In this paper, we discuss on the exact solutions of the nonlinear space-time fractional Burgerlike equation and also the nonlinear fractional fifth-order Sawada-Kotera equation with the expfunction method.We use the functional derivatives in the sense of Riemann-Jumarie derivative and fractional convenient variable transformation in this study. Further, we obtain some exact analytical solutions including hyperbolic function

Eskandari, Elahe M.

Taghizadeh, Nasir

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 15 Issue 2 Article 15 12-2020 Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method Elahe M. Eskandari University of Guilan Nasir Taghizadeh University of Guilan Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Ordinary Differential Equations and Applied Dynamics Commons, Other Physics Commons, and the Partial Differential Equations Commons Recommended Citation Eskandari, Elahe M. and Taghizadeh, Nasir (2020). Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method, Applications and Applied Mathematics: An International Journal (AAM), Vol. 15, Iss. 2, Article 15. Available at: https://digitalcommons.pvamu.edu/aam/vol15/iss2/15 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications and AppliedMathematics:An International Journal(AAM)Vol. 15, Issue 2 (December 2020), pp. 970 – 977Exact Solutions of Two Nonlinear Space-time Fractional DifferentialEquations by Application of Exp-function Method1Elahe Miri Eskandari and 2Nasir TaghizadehDepartment of MathematicsFaculty of Mathematical SciencesUniversity of GuilanP.O.Box 41335-19141Rasht, Iran1elahe-eskandari@phd.guilan.ac.ir; 2taghizadeh@guilan.ac.irReceived: February 29, 2020; Accepted: July 2, 2020AbstractIn this paper, we discuss on the exact solutions of the nonlinear space-time fractional Burger-like equation and also the nonlinear fractional fifth-order Sawada-Kotera equation with the exp-function method. We use the functional derivatives in the sense of Riemann-Jumarie derivative andfractional convenient variable transformation in this study. Further, we obtain some exact analyticalsolutions including hyperbolic function.Keywords: Exp-function method; