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Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method Saud Owyed a, M.A. Abdou b, Abdel-Haleem Abdel-Aty b,c,∗, W. Alharbi d, Ramzi Nekhili e a
Department of Mathematics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt d Physics Department, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia e College of Administrative Sciences, Applied Science University, Bahrain b c
a r t i c l e
i n f o
Article history: Received 28 August 2019 Revised 27 September 2019 Accepted 8 October 2019 Available online xxx
a b s t r a c t We construct an explicit and approximate solutions of fractional time-nonlinear fractional equations by using a new approach, namely, the reduced differential transform (RDTM) method. As a special cases, two models of fractional-time derivative of order α are used to find the solutions. The effect of the values of the parameter α on the behaviour of reduced differential transform (RDTM) method is investigated. The efficiency of the suggested algorithm is tested by using some numerical examples.
Keywords: Transform method Fractional calculus Adomian decomposition method Reduced differential Numerical solutions
1. Introduction Many of the life, nature and research problems and complex phenomena are described and simulated by fractional differential equations (FDE) such as biology, physics, economy and statistical applications [1–8]. More recently, another new areas of research are treated by the fractional calculus like signal processing, biomedical engineering, image and video processing and quantum applications which add more importance to the FDE as a tool help to improve and study of these applications [9–12]. Several groups struggled to find more efficient and powerful ways to provide analytical, exact and numerical solutions of the fractional differential equations PDEs [13–43] and so on. The fractional Schrödinger-Korteweg-de Vries equation (or Sch-KdV) equation which used in many applications in quantum physics was solved in many papers with different methods [44], like the solution of the coupled Schrödinger-KdV equation by the decomposition method [45], Adomian decomposition method (ADM) is used by Abdou and Elhanbaly [46], extended simple equation method [47], Homotopy Perturbation Method (HPM) [48], Ramswroop pre-
∗
Corresponding author. E-mail address: [email protected] (A.-H. Abdel-Aty).
© 2019 Elsevier Ltd. All rights reserved.
sented semi-analytical technique based on the homotopy analysis transform method (HATM) [49]. Here in this paper we motivated to introduce new soliton solutions for the time fractional nonlinear Schrodinger KdV equation. The rest sections of this article are arranged as follows. We can see in Section 2 the method of the fractional time reduced differential transform [21–24]. The models of our study, fractional time nonlinear evolution equation, coupled time frcational nonlinear Schrodinger-Korteweg-de Vries equation and coupled system of time fractional diffusion-reaction equations are described in Section 3. Finally, conclusion and remarks are introduced in the final Section 4. 2. Methodology and its applications This section provide a short discussion regarding the method which applied in this paper [21–23]. Firstly we start by supposing that the function contains two variables u(x, t) and this function is considered as a combination of two single-variable functions, i.e., u(x, t ) = f (x )g(t ). The equation u(x, t) is expressed as:
u(x, t ) =
∞
Uk (x )t kα ,
(1)
k=0
https://doi.org/10.1016/j.chaos.2019.109474 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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where the parameter α represents the order of the time fractional derivative in Caputo and sense the function Uk (x) represents the t-dimensional spectrum equation function of u(x, t).
Table 1 Reduced differentional transformation.
Definition 2.1. Let to define the function u(x, t) is analytic and differentiated continuously with respect to t and x in the domain of interest, then let
Uk (x ) =
∂ kα u(x, t ) |t=0 , (kα + 1 ) ∂ t kα 1
(2)
In this equation t represents dimensional spectrum function. The equation uk (x) is the transformed function. Here the small letter equation u(x, t) describe the main original function and the capital letter equation Uk (x) represents the transformed function. The next step by adding Eqs. (1) and (2) we obtain the following Eq. (3): ∞
∂ kα u(x, t ) |t=0t kα u(x, t ) = (kα + 1 ) ∂ t kα k=0 1
(3)
(4)
with of initial condition
u(x, 0 ) = f (x ),
(5)
α In this equation the L = ∂∂t α , R represents a linear operator, Nu(x,
t) can be considered as a nonlinear part and g(x, t) can be defined as an inhomogeneous function part.
