 2000, 35, 37–43. [Google Scholar] [CrossRef]
  21. Singh, J.; Kumar, D.; Kılıçman, A. Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. Abstr. Appl. Anal. 2013, 2013, 934060. [Google Scholar] [CrossRef]
  22. Dinkar, S.; Singh, P.; Chauhan, S. Homotopy Perturbation Sumudu Transform Method with He’s Polynomial for Solutions of Some Fractional Nonlinear Partial Differential Equations. Int. J. Nonlinear Sci. 2016, 21, 91–97. [Google Scholar]
  23. Touchent, K.A.; Belgacem, F.B.M. Nonlinear fractional partial differential equations systems solutions through a hybrid homotopy perturbation Sumudu transform method. Nonlinear Stud. 2015, 22, 591–600. [Google Scholar]
  24. Youssif, E.A.; Hamed, S.H.M. Solution of nonlinear fractional differential equations using the homotopy perturbation Sumudu transform method. Appl. Math. Sci. 2014, 8, 2195–2210. [Google Scholar] [CrossRef]
  25. Yang, X.-J.; Baleanu, D.; Zhong, W. Approximate solutions for diffusion equations on Cantor space-time. Proc. Rom. Acad. Ser. A 2013, 14, 127–133. [Google Scholar]
  26. Yang, X.-J.; Hristov, J.; Srivastava, H.M.; Ahmad, B. Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation. Abstr. Appl. Anal. 2014, 2014, 278672. [Google Scholar] [CrossRef]
  27. Liu, C.-F.; Kong, S.; Yuan, S. Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem. Therm. Sci. 2013, 17, 715–721. [Google Scholar] [CrossRef] [Green Version]
  28. Yang, X.-J.; Baleanu, D.; Khan, Y.; Mohyud-Din, S.T. Local fractional variational iteration method for diffusion and wave equations on Cantor sets. Rom. J. Phys. 2014, 59, 36–48. [Google Scholar]
  29. Srivastava, H.M.; Golmankhaneh, A.K.; Baleanu, D.; Yang, X.-J. Local fractional Sumudu transform with application to IVPs on Cantor sets. Abstr. Appl. Anal. 2014, 2014, 620529. [Google Scholar] [CrossRef]
  30. Yang, X.-J. Advanced Local Fractional Calculus and Its Applications; World Science: New York, NY, USA, 2012. [Google Scholar]
  31. Belgacem, F.B.M.; Karaballi, A.A. Sumudu transform fundamental properties investigations and applications. Int. J. Stochastic Anal. 2006, 2006, 91083. [Google Scholar] [CrossRef]
  32. Deeb, K.Q.; Belgacem, F.B.M. Applications of the Sumudu transform to fractional differential equations. Nonlinear Stud. 2011, 18, 99–112. [Google Scholar]
  33. Singh, J.; Kumar, D.; Nieto, J.J. A reliable algorithm for a local fractional tricomi equation arising in fractal transonic flow. Entropy 2016, 18, 206. [Google Scholar] [CrossRef]
  34. Su, W.-H.; Baleanu, D.; Yang, X.-J.; Jafari, H. Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method. Fix. Point Theory Appl. 2013, 2013, 89. [Google Scholar] [CrossRef] [Green Version]

Share and Cite

MDPI and ACS Style

Ait Touchent, K.; Hammouch, Z.; Mekkaoui, T.; Belgacem, F.B.M. Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs. Fractal Fract. 2018, 2, 22. https://doi.org/10.3390/fractalfract2030022

AMA Style

Ait Touchent K, Hammouch Z, Mekkaoui T, Belgacem FBM. Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs. Fractal and Fractional. 2018; 2(3):22. https://doi.org/10.3390/fractalfract2030022

Chicago/Turabian Style

Ait Touchent, Kamal, Zakia Hammouch, Toufik Mekkaoui, and Fethi B. M. Belgacem. 2018. "Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs" Fractal and Fractional 2, no. 3: 22. https://doi.org/10.3390/fractalfract2030022

Article Metrics

Back to TopTop