Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods
EExact solutions of the space time-fractional Klein-Gordon equation withcubic nonlinearities using some methods
Ayten ¨Ozkan, Erdo˘gan Mehmet ¨Ozkan
Abstract
Recently, finding exact solutions of nonlinear fractional differential equations has attracted greatinterest. In this paper, the space time-fractional Klein-Gordon equation with cubic nonlinearitiesis examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitarywave ansatz method. Some solutions are also illustrated by the computer simulations. Besides,the modified Kudryashov method is used to construct exact solutions of this equation.
Keywords:
Space time fractional Klein-Gordon equation, ansatz method, modified Kudryashovmethod, exact solutions
1. Introduction
Fractional differential equations are generalization of differential equations . In recent years,non-linear fractional differential equations (FDEs) have achieved importance in various disciplinesand have become popular. Recently, the theory and applications of FDEs have been the focusof many studies since they appear frequently in various applications in mathematics, physics,biology, engineering, signal processing, systems identification, control theory, finance, fractionaldynamics, and have increasingly fascinated the attention of many scientists. FDEs have beenstudied and many researchers published books and articles in this field [1, 2, 3]. Many methodshave been introduced to obtain exact solutions of FDEs. For instance the first integral method[4, 5, 6, 7, 8], exp-function method [9, 10, 11], ( G /G ) expansion method [12, 13, 14], sub-equationmethod [15, 16], functional variable method [17, 18], trial equation method [19, 20].A dependable and powerful method called the ansatz method has been put forward to search fortraveling wave solutions of nonlinear partial differential equations by Biswas [21, 22] . Although thismethod has been used by many authors, the applications of this method are very low in nonlinearFDEs. The installation of exact and analytical traveling wave solutions of nonlinear FDEs isone of the most significant and basic duties in nonlinear science, because they will characterizemiscellaneous natural case such as vibrations, solitons and finite speed distribution. The Ansatzmethod is one of the efficient methods used to obtain exact soliton solutions of FDEs.The solitary wave study has made important progress recently. In mathematics and physics,a soliton or a solitary wave is a self-reinforcing single wave that moves at a constant velocity,while maintaining its shape. Solitons represent solutions of the class of largely weak nonlineardistributive partial differential equations associated with physical systems. This field of study hasrecently made a huge progress [21, 23, 24, 25, 26, 27, 28, 29, 31, 30]. In the present study, FDEswill be converted into integer-order differential equations by fractional complex transformation, Preprint submitted to Journal Title June 11, 2020 a r X i v : . [ n li n . S I] J un nd then various exact solutions will be obtained to determine singular soliton solutions, darksoliton solutions and bright soliton solutions [32, 33].One of the approaches that led to creating exact solutions of fractional differential equations isa modified version of the Kudryashov method [34]. The modified Kudryashov method is a powerfulsolution method for finding exact solutions of nonlinear partial differential equations (PDEs) inmathematical physics and biology. This method was first applied in fractional differential equationsby Ege and Misirli [35]. Recently, this method has gained considerable attention due to the abilityof PDEs to extract new complete solutions both in integer order and in fractional order [36, 37, 38].Nonlinear Klein - Gordon equations have important application areas in science and engineeringsuch as solid state physics, nonlinear optics and quantum field theory [39]. This equation is arelativistic field equation for scalar particles and is a relativistic generalization of the well-knownSchrÃűdinger equation. Despite other relative wave equations, the Klein-Gordon equation is themost frequently studied equation in quantum field theory, since it is used to describe particledynamics [40]. They have been studied by many researchers and various methods have beenused to solve them. Some of these studies can be listed as follows : Homotopy perturbationmethod [41], a semi-analytical method called fractional-reduced differential transformation methodwith the appropriate initial condition [42], modified Kudryashov method [43], fractional complextransformation, ( G /G ) and ( w/g ) expansion methods [44], the well-organized ansatz method [45], adirect analytic method [46], the modified expanded Tanh method [47]. In this study, ansatz methodand modified Kudryashov method are applied to find out several new explicit exact solutions ofthe space time-fractional KleinâĂŞGordon equation with cubic nonlinearities.