(kα + Nα + 1 ) Uk+N (x ) = Gk (x ) − R(Uk (x )) − N (Uk (x )), (kα + 1 )
(6)
where Uk (x), R(Uk (x)), N(Uk (x)) and Gk (x) represent the transformation methods of the equations Lu(x, t), Ru(x, t), Nu(x, t) and g(x, t) respectively. By using the initial condition, resulting the following equations:
U0 (x ) = f (x ),
(7)
Inserting Eqs. (7), Eq. (6) with the iterative formula, one can directly obtain to the different values of Uk (x). By using the inverse transformation for the group of the equations Uk (x )n k=0 , the approximation solution can be written as,
u∗ (x, t ) =
n
Uk (x )t kα ,
k=0
By the substitution from the previous equation, the exact solution of model can be found as:
(9)
3. New applications
3.1. Fractional time of coupled nonlinear Schrodinger-Korteweg-de Vries equation Let us firstly, suppose that our system is a coupled nonlinear Schrodinger-Korteweg-de Vries model [29],
(10)
kα
(11)
with the initial conditions as
√ u(x, 0 ) = 6 2eiβ x k2 sech2 (kx ),
β + 16k2
(12)
− 6k2 tanh2 (kx ),
3
(13)
α
where the equation ∂∂t α is the fractional time derivative operator of order α and β and k are constants. In view of RDTM table [1], Eqs. (10) and (11) can be written as
(kα + α + 1 ) Uk+1 (x ) = −i[uxx + uv], (kα + 1 ) (kα + α + 1 ) Uk+1 (x ) = −[6uvx + vxxx − (|u| )2 )x ] (kα + 1 )
(14) (15)
Eqs. (14) and (15) with Eqs. (12) and (13) admits to
(kα + α + 1 ) Uk+1 (x ) = −i[uxx + N (u, v )], (kα + 1 ) (kα + α + 1 ) Uk+1 (x ) = −[6M (u, v ) + vxxx − R(u )], (kα + 1 )
(16) (17)
and nonlinear operators N (u, v ) = uv, M (u, v ) = uvx and R(u, v ) = (|u|2 )x by the infinite series of Adomian polynomials given by ∞
N (u, v ) = M (u, v ) =
An ,
n=0 ∞
(18)
Bn ,
(19)
n=0
R (u ) =
∞
Cn ,
(20)
n=0
In the previous equation the parameters An , Bn and Cn represent the approximate Adomian’s polynomials. The definition of these polynomials are quantified in [27,28],
An ( u0 , . . . , un , v0 , . . . , vn )
The properties, efficiency and the importance of the suggested method are clarified by using two models playing an important role in the physics and it’s application are fractional nonlinear evolution equations arising in physics are chosen, namely, time fractional coupled system of diffusion-reaction equation and coupled nonlinear Schrodinger-Korteweg-de Vries equation.
∂ α u(x, t ) i = uxx + uv, ∂tα
Uk = (kα1 +1) [ ∂ ∂ut k(αx,t ) ]t=0 Wk (x ) = Uk (x ) ± Vk (x ) Wk (x ) = αUk (x ), α is a constant Wk (x ) = xm δ (kα − n ) Wk (x ) = xmU (kα − n ) Wk (x ) = kr=0 Ur (x )Vk−r (x ) kα +Nα +1 ) Uk+N (x ) Wk (x ) = ((( kα +1 ) m Wk (x ) = ∂∂xm Uk (x )
∂ α v(x, t ) = −6uvx − vxxx + (|u|2 )x , ∂tα
(8)
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
Transformed
u(x, t) w(x, t ) = u(x, t ) ± v (x, t ) w(x, t ) = α u(x, t ) w(x, t ) = xm t n w(x, t ) = xm t n u(x, t ) w(x, t ) = u(x, t )v (x, t ) Nα w(x, t ) = ∂∂t Nα u(x, t ) m w(x, t ) = ∂∂xm u(x, t )
v(x, 0 ) =
Let us first take the nonlinear evolution equation as follows:
L(u(x, t )) + R(u(x, t )) + N (u(x, t )) = g(x, t ),
Functional
1 = n!
dn N dλn
∞
λ uk , k
k=0
∞ k=0
λ vk k
,n > 0
(21)
λ=0
The equation parameters An , Bn and Cn the Adomian polynomials can be obtained form the following equations:
A0 = u0 v0 , A1 = u1 v0 + v1 u0 , A2 = u1 v1 + v0 u2 + u0 v2 , B 0 = u 0 v0x , B 1 = u 1 v0x + v1x u 0 , B 2 = u 2 v0x + v1x u 1 + u 0 v2x , C0 = (u20 )x , C1 = (2u1 u0 )x , C2 = (2u2 u0 + u21 )x ,
(22)
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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3
Fig. 1. The exact and numerical solution of u(x,t) and v(x,t) for different values of α =1 for a coupled Schrodinger-KdV equation.
