Topological 1-soliton solutions to some conformable fractional partial differential equations

Transcription

1 Toological 1-soliton solutions to some conformable fractional artial differential equations Gökhan Koyunlu arxiv: v2 [nlin.si] 8 Se 2017 Deartment of Comuter Engineering Nile University of Nigeria. Abuja, Nigeria. Setember 11, 2017 Abstract Toological 1-soliton solutions to various conformable fractional PDEs in both one and more dimensions are constructed by using simle hyerbolic function ansatz. Suitable traveling wave transformation reduces the fractional artial differential equations to ordinary ones. The next ste of the rocedure is to determine the ower of the ansatz by substituting it into the ordinary differential equation. Once the ower is determined, if ossible, the ower determined form of the ansatz is substituted into the ordinary differential equation. Rearranging the resultant equation with resect to the owers of the ansatz and assuming the coefficients are zero leads to an algebraic system of equations. The solution of this system gives the relation between the arameters used in the ansatz. Keywords: Conformable derivative, fractional modified EW equation, fractional Klein-Gordon equation, fractional otential Kadomtsev-Petviashvili equation, toological 1-soliton solution. PACS: Jr, Wz, Fg. AMS2010: 5C07;35R11;35Q53. g.koyunlu@nileuniversity.edu.ng 1

2 1 Introduction In the literature of fractional artial differential equations, several definitions are used to construct the equations. One of the recent definitions is the conformable fractional derivative. The main descrition and some useful roerties of this derivative is given in the next section. This new definition of the fractional derivative has been used to derive new forms, robably a more general ones, of the nonlinear artial differential equations. Thus, exact solutions of these equations have become more significant in both theoretical studies and real world alications. There exist lenty of techniques in the literature to solve nonlinear artial differential equations. Lately, most of them have been adated to solve fractional artial differential equations when ossible. When the exact solutions to nonlinear artial differential equations are examined, one recognize that a method may be sufficient to generate an exact solutions to some equations but does not guarantee exact solutions to all nonlinear artial differential equations. The non linearity is a general concet and characteristics of each non linearity can be comletely different from the other. This ersective of non linearity forces researchers to imlement the known methodsof solution to each non linear artial differential equations. In the literature, there are many classical methods to solve non linear artial differential equations to set u exact solutions of various tyes. Simle equation methods [1 4], first integral method [5 7], variations of Kudryashov methods [8 10], (G /G-)exansion methods [11 13],and various hyerbolic function ansatz techniques [14 16] can be listed as some of well known methods to develo exact solutions to non linear artial differential equations. These techniques can also be imlemented to fractional artial differential equations for some tyes of equations. This study aims to imlement hyerbolic tangent ansatz methods to develo toological 1-soliton solutions to some conformable time fractional artial differential equations in one and two sace dimensions, namely the fractional modified Equal- Width (fmew), the fractional Klein-Gordon (fkg), and the fractional otential Kadomtsev-Petviashvili (fkp) equations. All the fractional derivatives used in these equations are chosen as newly defined conformable derivative. In order not to make restrictions in sace domain, the only time fractional forms of the equations are considered throughout the study. Before setting u the toological 1-soliton solutions, some significant roerties of conformable fractional derivative are briefly described in the next section. The considered equations are described briefly and toological 1-soliton solutions are develoed in the following sections. 2