2. The modified Riemann-Liouville derivative and methodology of solution
With recent studies, it is well known that the dynamics of many physical processes are accu-rately described using FDEs having different kinds of fractional derivatives. The most popular onesare the Caputo derivative, the Riemann-Liouville derivative and Gr¨unwald-Letnikov derivative. Adifferent definition of the fractional derivative is given by Jumarie with a little modification ofthe Riemann-Liouville derivative. In [48], f : R → R , ω → f ( ω ) as a continuous function (notnecessarily differentiable), the modified Riemann-Liouville derivative of order α is given as follows D αω f ( ω ) = − α ) ddω R ω f ( τ ) − f (0)( ω − τ ) α dτ , < α < f ( n ) ( ω )) ( α − n ) , n ≤ α ≤ n + 1 , n ≥ . ) is the Gamma function. In addition, some important properties of the fractional mod-ified Riemann-Liouville derivative (mRLd) are listed as follows [49]: D αω ω γ = Γ(1 + γ )Γ(1 + γ − α ) , γ > , (2.2) D αω ( c ) = 0 ( c constant) , (2.3) D αω ( af ( ω ) + bg ( ω )) = aD αω f ( ω ) + bD αω g ( ω ) , (2.4)where a = 0 and b = 0 are constants.Now, we will take into account the following nonlinear space-time FDE of the type H ( u, D αt u, D αx u, D αtt u, D αxx u, D αt D αx u... ) = 0 , < α < u is an unknown functions, H is a polynomial of u and its partial fractional derivatives, and α is order of the mRLd of the function u = u ( x, t ).The traveling wave transformation is u ( x, t ) = U ( ε ) ,ε = kx α Γ(1 + α ) − ct α Γ(1 + α ) , (2.6)with k = 0 and c = 0 are constants. We use the chain rule D αt u = σ t ∂U∂ε D αt ε,D αx u = σ x ∂U∂ε D αx ε, (2.7)with σ t , σ x are sigma indexes [50] and they can be σ t = σ x = L , where L is a constant.Substituting (2.6) and applying (2.2) and (2.7) to (2.5), we get following nonlinear ODE N ( U, dUdε , d Udε , d Udε , ... ) = 0 . (2.8)
3. Modified Kudryashov method
Let the exact solution of (2.8) can be showed as follows U ( ε ) = a + a Q ( ε ) + ... + a N Q ( ε ) N , (3.1)where a i values ( i = 0 , , , ..., N ) are arbitrary constants to be found later, but a N = 0. Q ( ε ) hasthe form Q ( ε ) = 11 + dA ε (3.2)which is a solution to the Riccati equation Q ( ε ) = ( Q ( ε ) − Q ( ε )) lnA (3.3)where d and A are nonzero constants with A > A = 1. N is revealed by balancing thehighest order derivative and nonlinear terms in (2.8). Substituting (3.1) into (2.8) and comparingthe results of the terms with a series of nonlinear equations, new exact solutions will be taken for(2.5).
