On The Prototype Solutions of Symmetric Regularized Long Wave Equation by Generalized Kudryashov Method

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1 Mathematics Letters 5; (): -6 Published online December, 5 ( doi: 648/jml5 On The Prototype Solutions of Symmetric Regularized Long Wave Equation by Generalized Kudryashov Method Hasan ulut, Haci Mehmet asonus, Eren Cüvele Department of Mathematics, Firat University, Elazig, Turey Department of Computer Engineering, Tunceli University, Tunceli, Turey address: hbulut@firatedutr (H ulut), hmbasonus@gmailcom (H M asonus), erencuvele@gmailcom (E Cüvele) To cite this article: Hasan ulut, Haci Mehmet asonus, Eren Cüvele On The Prototype Solutions of Symmetric Regularized Long Wave Equation y Generalized Kudryashov Method Mathematics Letters Vol, No, 5, pp -6 doi: 648/jml5 Abstract: In this study, we have applied the generalized udryashov method to the symmetric regularized long wave equation for obtaining some new analytical solutions such as trigonometric function solution, eponential function solution, complel function solution, hyperbolic function solution after giving the fundamental properties of method Afterwards, we have observed that these analytical solutions are verified the symmetric regularized long wave equation by means of Wolfram Mathematica 9 Then, we have drawn two and three dimensional surfaces of analytical solutions Finally, we have submitted a conclusion to literature Keywords: Generalized Kudryashov Method, Symmetric Regularized Long Wave Equation, Eponential Function Solution, Comple Function Solution, Hyperbolic Function Solution, Trigonometric Function Solution Introduction All over the world, some researchers have submitted to literature some analytical solutions of many nonlinear partial differential equations Many powerful methods such as the transformation techniques, the sine cosine technique, the eponential function method, the standard tanh and developed tanh methods, the jacobi function method, trial equation method, darbou transformation, homotopy perturbation method, sumudu transform method, udryashov method and so on have been applied successfully [-] When it comes to this paper, we have presented the general properties of generalized udryashov method (GKM) [] in section In section, we have applied GKM to the symmetric regularized long wave equation (SRLW) defined by [] u + u + uu + uu + u () tt t t tt Finally, we have submitted a conclusion including some important remars in section 4 General Properties of Generalized Kudryashov Method Recently, some authors have constructed the generalized udryashov method [] We consider the following nonlinear partial differential equation for a functionu of two real variables, space and timet : P u, ut, u, u, u, () The basic phases of the generalized udryashov method are epressed as the follows: Step First of all, we must get the travelling wave solution of Eq() as following form; ( ρ) u, t u, ρ ct, () where and c are arbitrary constants Eq() was converted into a nonlinear ordinary differential equation of the form: N ( u, u, u, u, ), () where the prime indicates differentiation with respect to ρ Step Suggest that the eact solutions of Eq() can be written as the following form; u ( ρ) N i aq i ρ A Q( ρ i ) M (4) j b Q Q( ρ ) j j ρ

2 Hasan ulut et al: On The Prototype Solutions of Symmetric Regularized Long Wave Equation y Generalized Kudryashov Method whereq is We note that the function Q is solution of equation [] ± e ρ Q Q Q ρ (5) Taing into consideration Eq(4), we obtain AQ AQ u ( ρ) ( Q Q) A A A A Q Q Q u Q A A Q Q + [ ( A A ) A + A ]], ( ρ) [( )( ) ( Q Q) u ( ρ) ( Q Q) ( Q ) + ( Q Q)( Q Q ) ( ) A A A A 6( A + A ) 6A( ) + 4 ( ) A A A + A( ) A A , (6) (7) (8) Step Under the terms of proposed method, we suppose that the solution of Eq() can be eplained in the form of the following: u ( ρ) N a + a Q + a Q + + a N Q + M b + bq + b Q + + b M Q + (9) To calculate the values M and N in Eq(9) that is the pole order for the general solution of Eq(), we progress conformably as in the classical udryashov method on balancing the highest order nonlinear terms in Eq() and we can determine a formula of M and N We can receive some values of M and N Step 4 Replacing Eq(4) into Eq() provides a polynomial R ( Q) of Q Establishing the coefficients of R ( Q) to zero, we acquire a system of algebraic equations Solving this system, we can describe ρ and the variable coefficients ofa, a, a,, an, b, b, b,, bm In this way, we attain the eact solutions to Eq() Implementation of Method Proposed For symmetric regularized long wave equation [] u + u + uu + uu + u, () tt t t tt with Eq() transformation, nonlinear differential form for Eq() is obtained as the following; u and c + + () c u u cu u terms is used according to balance principle And then, balance term is obtained as the following; N M+ () Case : ForM andn when Eq(7) is rewritten with Eq(4), we can find follows; u ( ρ) N i aq i ρ A Q( ρ i ) M j, (4) b Q Q ρ j j ρ Q Q u Q A A Q Q + [ ( A A ) A + A ]], ( ρ) [( )( ) (5) wherea, b, If Eq(4) and Eq(5) are considered in Eq(), a new equation is occurred withq Solving this equation with Mathematica 9, it yields us the following coefficients; Case- b (6 b b) a, a, + + ( b + b) b a, a, + + c + (6) If Eq(6) is written in Eq(4), we can obtain the hyperbolic function solution as the following: t ( + sec h( ( + ))) + u( t,) + (7)

