Generalized Differential Transform Method to Space- Time Fractional Non-linear Schrodinger Equation

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1 International Journal of Latest Engineering Research and Applications (IJLERA) ISSN: Volume, Issue, December 7, PP 7-3 Generalized Differential Transform Method to Space- Time Fractional Non-linear Schrodinger Equation Deepanjan Das Department of Mathematics, Ghani Khan Choudhury Institute of Engineering and Technology, Narayanpur, Malda, West Bengal-734,India Abstract: In the present paper, Generalized Differential Transform Method (GDTM) is used for obtaining the approximate analytic solution of Space-Time Fractional Non-linear Schrodinger Equation. The fractional derivatives are described in the Caputo sense. Keywords: Fractional differential equation; Caputo fractional derivative; Generalized Differential transform method; Analytic solution. Mathematical Subject Classification (): 6A33, 34A8, 35A, 35R, 35C, 74H.. Introduction Differential equations with fractional order are generalizations of classical differential equations of integer order and have recently been proved to be valuable tools in the modeling of many physical phenomena in various fields of science and engineering. By using fractional derivatives a lot of works have been done for a better description of considered material properties. Based on enhanced rheological models Mathematical modeling naturally leads to differential equations of fractional order and to the necessity of the formulation of the initial conditions to such equations. Recently, various analytical and numerical methods have been employed to solve linear and nonlinear fractional differential equations. The differential transform method (DTM) was proposed by Zhou [] to solve linear and nonlinear initial value problems in electric circuit analysis. This method has been used for solving various types of equations by many authors [-4]. DTM constructs an analytical solution in the form of a polynomial and different from the traditional higher order Taylor series method. For solving two-dimensional linear and nonlinear partial differential equations of fractional order DTM is further developed as Generalized Differential Transform Method (GDTM) by Momani, Odibat, and Erturk in their papers [5-7]. Recently, Vedat Suat Ertiirka and Shaher Momanib applied generalized differential transform method to solve fractional integro-differential equations [8]. The GDTM is implemented to derive the solution of space-time fractional telegraph equation by Mridula Garg,Pratibha Manohar and Shyam L.Kalla [9]. Manish Kumar Bansal,Rashmi Jain applied generalized differential transform method to solve fractional order Riccati differential equation []. Aysegul Cetinkaya, Onur Kiymaz and Jale Camli applied generalized differential transform method to solve non linear PDE s of fractional order [].. Mathematical Preliminaries on Fractional Calculus Many definitions of fractional calculus have been developed to solve the problems of fractional differential equations. The most frequently encountered definitions include Riemann-Liouville, Caputo, Wely, Rize fractional operator. Introducing the following definitions [, 3] in the present analysis:. Definition: Let α R +. The integral operator I α defined on the usual Lebesgue space L (a, b) by x d f (x) I α f(x) = x t f (t)dt dx ( ) I f x = f x, for x a, b is called Riemann-Liouville fractional integral operator of order α ( ). It has the following properties: (i) I α f(x)exists for any xε a, b ii I α I β f x = I α+β f(x) (iii) I α I β f x = I β I α f x (iv) I α x γ = Γ(γ+) Γ(α+γ+) xα+γ, where f x εl a, b, α, β, γ > 7 IJLERA All Right Reserved 7 Page