The equations un (x, t) and vn (x, t), n > 1 can determined via the recurrence relations by taking the zeroth component u(0, t) and v(0, t) by substituting of the initial condition.
(kα + α + 1 ) Uk+1 (x, t ) = −i[ukxx + Ak ], k > 0 (kα + 1 )
(23)
(kα + α + 1 ) Uk+1 (x, t ) = −[6Bk + vkxxx − Ck ], k > 0 (kα + 1 )
(24)
By using Eqs. (23) and (24), in terms of Un (x, t) and Vn (x, t) can be solved directly. Having determined the components Un (x, t) and Vn (x, t), the RDM series of u(x, t) and v(x, t) can be written directly as
u∗ (x, t ) =
n
Uk (x, t )t kα ,
(25)
Vk (x, t )t kα ,
(26)
k=0
v∗ (x, t ) =
n k=0
3
3
(30) (31)
As a comparison between the results to solutions, they are much good as we can see in Fig. 1(c–f). 3.2. Fractional time coupled system of diffusion-reaction equations The second model in this paper is the time fractional coupled system of diffusion-reaction equation [29],
∂ α u(x, t ) = u(1 − u − v ) + uxx , ∂tα α ∂ v(x, t ) = vxx − uv, ∂tα
(32) (33)
with
where n is order of approximation solution. Therefore, the approximate solutions of u(x, t) and v(x, t) are given by
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
(27)
v(x, t ) = limn−→∞ v∗n (x, t ), i = 1, 2, . . . ., n
(28)
Fig. 1(a,b) present the dynamics and the behaviour of RDTM solutions of u(x, t) and v(x, t) under the effect of the changing in the values of the fractional order α . In the case of the fractional order (α = 1), the RDTM solutions is compared with the exact solution of Schrodinger-KdV equation [29],
√ u(x, t ) = 6 2ei sech2 (kξ ),
β + 16k2 v(x, t ) = − 6k2 tanh2 (kξ ), 3
10k2 t βt = + β 2t − + β x , ξ = x + 2β t
(29)
ekx , [1 + e0.5kx ]2 1 v(x, 0 ) = , [1 + e0.5kx ] u(x, 0 ) =
(34) (35)
where k is constants. This model of equations have importance in the modeling and simulations of chemical reaction or ecology, and such fields arising in fields of physics. In view RDTM table [1], Eqs. (32) and (33) reduces to
(kα + α + 1 ) Uk+1 (x, t ) = [u − N (u ) − K (u, v ) + uxx ], (kα + 1 )
(36)
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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(kα + α + 1 ) Uk+1 (x, t ) = [vxx − K (u, v )] (kα + 1 )
(37)
where the relations N(u), K(u, v) have a relation with the nonlinear terms of the main equations that can be written by Adomian polynomials as in the following equations:
N (u ) = u = 2
∞
An ( u ),
K (u, v ) = uv =
u∗ (x, t ) =
n
Uk (x, t )t kα
(43)
Vk (x, t )t kα
(44)
k=0
v∗ (x, t ) = (38)
n=0 ∞
given by
n k=0
The approximate solutions of u(x, t) and v(x, t) are given by
Bn ( uv )
(39)
n=0
The first three components of Adomian polynomials reads
A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 , B0 = u0 v0 , B1 = v0 v1 + u0 v1 , B2 = v0 u2 + u0 v2 ,
(40)
Eqs. (36) and (37) are reformulated into a group of recursive relations as you see in the following equations:
(kα + α + 1 ) Uk+1 (x, t ) = [uk − Ak − Bk + ukxx ], k > 0 (kα + 1 ) (kα + α + 1 ) Uk+1 (x, t ) = [vkxx − Bk ], k > 0 (kα + 1 )
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
(45)
v(x, t ) = limn−→∞ v∗n (x, t ), i = 1, 2, . . . ., n
(46)
The dynamics of the numerical and approximate solutions of RDTM with the changing in the values of the fractional parameter α can be found in Fig. 2(a,b). It is clear that when α = 1, the RDTM solution is go back to the exact solution Eq. (4),
u (z ) =
(41)
v (z ) = (42)
Knowing the above recursive relationship Eqs. (41) and (42), all terms of Un (x, t) and Vn (x, t) are obtained. Knowing the components Un (x, t) and Vn (x, t), the RDTM series of u(x, t) and v(x, t) are
ekz , (1 + e0.5kz )2 1
( 1 + e 0. 5k )
,
(47)
(48)
These two equation can be considered as a result of the classical form of diffusion-reaction equation where z = x + ct. The dynamics of the exact solutions of equation u(x, t) and v(x, t) where k = c = 1 is shown in Fig. 2(c–f).