3 2 Conformable Derivative The conformable fractional derivative definition is given by Khalil [17] as D α (u(t)) = lim h 0 u(t + ht 1 α ) u(t) h (1) where α (0, 1] and u : [0, ) R in the half sace t > 0. This fractional derivative suorts lenty of roerties given below under the assumtions that the order is α (0, 1] and that u = u(t) and v = v(t) are sufficiently α-differentiable for all t > 0. Then, D α (a 1 u + a 2 v) = a 1 D α (u) + a 2 D α (v) D α (t k ) = kt k α, k R D α (λ) = 0, for all constant λ D α (uv) = ud α (v) + vd α (u) D α ( u v ) = vdα (u) ud α (v) v 2 D α 1 α du (u)(t) = t dt for a 1, a 2 R [18,19]. The conformable derivative gives suort to Lalace transformations, exonential function roerties, chain rule, Taylor Series exansion, etc. [20]. Probably the most useful roerty indicates the relation between the conformable derivative and common derivative. Theorem 1. Let u be an α-conformable differentiable function, and v is also differentiable function defined in the range of u. Then, D α (u v)(t) = t 1 α v (t)u (v(t)) (2) 3 The Main Frame of Method Consider a fractional order artial differential equation in a general form F 1 (u, Dt α u, u x, Dt α u x, u xx,...) = 0 (3) The traveling wave transformation u(x, y, t) = U(ξ), ξ = ax + by ν α tα (4) 3

4 reduces the fractional artial differential equation (3) to F 2 (U, U, U,...) = 0 (5) where denotes the ordinary derivative. In the traveling wave transformation (4), a, b, and ν are assumed as constants. The next ste of the method is to suose that (5) has a solution of the form U(ξ) = A tanh B ξ (6) where A 0 and B Z + are constants to be determined. Substituting this solution into (5) and rearranging the resultant algebraic form for the owers of tanh function gives an equation. The ower constant B is determined by equating the owers of the tanh function, if ossible. After determination of B, the solution (6) is substituted into (5) with writing the value of B. Rearranging the resultant equation for owers of tanh and equating the coefficients of them to zero give an algebraic equation system to be solved for A, ν, a, b. Note that when the dimension of sace domain of u is 2, then, one of a, b is removed from the wave transformation (4). Similarly, the wave transformation (4) is modified in a comatible form while studying in one sace dimension. 4 Toological 1-Soliton Solutions to Time Fractional Modified EW (fmew) Equation The fmew equation is given as D α t u(x, t) + u(x, t)u x (x, t) + qd α t u xx (x, t) = 0 (7) where Dt α is the αth order conformable fractional derivative and the subscrits denote the ordinary derivative. The traveling wave transformation (4) for b = 0 converts (7) to νu + au 2 U qνa 2 U = 0 (8) Integrating this equation once leads νu au 3 qνa 2 U = K (9) where K is constant of integration. Substituting the hyerbolic tangent ansatz tanh B (ξ) into (9) gives 1 3 A3 a tanh 3 B ξ + ( 2 ν qa 2 AB 2 ν A ) tanh B ξ + ( ν qa 2 AB 2 + ABa 2 ν q ) tanh B 2 ξ + ( ν qa 2 AB 2 ABa 2 ν q ) tanh B+2 ξ = K (10) 4

5 Equating the owers 3B = B + 2 results B = 1. Thus, substituting tanh ξ into (9) results ( 1/3 aa 3 2 a 2 ν Aq ) tanh 3 ξ + ( 2 a 2 ν Aq ν A ) tanh ξ K = 0 (11) Since tanh ξ is a nonzero solution,then, the coefficients of tanh ξ and tanh 3 ξ should be zero in addition to the zero integration constant, K = 0. Thus, the solution of the system of algebraic equations 1/3 aa 3 2 a 2 ν Aq = 0 2 a 2 ν Aq ν A = 0 (12) gives and 2 q 1 ν q A = ± 3 a = 1/2 2 (13) q 1 2 q 1 ν q A = ± ±3 a = 1/2 2 (14) q 1 for arbitrarily chosen ν. These values of A and a gives several solutions to (9) as 2 q 1 ν q U 1,2 (ξ) = ± 3 tanh ξ (15) 2 q 1 ν q U 3,4 (ξ) = ± 3 tanh ξ Returning the original variables makes the solutions ( 2 q 1 ν q u 1,2 (x, t) = ± 3 tanh 1/2 2 ) q 1 x ν tα α ( 2 q 1 ν q u 3,4 (x, t) = ± 3 tanh 1/2 2 ) q 1 x ν tα α (16) for 0 and q 0. The grahical illustration of u 1 (x, t) for articular choices of, q and ν is given in Fig 1(a)-1(d) for various values of α. 5