2 International Journal of Latest Engineering Research and Applications (IJLERA) ISSN: Volume, Issue, December 7, PP 7-3. Definition: The Riemann-Liouville definition of fractional order derivative is RL D α x f x = dn n α I dx n x f x = d n x x t n α f(t) dt, Γ(n α) dx n where n is an integer that satisfies α n, n.3 Definition: A modified fractional differential operator Dx α proposed by Caputo is given by c Dx α d n x n α f x = I x dx n f x = x t n α f n (t) dt, Γ(n α) where α R + is the order of operation and n is an integer that satisfies α n, n. It has the following two basic properties[4]: (i) If f L (a, b) or f C a, b and α > then c Dx α α I x f x = f(x) (ii) If f C n a, b and if α >,then α c I x D x α f x = f x D t α u x, t = α u(x, t) t α = n f k ( + ) k= k! x k ; α n, n.4 Definition: For m being the smallest integer that exceeds α, the Caputo time-fractional derivative operator of order α >, is defined as in [5] m u x, ξ ξ m ; α = mεn Γ(m α) t t ξ m α m u(x, ξ) ξ m c dξ ; m α < m Relation between Caputo derivative and Riemann-Liouville derivative: RL m f k ( + ) c Dt α f x = D α t f x k= x k α ; α m, m. Γ(k α+) Integrating by parts, we get the following formulae as given in [6]: b c (i) g(x) D a a x α b RL α f x dx = f x x RL α +j n n j b n RL D b g x dx + D a j = x b g x xd b f x a b c (ii) For n =, g x a Dx α b RL α f x dx = f x xd b g x dx + xib α b g x. f(x) a a a 3. Generalized two dimensional differential transform method Consider a function of two variables u x, y be a product of two single-variable functions, i.e. u x, y = f x g(y), which is analytic and differentiated continuously with respect to x and y in the domain of interest. Then the generalized two-dimensional differential transform of the function u x, y is given by [6-8] β U α,β k, = D Γ αk + Γ β+ y u(x, y) x,y where < α, β ; U α,β k, = F α k G β () is called the spectrum of u(x, y) and D α x, D α α x,., D x (k-times). α D k x (3.) The inverse generalized differential transform of U α,β k, is given by u x, y = k= = U α,β k, x x kα y y β (3.) It has the following properties: (i) if u x, y = v(x, y) ± w(x, y) then U α,β k, = V α,β k, ± W α,β k, (ii) if x, y = av(x, y),a R then U α,β k, = av α,β k, (iii) if u x, y = v(x, y)w(x, y) then U α,β k, = s= V α,β r, s W α,β k r, s (iv) if u x, y = x x nα y y mβ then U α,β k, = δ k n δ(h m) k r= (v) if u x, y = D α x v x, y, < α then U α,β k, = Γ α k+ + Γ αk+ V α,β k +, (vi) if u x, y = D γ y v x, y, < γ then U α,β k, = Γ βh+γ+ V Γ βh+ α,β k, + γ β α D k x = 7 IJLERA All Right Reserved 8 Page

3 International Journal of Latest Engineering Research and Applications (IJLERA) ISSN: Volume, Issue, December 7, PP 7-3 (vii) if ux, t vx, twx, tqx, t k kr h hs,,,, U k h V r h s p W t s Q k r t p,,,, r t s p (viii) if u( x, y) f ( x) g( y) and the function f ( x) x h( x) where, ( ) n n and n Taylor series expansion h( x) a x x (a) (b) 7 IJLERA All Right Reserved 9 Page then hx has the generalized and is arbitrary or, is arbitrary and a for n,,,... m, where m m. Then (3.) becomes h k U k, h Dx D, y u x y, k h (ix) if vx, y f x g y, the function ( ) ux, y Dx vx, y, m m then k U k, h V k, h, k, where U α,β k,, V α,β k,,w α,β k, and Q, k, h u x, y, v(x, y),w(x, y) and qx, y respectively and δ k n = ; k = n ; k n n x, y f x satisfies the conditions given in (viii) and are the differential transformations of the functions 4. Solution of Space-Time Fractional Non-linear Schrodinger Equation by Generalized Differential Transform Method In this section, we consider Space-Time Fractional Non-linear Schrodinger Equation in the form., u x, t u x t i p q u x, t u x, t t x u, t c (constant) subject to initial condition x where and t. The function, are the fractional differential operators(caputo derivative) of order (4.) u x t is a complex valued function of the spatial coordinates x and the time t. p and q are real parameters. Applying generalized two-dimensional differential transform (3.) with x, t =, on (4.) we obtain k h U, k, h i U, k, h p k h and k kr h hs q U r h s p U t s U k r t p r t s p,,,,,, (4.) U,, h c h W (4.3) Now utilizing the recurrence relation (4.) and the initial condition (4.3), we obtain after a little simplification the following values of U α,α k, for k =,,, and =,,,3

4 International Journal of Latest Engineering Research and Applications (IJLERA) ISSN: Volume, Issue, December 7, PP 7-3 U,, h h W U,, ic q c c p ; U,, ic 3q c c; p 3 U,, ic 6q c c; p 4 U,,3 ic q c c; U, 3, h h W ; p 3 U, 4, 3 4 i p p ic q c c q c ic p qc c c c p ic qc c and so on Now, from (3.), we have k h u x, t U, k, h x t k h Using the above values of U α,α k, in (4.4), the solution of (4.) is obtained as u x 3 4, t t t t t c 3 ic q c c ic p p q c c t p ic q c c t ic p 3 q c c t x 3 4 i p p ic q c c ; (4.4) 3 q 4... p c ic qc c c c p ic qc c x 5. Conclusions In the present study, we have applied the Generalized Differential Transform Method (GDTM) to find the approximate analytic solution of Space-Time Fractional Non-linear Schrodinger Equation. It may be concluded that GDTM is a reliable technique to handle linear and nonlinear fractional differential equations. Compared with other approximate methods this technique provides more realistic series solutions. References (4.5) 7 IJLERA All Right Reserved 3 Page

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