Fig. 2. Exact and numerical solutions of the equations u(x,t) and v(x,t) by RDTM for different values of α for the chemical reaction model.
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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4. Conclusion The work is concerned with introduce and propose a new method to find the numerical and approximate solutions of coupled systems of fractional coupled system of diffusion-reaction equation and coupled nonlinear Schrodinger-Korteweg-de Vries equation. The obtained results provided the efficiency and the power of the RTDM in obtaining the solutions of the time fractional nonlinear differential equations both numerically and analytically. Also, we can say that the solutions are more general where we can obtain the classical one when α = 1. Moreover, the presence of free parameters give more dynamics of the solutions and can deal with initial and boundary value problem with fractional order. Declaration of Competing Interest I declare that I have no significant competing financial, professional, or personal interests that might have influenced the performance or presentation of the work. References [1] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974. [2] Podlubny I. Fractional differential equations, mathematics in science and engineering. San Diego, CA: Academic Press; 1999. [3] Sugimoto N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J Fluid Mec 1991;225:631. [4] Saad KM, Atangana A, Baleanu D. New fractional derivatives with non-singular kernel applied to the burgers equation. Chaos 2018;28:063109. [5] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci 2016;20. 763-9 [6] Owyed S, Abdou MA, Abdel-Aty A, Ray SS. New optical soliton solutions of nolinear evolution equation describing nonlinear dispersion. Commun Theor Phys 2019;71:1063–8. [7] Ismail GM, Abdl-Rahim HR, Abdel-Aty A, Kharabsheh R, Alharbi W, Abdel-Aty M. An analytical solution for fractional oscillator in a resisting medium. Chaos Solitons Fractals 2020;130:109395. [8] Elgendy AT, Abdel-Aty A, Youssef AA, Khder MAA, Lotfy K, Owyed S. Exact solution of arrhenius equation for non-isothermal kinetics at constant heating rate and n-th order of reaction. J Math Chemistry 2020. doi:10.1007/ s10910- 019- 01056- 7. [9] Cruz-Duarte M, Rosales-Garciaa J, Correa-Cely CR, Garcia-Perez A, Avina-Cervantes JG. A closed form expression for the gaussian-based caputo-fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci Numer Simul 2018;61:138–48. [10] Abdel-Aty AM, Soltan A, Ahmed WA, Radwan AG. On the analysis and design of fractional-order chebyshev complex filter. Circuits Systems, andSignal Process 2018;37:915–38. [11] Yang F, Mou J, Yan H, Hu J. Dynamical analysis of a novel complex chaotic system and application in image diffusion. IEEE Access 2019;7:118188–202. [12] Pu Y-F. A fractional-order variational framework for retinex: fractional-order partial differential equation-based formulation for multi-scale nonlocal contrast enhancement with texture preserving. IEEE Trans Image Process 2018;27:1214. [13] He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39. [14] Momani S, Odibat Z. Comparison between the hompotopy perturbation method and the VIM for linear fractiona partial differential equations. Comut Math Appl 2007;54:910–19. [15] Momani S, Odibat Z. Generalized differential transform method for solving a space-time fractional diffusion equation. Phys Lett A 2007;370:379–87. [16] Lu D, Seadawy AR, Khater MMA. Structures of exact and solitary optical solutions for the higher-order nonlinear schrödinger equation and its applications in mono-mode optical fibers. Mod Phys Lett B 2019;33:1950279. [17] Osman MS, Lu D, Khater MMA. A study of optical wave propagation in the nonautonomous schrödinger-hirota equation with power-law nonlinearity. Results Phys 2019;13:102157. [18] Khater MMA, Lu D, Attia RAM. Dispersive long wave of nonlinear fractional wu-zhang system via a modified auxiliary equation method. AIP Adv 2019;9:025003.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method Saud Owyed a, M.A. Abdou b, Abdel-Haleem Abdel-Aty b,c,∗, W. Alharbi d, Ramzi Nekhili e a
Department of Mathematics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt d Physics Department, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia e College of Administrative Sciences, Applied Science University, Bahrain b c
a r t i c l e
i n f o
Article history: Received 28 August 2019 Revised 27 September 2019 Accepted 8 October 2019 Available online xxx
a b s t r a c t We construct an explicit and approximate solutions of fractional time-nonlinear fractional equations by using a new approach, namely, the reduced differential transform (RDTM) method. As a special cases, two models of fractional-time derivative of order α are used to find the solutions. The effect of the values of the parameter α on the behaviour of reduced differential transform (RDTM) method is investigated. The efficiency of the suggested algorithm is tested by using some numerical examples.