6 (a) α = 0.25 (b) α = 0.50 (c) α = 0.75 (d) α = 1.00 Figure 1: Illustration of the solutions u 1 (x, t) for = q = 1 and ν = 3 6

7 5 Toological 1-Soliton Solutions to Time Fractional Klein-Gordon Equation The conformable fractional Klein-Gordon equation can transformed into D 2α t u u xx u qu 3 = 0 (17) ν 2 U a 2 U U qu 3 = 0 (18) via the comatible wave transformation (4) for ξ = ax ν tα α. Substituting tanhb ξ into 18 gives A 3 q tanh 3 B ξ + ( AB 2 a 2 + AB 2 ν 2 + ABa 2 ABν 2) tanh B 2 ξ + ( 2 AB 2 a 2 2 AB 2 ν 2 A ) tanh B ξ + ( AB 2 a 2 + AB 2 ν 2 ABa 2 + ABν 2) tanh B+2 ξ = 0 (19) Equating 3B = B + 2 gives B = 1. When the solution tanh ξ is substituted into (18), it takes the form ( A 3 q 2 Aa Aν 2) tanh 3 ξ + ( 2 Aa 2 2 Aν 2 A ) tanh ξ = 0 (20) When the coefficients are equated to zero, the algebraic system A 3 q 2 Aa Aν 2 = 0 2 Aa 2 2 Aν 2 A = 0 (21) Solving this system for A and a A = ± q ν (22) a = ± Thus, the solutions to (18) is determined as U 5,6 (ξ) = ± tanh(ξ) (23) q Therefore, toological 1-soliton solutions to the fkg equation become u 5,6,7,8 (x, t) = ± q tanh(± ν x ν tα α ) (24) 7

8 where q 0. The grahical illustration of u 5 (x, t) for various choices of, q and ν is given in Fig 2(a)-2(d) for different values of α. (a) α = 0.25 (b) α = 0.50 (c) α = 0.75 (d) α = 1.00 Figure 2: Illustration of the solutions u 5 (x, t) for = 1, q = 1 and ν = 2 8

9 6 Toological 1-Soliton Solutions to the fractional otential Kadomtsev-Petviashvili (fkp) Equation The time fractional otential Kadomtsev-Petviashvili (fkp) equation of the form D α t u x + u xx u x + qu xxxx + ru yy = 0 (25) is reduced to the ordinary differential equation νau + a 3 U U + qa 4 U + rb 2 U = 0 (26) by using the wave transformation (4). Assuming (26) has a solutions of the form tanh B ξ and substituting this solution into (26) gives ( 6 AB 4 a 4 q + 10 AB 2 a 4 q 2 AB 2 b 2 r + 2 AB 2 aν ) tanh B ξ + ( AB 4 a 4 q 6 AB 3 a 4 q + 11 AB 2 a 4 q 6 ABa 4 q ) tanh B 4 ξ + ( 4 AB 4 a 4 q + 12 AB 3 a 4 q 16 AB 2 a 4 q + 8 ABa 4 q + AB 2 b 2 r AB 2 aν ABb 2 r + ABaν ) tanh B 2 ξ + ( 4 AB 4 a 4 q 12 AB 3 a 4 q 16 AB 2 a 4 q 8 ABa 4 q + AB 2 b 2 r AB 2 aν + ABb 2 r ABaν ) tanh B+2 ξ + ( AB 4 a 4 q + 6 AB 3 a 4 q + 11 AB 2 a 4 q + 6 ABa 4 q ) tanh B+4 ξ + ( A 2 B 3 a 3 A 2 B 2 a 3 ) tanh 2 B 3 ξ + ( 3 A 2 B 3 a 3 + A 2 B 2 a 3 ) tanh 2 B 1 ξ + ( 3 A 2 B 3 a 3 + A 2 B 2 a 3 ) tanh 2 B+1 ξ + ( A 2 B 3 a 3 A 2 B 2 a 3 ) tanh 2 B+3 ξ = 0 (27) Equating the ower 2B 3 to B 2 gives B = 1. Thus, the redicted solution becomes tanh ξ. Substituting this solution into (26) gives ( 2 A 2 a Aa 4 q ) tanh 5 ξ + ( 4 A 2 a 3 40 Aa 4 q + 2 Ab 2 r 2 Aaν ) tanh 3 ξ + ( 2 A 2 a Aa 4 q 2 Ab 2 r + 2 Aaν ) tanh ξ = 0 (28) Since we seek a nonzero solution, the only coefficients of the owers of tanh ξ should be zero. Therefore, the algebraic system of equations is obtained 2 A 2 a Aa 4 q = 0 4 A 2 a 3 40 Aa 4 q + 2 Ab 2 r 2 Aaν = 0 2 A 2 a Aa 4 q 2 Ab 2 r + 2 Aaν = 0 The solutions of this system for A, ν, a and b gives A = 12aq ν = 4a4 q + b 2 r a (29) (30) 9