4. Applications
We consider the space-time fractional Klein-Gordon equation of the form D αtt u − a D αxx + b u − λu = 0 , ( t > , < α ≤ , (4.1)where a, b, λ are constants [46]. The bright, dark and singular soliton solutions will be applied to thesolitary wave ansatz method. In order to solve Eq.(4.1), using the traveling wave transformation(2.6), we obtain to an ODE L ( a k − c ) U − b U + λU = 0 , (4.2)with U = dUdε . 3 .1.1. The bright soliton solution For the bright soliton solution, we let
A, k and, c be abritrary constants. Then suppose U ( ε ) = A sech p ( ε ) , (4.3)where ε = kx α Γ(1 + α ) − ct α Γ(1 + α ) . (4.4)It follows from ansatz (4.3) and (4.4) that d Udε = Ap sech p ( ε ) − Ap ( p + 1)sech p +2 ( ε ) , (4.5)and U = A sech p ( ε ) . (4.6)Substituting the ansatz (4.3)-(4.6) into (4.2), the following equation is obtained L ( a k − c ) Ap sech p ( ε ) − L ( a k − c ) Ap ( p + 1)sech p +2 ( ε ) − b A sech p ( ε )+ λA sech p ( ε ) = 0 . (4.7)From (4.7), we suppose the exponents p + 2 and 3 p are equal and from that p is determined as 1.When this value is placed in (4.7), it is reduced to the following equation L ( a k − c ) A sech( ε ) − L ( a k − c ) A sech ( ε ) − b A sech( ε ) + λA sech ( ε ) = 0 . (4.8)From (4.8), we obtain the following system of algebraic equations ( λA − L ( a k − c ) = 0 ,L ( a k − c ) − b = 0 . Solving this system, we get A = ∓ s L ( a k − c ) λ ,c = ∓ s L a k − b L . (4.9)Finally, we obtain the bright soliton solution for the Fractional Klein-Gordon as follows u ( x, t ) = ∓ s L ( a k − c ) λ sech (cid:16) kx α Γ(1 + α ) ∓ s L a k − b L t α Γ(1 + α ) (cid:17) . (4.10)The solution (4.10) is displayed in Figure 1, in the interval 0 < x <
10 and 0 < t < igure 1: The solution u ( x, t ) for equation (4.10) when a = 2 , k = 1 , b = 1 , L = 1 , λ = 1. .1.2. The dark soliton solution To obtain dark soliton solution , suppose that U ( ε ) = A tanh p ( ε ) , (4.11)where ε = kx α Γ(1 + α ) − ct α Γ(1 + α ) , (4.12)which k, c and A are nonzero constant coefficients. From ansatz (4.11) and (4.12), we get d Udε = Ap ( p − p − ( ε ) − Ap tanh p ( ε ) + Ap ( p + 1)tanh p +2 ( ε ) , (4.13)and U = A tanh p ( ε ) . (4.14)Thus, substituting the ansatz (4.11)-(4.14) into (4.2), it is achieved L ( c − a k )[ Ap ( p − p − ( ε ) − Ap tanh p ( ε ) + Ap ( p + 1)tanh p +2 ( ε )]+ b A tanh p ( ε ) − λA tanh p ( ε ) = 0 . (4.15)From (4.15), equating exponents p +2 and 3 p , that gives rise to p =1. By using this value, Eq.(4.15) reduces to L ( c − a k )[ − A tanh( ε ) + 2 A tanh ( ε )] + b A tanh( ε ) − λA tanh ( ε ) = 0 . (4.16)From (4.16), we find the algebraic system2 L ( c − a k ) − λA = 0 , − L ( c − a k ) + b = 0 . (4.17)Solving the system (4.17) A = ∓ s L ( c − a k ) λ ,c = ∓ s b + 2 L a k L . (4.18)Finally, we get the dark soliton solution for the Fractional Klein-Gordon as follows: u ( x, t ) = ∓ s L ( c − a k ) λ tanh (cid:16) kx α Γ(1 + α ) ∓ s b + 2 L a k L t α Γ(1 + α ) (cid:17) . (4.19)The solution (4.19) is displayed in Figure 2, in the interval 0 < x <