Keywords: Transform method Fractional calculus Adomian decomposition method Reduced differential Numerical solutions
1. Introduction Many of the life, nature and research problems and complex phenomena are described and simulated by fractional differential equations (FDE) such as biology, physics, economy and statistical applications [1–8]. More recently, another new areas of research are treated by the fractional calculus like signal processing, biomedical engineering, image and video processing and quantum applications which add more importance to the FDE as a tool help to improve and study of these applications [9–12]. Several groups struggled to find more efficient and powerful ways to provide analytical, exact and numerical solutions of the fractional differential equations PDEs [13–43] and so on. The fractional Schrödinger-Korteweg-de Vries equation (or Sch-KdV) equation which used in many applications in quantum physics was solved in many papers with different methods [44], like the solution of the coupled Schrödinger-KdV equation by the decomposition method [45], Adomian decomposition method (ADM) is used by Abdou and Elhanbaly [46], extended simple equation method [47], Homotopy Perturbation Method (HPM) [48], Ramswroop pre-
∗
Corresponding author. E-mail address: [email protected] (A.-H. Abdel-Aty).
© 2019 Elsevier Ltd. All rights reserved.
sented semi-analytical technique based on the homotopy analysis transform method (HATM) [49]. Here in this paper we motivated to introduce new soliton solutions for the time fractional nonlinear Schrodinger KdV equation. The rest sections of this article are arranged as follows. We can see in Section 2 the method of the fractional time reduced differential transform [21–24]. The models of our study, fractional time nonlinear evolution equation, coupled time frcational nonlinear Schrodinger-Korteweg-de Vries equation and coupled system of time fractional diffusion-reaction equations are described in Section 3. Finally, conclusion and remarks are introduced in the final Section 4. 2. Methodology and its applications This section provide a short discussion regarding the method which applied in this paper [21–23]. Firstly we start by supposing that the function contains two variables u(x, t) and this function is considered as a combination of two single-variable functions, i.e., u(x, t ) = f (x )g(t ). The equation u(x, t) is expressed as:
u(x, t ) =
∞
Uk (x )t kα ,
(1)
k=0
https://doi.org/10.1016/j.chaos.2019.109474 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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where the parameter α represents the order of the time fractional derivative in Caputo and sense the function Uk (x) represents the t-dimensional spectrum equation function of u(x, t).
Table 1 Reduced differentional transformation.
Definition 2.1. Let to define the function u(x, t) is analytic and differentiated continuously with respect to t and x in the domain of interest, then let
Uk (x ) =
∂ kα u(x, t ) |t=0 , (kα + 1 ) ∂ t kα 1
(2)
In this equation t represents dimensional spectrum function. The equation uk (x) is the transformed function. Here the small letter equation u(x, t) describe the main original function and the capital letter equation Uk (x) represents the transformed function. The next step by adding Eqs. (1) and (2) we obtain the following Eq. (3): ∞
∂ kα u(x, t ) |t=0t kα u(x, t ) = (kα + 1 ) ∂ t kα k=0 1
(3)
(4)
with of initial condition
u(x, 0 ) = f (x ),
(5)
α In this equation the L = ∂∂t α , R represents a linear operator, Nu(x,
t) can be considered as a nonlinear part and g(x, t) can be defined as an inhomogeneous function part.