3 Mathematics Letters 5; (): Figure Three dimensional surfaces of (9) the being comple trigonometric function solutions fort Figure Two and three dimensional surfaces of (7) hyperbolic function solutions for andt for two dimensional surfaces Case- 4 6 a, a 6i b, a 6i ( b b), i a 6i b,, c (8) If Eq(8) is written in Eq(4), we can obtain being comple trigonometric function solution as the following: i u( t,) t + cos( + i) (9) Figure Two dimensional surfaces of (9) being comple trigonometric function solutions fort Case- a, a 6i b, a 6i ( b + b), a i 6i b,, c ()

4 Hasan ulut et al: On The Prototype Solutions of Symmetric Regularized Long Wave Equation y Generalized Kudryashov Method If Eq() is written in Eq(4), we can obtain being comple trigonometric function solution as the following: 6 4 i u( t,) t + cos( i) () Figure 5 Two dimensional surfaces of () being comple trigonometric function solutions fort Case-4 a, a i b, a 6i b, a 6i b, b, ic, () If Eq() is written in Eq(4), we can obtain comple hyperbolic function solution as the following: i( + sec h( ( t+ i))) u 4 4( t,) () Figure 4 Three dimensional surfaces of () being comple trigonometric function solutions fort Figure 6 Three dimensional surfaces of () comple hyperbolic function solutions fort

5 Mathematics Letters 5; (): Imaginary Part Figure 8 Three dimensional surfaces of (5) comple eponential function solutions for, t 8584 Figure 7 Two dimensional surfaces of () comple hyperbolic function solutions fort Case-5 b ( b b) a, a, a, b a, c (4) If Eq(4) is written in Eq(4), we can obtain comple eponential function solution as the following: 5 t ( + ) e u t, t + ( e e ) (5) Figure 9 Two dimensional surfaces of (5) comple eponential function solutions for, t Case-6 a i b, a i ( 6 b + b), a 6i ( b b), (6) a 6i b, ic, If Eq(6) is written in Eq(4), we can obtain comple hyperbolic function solution as the following:

6 5 Hasan ulut et al: On The Prototype Solutions of Symmetric Regularized Long Wave Equation y Generalized Kudryashov Method i( sec h( ( t i))) u 4 6( t,) (7) Figure Two dimensional surfaces of (7) comple hyperbolic function solutions fort 4 Conclusions In this paper, symmetric regularized long wave equation has been solved by using GKM We have obtained the prototype solutions such as comple function, trigonometric function, eponential function and hyperbolic function solutions It has been observed that these analytical solutions have verified to the SRLW Eq() by using Wolfram Mathematica 9 These analytical solutions obtained by using GKM in this paper are new comple, trigonometric, eponential and hyperbolic function solutions when we compare these solutions with analytical solutions obtained by Jalil Manafian and Isa Zamanpour [] Then, two and three dimensional surfaces of these analytical solutions have been plotted by using Wolfram Mathematica 9 According to solutions and graphics, one can see that GKM is a powerful tool for obtaining some new analytical solutions for such problems Disclosure Statement No potential conflict of interest was reported by the authors References Figure Three dimensional surfaces of (7) comple hyperbolic function solutions fort 5 [] G K Watugala, Sumudu Transform: A New Integral Transform to Solve Differantial Equations and Control Engineering Problems, International Journal of Mathematical Education in Science and Technology, 99, 4, 5-4 [] Y Pandir, New eact solutions of the generalized Zaharov Kuznetsov modified equal-width equation, Pramana journal of physics, 4, 8(6), [] H ulut, H M asonus and F M elgacem, The Analytical Solutions of Some Fractional Ordinary Differential Equations by Sumudu Transform Method, Abstract and Applied Analysis, [4] A M Wazwaz, The tanh method: solitons and periodic solutions for Dodd-ullough-Mihailov and Tzitzeica- Dodd-ullough equations, Chaos, Solitons and Fractals, 5, 5, [5] CS Liu, A new trial equation method and its applications, Communications in Theoretical Physics,6, 45(), 95-97

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