10 and 0 < t < igure 2: The solution u ( x, t ) for equation (4.10) when a = 2 , k = 1 , b = 1 , L = 1 , λ = 1. .1.3. The singular soliton solution In finding singular soliton solution we assume U ( ε ) = A csch p ( ε ) , (4.20)with ε = kx α Γ(1 + α ) − ct α Γ(1 + α ) , (4.21)where k, c and A are nonzero constant coefficients. From ansatz (4.20) and (4.21), we find d Udε = Ap csch p ( ε ) + Ap ( p + 1)csch p +2 ( ε ) , (4.22)and U = A csch p ( ε ) . (4.23)Substituting ansatz (4.20)-(4.23) into (4.2), yields L ( c − a k ) Ap csch p ( ε ) + L ( c − a k ) Ap ( p + 1)csch p +2 ( ε ) + b A csch p ( ε ) − λA csch p ( ε ) = 0 . (4.24)In (4.24), when equating exponents p +2 and 3 p , leads p =1. Similarly using p = 1, equation (4.24)reduces to L ( c − a k ) A csch( ε ) + 2 L ( c − a k ) A csch ( ε ) + b A csch( ε ) − λA csch ( ε ) = 0 . (4.25)From (4.25), we find the algebraic equation system ( L ( c − a k ) − λA = 0 ,L ( c − a k ) + b = 0 . Solving this system, we get A = ∓ s L ( c − a k ) λ ( c − a k > , λ < ,c = ∓ s L a k − b L ( L a k − b > . (4.26)Finally, we find the singular soliton solution for the Fractional Klein-Gordon as follows u ( x, t ) = ∓ s L ( c − a k ) λ csch (cid:16) kx α Γ(1 + α ) ∓ s L a k − b L t α Γ(1 + α ) (cid:17) . (4.27)The solution (4.27) is displayed in Figure 3, in the interval 0 < x <
10 and 0 < t < igure 3: The solution u ( x, t ) for equation (4.27) when a = 2 , k = 1 , b = 1 , L = 1 , λ = − .2. Application of modified Kudryashov method to space time fractional Klein-Gordon equation We consider the space-time fractional Klein-Gordon equation of the form (4.1). In order to solveEq.(4.1), using the traveling wave transformation (2.6), we obtain to an ODE L ( c − a k ) U + b U − λU = 0 , (4.28)with U = dUdε . The balance of U and U gives N = 1. Therefore, we have U ( ε ) = a + a Q ( ε ) , a = 0 . (4.29)Substituting the solution (4.29) and its derivative into (4.28) gets (cid:16) a L ( c − a k )( lnA ) − λa (cid:17) Q ( ε ) − (cid:16) a L ( c − a k )( lnA ) + λa a (cid:17) Q ( ε )+ (cid:16) a L ( c − a k )( lnA ) + b a − λa a (cid:17) Q ( ε ) + b a − λa = 0 . (4.30)Equating the coefficients of each power of Q ( ε ) and the constant term to zero, solving the resultingsystem of algebraic equations, we get the following solutions.Case 1: a = − bλ q λ , a = 2 b r λ , c = ∓ q ( lnA ) a k L + 2 b (4.31)Substuting (4.31) into (4.29), we have U ( ε ) = − bλ q λ + 2 b r λ (cid:16)
11 + dA ε (cid:17) , ( λ > . (4.32)Finally, we obtain the exact solution of (4.1) u ( x, t ) = − bλ q λ + 2 b r λ (cid:16)
11 + dA kxα Γ(1+ α ) ∓ √ ( lnA )2 a k L b tα Γ(1+ α ) (cid:17) , ( λ > . (4.33)Case 2: a = bλ q λ , a = − b r λ , c = ∓ q ( lnA ) a k L + 2 b (4.34)Substuting (4.34) into (4.29), we get U ( ε ) = bλ q λ − b r λ (cid:16)
11 + dA ε (cid:17) , ( λ > . (4.35)Finally, we obtain the exact solution of (4.1) u ( x, t ) = bλ q λ − b r λ (cid:16)
11 + dA kxα Γ(1+ α ) ∓ √ ( lnA )2 a k L b tα Γ(1+ α ) (cid:17) , ( λ > . (4.36)10 . Conclusion In this article, the space time-fractional Klein-Gordon equation with cubic nonlinearities isinvestigated for soliton and exact solutions. Complex fractional transformation is utilized to attainthe nonlinear ODE from this equation. Bright, dark and singular soliton solutions are obtained withsolitary wave ansatz method. Moreover, some exact solutions are found with modified Kudryashovmethod. The results are proof that these methods are accurate and effective. In addition, graphsof all soliton solutions are drawn for the appropriate coefficients.
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