(kα + Nα + 1 ) Uk+N (x ) = Gk (x ) − R(Uk (x )) − N (Uk (x )), (kα + 1 )
(6)
where Uk (x), R(Uk (x)), N(Uk (x)) and Gk (x) represent the transformation methods of the equations Lu(x, t), Ru(x, t), Nu(x, t) and g(x, t) respectively. By using the initial condition, resulting the following equations:
U0 (x ) = f (x ),
(7)
Inserting Eqs. (7), Eq. (6) with the iterative formula, one can directly obtain to the different values of Uk (x). By using the inverse transformation for the group of the equations Uk (x )n k=0 , the approximation solution can be written as,
u∗ (x, t ) =
n
Uk (x )t kα ,
k=0
By the substitution from the previous equation, the exact solution of model can be found as:
(9)
3. New applications
3.1. Fractional time of coupled nonlinear Schrodinger-Korteweg-de Vries equation Let us firstly, suppose that our system is a coupled nonlinear Schrodinger-Korteweg-de Vries model [29],
(10)
kα
(11)
with the initial conditions as
√ u(x, 0 ) = 6 2eiβ x k2 sech2 (kx ),
β + 16k2
(12)
− 6k2 tanh2 (kx ),
3
(13)
α
where the equation ∂∂t α is the fractional time derivative operator of order α and β and k are constants. In view of RDTM table [1], Eqs. (10) and (11) can be written as
(kα + α + 1 ) Uk+1 (x ) = −i[uxx + uv], (kα + 1 ) (kα + α + 1 ) Uk+1 (x ) = −[6uvx + vxxx − (|u| )2 )x ] (kα + 1 )
(14) (15)
Eqs. (14) and (15) with Eqs. (12) and (13) admits to
(kα + α + 1 ) Uk+1 (x ) = −i[uxx + N (u, v )], (kα + 1 ) (kα + α + 1 ) Uk+1 (x ) = −[6M (u, v ) + vxxx − R(u )], (kα + 1 )
(16) (17)
and nonlinear operators N (u, v ) = uv, M (u, v ) = uvx and R(u, v ) = (|u|2 )x by the infinite series of Adomian polynomials given by ∞
N (u, v ) = M (u, v ) =
An ,
n=0 ∞
(18)
Bn ,
(19)
n=0
R (u ) =
∞
Cn ,
(20)
n=0
In the previous equation the parameters An , Bn and Cn represent the approximate Adomian’s polynomials. The definition of these polynomials are quantified in [27,28],
An ( u0 , . . . , un , v0 , . . . , vn )
The properties, efficiency and the importance of the suggested method are clarified by using two models playing an important role in the physics and it’s application are fractional nonlinear evolution equations arising in physics are chosen, namely, time fractional coupled system of diffusion-reaction equation and coupled nonlinear Schrodinger-Korteweg-de Vries equation.
∂ α u(x, t ) i = uxx + uv, ∂tα
Uk = (kα1 +1) [ ∂ ∂ut k(αx,t ) ]t=0 Wk (x ) = Uk (x ) ± Vk (x ) Wk (x ) = αUk (x ), α is a constant Wk (x ) = xm δ (kα − n ) Wk (x ) = xmU (kα − n ) Wk (x ) = kr=0 Ur (x )Vk−r (x ) kα +Nα +1 ) Uk+N (x ) Wk (x ) = ((( kα +1 ) m Wk (x ) = ∂∂xm Uk (x )
∂ α v(x, t ) = −6uvx − vxxx + (|u|2 )x , ∂tα
(8)
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
Transformed
u(x, t) w(x, t ) = u(x, t ) ± v (x, t ) w(x, t ) = α u(x, t ) w(x, t ) = xm t n w(x, t ) = xm t n u(x, t ) w(x, t ) = u(x, t )v (x, t ) Nα w(x, t ) = ∂∂t Nα u(x, t ) m w(x, t ) = ∂∂xm u(x, t )
v(x, 0 ) =
Let us first take the nonlinear evolution equation as follows:
L(u(x, t )) + R(u(x, t )) + N (u(x, t )) = g(x, t ),
Functional
1 = n!
dn N dλn
∞
λ uk , k
k=0
∞ k=0
λ vk k
,n > 0
(21)
λ=0
The equation parameters An , Bn and Cn the Adomian polynomials can be obtained form the following equations:
A0 = u0 v0 , A1 = u1 v0 + v1 u0 , A2 = u1 v1 + v0 u2 + u0 v2 , B 0 = u 0 v0x , B 1 = u 1 v0x + v1x u 0 , B 2 = u 2 v0x + v1x u 1 + u 0 v2x , C0 = (u20 )x , C1 = (2u1 u0 )x , C2 = (2u2 u0 + u21 )x ,
(22)
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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3
Fig. 1. The exact and numerical solution of u(x,t) and v(x,t) for different values of α =1 for a coupled Schrodinger-KdV equation.