10 for arbitrarily chosen a 0 and b where 0. Thus, the solution to (26) is constructed as U(ξ) = 12aq tanh ξ (31) Returning to the original variables gives the solution to the fkp as u 9 (x, y, t) = 12aq ( ) tanh ax + by 4a4 q + b 2 r t α a α where 0 and a 0. The grahical illustration of u (x, t) for various choices of, q, r, a and b is given in Fig 3(a)-3(d) for different values of α. (32) 10

11 (a) α = 0.25 (b) α = 0.50 (c) α = 0.75 (d) α = 1.00 Figure 3: Projection of the solution u 9 (x, t) for = q = a = 1, r = 1 and y = 0. 11

12 7 Conclusion In the resent study, a simle hyerbolic tangent ansatz is used to derive toological 1-soliton solutions to some one and two dimensional fractional nonlinear artial differential equations. The conformable derivative suorts the chain rule. Comatible traveling wave transformation reduces the fractional artial differential equations to some ordinary differential equations. The hyerbolic tangent ansatz is a redicted solution and includes some arameters to be determined in an order. The first arameter to be determined is the ositive integer ower arameter. The determination of this arameter is followed by the other arameters in the solution by solving some algebraic equation systems. The time fractional modified EW, Klein-Gordon and otential Kadomtsev-Petviashvili equations are solved by using the hyerbolic tangent ansatz. Exlicit solutions are derived and grahical illustrations are reresented in 3D lots by assist of comuter. References [1] Zayed, E. M., & Al-Nowehy, A. G. (2017). Solitons and other exact solutions for variant nonlinear Boussinesq equations. Otik-International Journal for Light and Electron Otics. [2] Roshid, H. O., Roshid, M. M., Rahman, N., & Pervin, M. R. (2017). New solitary wave in shallow water, lasma and ion acoustic lasma via the GZK- BBM equation and the RLW equation. Proulsion and Power Research. [3] Kalan, M., & Bekir, A. (2016). The modified simle equation method for solving some fractional-order nonlinear equations. Pramana, 87(1), 1-5. [4] Kalan, M., Bekir, A., Akbulut, A., & Aksoy, E. (2015). The modified simle equation method for nonlinear fractional differential equations. Romanian J. Phys, 60(9-10), [5] Eslami, M., Mirzazadeh, M., Vajargah, B. F., & Biswas, A. (2014). Otical solitons for the resonant nonlinear Schrödinger s equation with timedeendent coefficients by the first integral method. Otik-International Journal for Light and Electron Otics, 125(13), [6] Lu, B., Zhang, H., & Xie, F. (2010). Travelling wave solutions of nonlinear artial equations by using the first integral method. Alied Mathematics and Comutation, 216(4), [7] Bekir, A., & Ünsal, Ö. (2012). Analytic treatment of nonlinear evolution equations using first integral method. Pramana, 79(1),

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