The equations un (x, t) and vn (x, t), n > 1 can determined via the recurrence relations by taking the zeroth component u(0, t) and v(0, t) by substituting of the initial condition.
(kα + α + 1 ) Uk+1 (x, t ) = −i[ukxx + Ak ], k > 0 (kα + 1 )
(23)
(kα + α + 1 ) Uk+1 (x, t ) = −[6Bk + vkxxx − Ck ], k > 0 (kα + 1 )
(24)
By using Eqs. (23) and (24), in terms of Un (x, t) and Vn (x, t) can be solved directly. Having determined the components Un (x, t) and Vn (x, t), the RDM series of u(x, t) and v(x, t) can be written directly as
u∗ (x, t ) =
n
Uk (x, t )t kα ,
(25)
Vk (x, t )t kα ,
(26)
k=0
v∗ (x, t ) =
n k=0
3
3
(30) (31)
As a comparison between the results to solutions, they are much good as we can see in Fig. 1(c–f). 3.2. Fractional time coupled system of diffusion-reaction equations The second model in this paper is the time fractional coupled system of diffusion-reaction equation [29],
∂ α u(x, t ) = u(1 − u − v ) + uxx , ∂tα α ∂ v(x, t ) = vxx − uv, ∂tα
(32) (33)
with
where n is order of approximation solution. Therefore, the approximate solutions of u(x, t) and v(x, t) are given by
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
(27)
v(x, t ) = limn−→∞ v∗n (x, t ), i = 1, 2, . . . ., n
(28)
Fig. 1(a,b) present the dynamics and the behaviour of RDTM solutions of u(x, t) and v(x, t) under the effect of the changing in the values of the fractional order α . In the case of the fractional order (α = 1), the RDTM solutions is compared with the exact solution of Schrodinger-KdV equation [29],
√ u(x, t ) = 6 2ei sech2 (kξ ),
β + 16k2 v(x, t ) = − 6k2 tanh2 (kξ ), 3
10k2 t βt = + β 2t − + β x , ξ = x + 2β t
(29)
ekx , [1 + e0.5kx ]2 1 v(x, 0 ) = , [1 + e0.5kx ] u(x, 0 ) =
(34) (35)
where k is constants. This model of equations have importance in the modeling and simulations of chemical reaction or ecology, and such fields arising in fields of physics. In view RDTM table [1], Eqs. (32) and (33) reduces to
(kα + α + 1 ) Uk+1 (x, t ) = [u − N (u ) − K (u, v ) + uxx ], (kα + 1 )
(36)
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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(kα + α + 1 ) Uk+1 (x, t ) = [vxx − K (u, v )] (kα + 1 )
(37)
where the relations N(u), K(u, v) have a relation with the nonlinear terms of the main equations that can be written by Adomian polynomials as in the following equations:
N (u ) = u = 2
∞
An ( u ),
K (u, v ) = uv =
u∗ (x, t ) =
n
Uk (x, t )t kα
(43)
Vk (x, t )t kα
(44)
k=0
v∗ (x, t ) = (38)
n=0 ∞
given by
n k=0
The approximate solutions of u(x, t) and v(x, t) are given by
Bn ( uv )
(39)
n=0
The first three components of Adomian polynomials reads
A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 , B0 = u0 v0 , B1 = v0 v1 + u0 v1 , B2 = v0 u2 + u0 v2 ,
(40)
Eqs. (36) and (37) are reformulated into a group of recursive relations as you see in the following equations:
(kα + α + 1 ) Uk+1 (x, t ) = [uk − Ak − Bk + ukxx ], k > 0 (kα + 1 ) (kα + α + 1 ) Uk+1 (x, t ) = [vkxx − Bk ], k > 0 (kα + 1 )
u(x, t ) = limn−→∞ u∗n (x, t ), i = 1, 2, . . . ., n
(45)
v(x, t ) = limn−→∞ v∗n (x, t ), i = 1, 2, . . . ., n
(46)
The dynamics of the numerical and approximate solutions of RDTM with the changing in the values of the fractional parameter α can be found in Fig. 2(a,b). It is clear that when α = 1, the RDTM solution is go back to the exact solution Eq. (4),
u (z ) =
(41)
v (z ) = (42)
Knowing the above recursive relationship Eqs. (41) and (42), all terms of Un (x, t) and Vn (x, t) are obtained. Knowing the components Un (x, t) and Vn (x, t), the RDTM series of u(x, t) and v(x, t) are
ekz , (1 + e0.5kz )2 1
( 1 + e 0. 5k )
,
(47)
(48)
These two equation can be considered as a result of the classical form of diffusion-reaction equation where z = x + ct. The dynamics of the exact solutions of equation u(x, t) and v(x, t) where k = c = 1 is shown in Fig. 2(c–f).
Fig. 2. Exact and numerical solutions of the equations u(x,t) and v(x,t) by RDTM for different values of α for the chemical reaction model.
Please cite this article as: S. Owyed, M.A. Abdou and A.-H. Abdel-Aty et al., Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method, Chaos, Solitons and Fractals, https://doi.org/10.1016/ j.chaos.2019.109474
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4. Conclusion The work is concerned with introduce and propose a new method to find the numerical and approximate solutions of coupled systems of fractional coupled system of diffusion-reaction equation and coupled nonlinear Schrodinger-Korteweg-de Vries equation. The obtained results provided the efficiency and the power of the RTDM in obtaining the solutions of the time fractional nonlinear differential equations both numerically and analytically. Also, we can say that the solutions are more general where we can obtain the classical one when α = 1. Moreover, the presence of free parameters give more dynamics of the solutions and can deal with initial and boundary value problem with fractional order. Declaration of Competing Interest I declare that I have no significant competing financial, professional, or personal interests that might have influenced the performance or presentation of the work. References [1] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974. [2] Podlubny I. Fractional differential equations, mathematics in science and engineering. San Diego, CA: Academic Press; 1999. [3] Sugimoto N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J Fluid Mec 1991;225:631. [4] Saad KM, Atangana A, Baleanu D. New fractional derivatives with non-singular kernel applied to the burgers equation. Chaos 2018;28:063109. [5] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci 2016;20. 763-9 [6] Owyed S, Abdou MA, Abdel-Aty A, Ray SS. New optical soliton solutions of nolinear evolution equation describing nonlinear dispersion. Commun Theor Phys 2019;71:1063–8. [7] Ismail GM, Abdl-Rahim HR, Abdel-Aty A, Kharabsheh R, Alharbi W, Abdel-Aty M. An analytical solution for fractional oscillator in a resisting medium. Chaos Solitons Fractals 2020;130:109395. [8] Elgendy AT, Abdel-Aty A, Youssef AA, Khder MAA, Lotfy K, Owyed S. Exact solution of arrhenius equation for non-isothermal kinetics at constant heating rate and n-th order of reaction. J Math Chemistry 2020. doi:10.1007/ s10910- 019- 01056- 7. [9] Cruz-Duarte M, Rosales-Garciaa J, Correa-Cely CR, Garcia-Perez A, Avina-Cervantes JG. A closed form expression for the gaussian-based caputo-fabrizio fractional derivative for signal processing applications. Commun Nonlinear Sci Numer Simul 2018;61:138–48. [10] Abdel-Aty AM, Soltan A, Ahmed WA, Radwan AG. On the analysis and design of fractional-order chebyshev complex filter. Circuits Systems, andSignal Process 2018;37:915–38. [11] Yang F, Mou J, Yan H, Hu J. Dynamical analysis of a novel complex chaotic system and application in image diffusion. IEEE Access 2019;7:118188–202. [12] Pu Y-F. A fractional-order variational framework for retinex: fractional-order partial differential equation-based formulation for multi-scale nonlocal contrast enhancement with texture preserving. IEEE Trans Image Process 2018;27:1214. [13] He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39. [14] Momani S, Odibat Z. Comparison between the hompotopy perturbation method and the VIM for linear fractiona partial differential equations. Comut Math Appl 2007;54:910–19. [15] Momani S, Odibat Z. Generalized differential transform method for solving a space-time fractional diffusion equation. Phys Lett A 2007;370:379–87. [16] Lu D, Seadawy AR, Khater MMA. Structures of exact and solitary optical solutions for the higher-order nonlinear schrödinger equation and its applications in mono-mode optical fibers. Mod Phys Lett B 2019;33:1950279. [17] Osman MS, Lu D, Khater MMA. A study of optical wave propagation in the nonautonomous schrödinger-hirota equation with power-law nonlinearity. Results Phys 2019;13:102157. [18] Khater MMA, Lu D, Attia RAM. Dispersive long wave of nonlinear fractional wu-zhang system via a modified auxiliary equation method. AIP Adv 2019;9:025003.
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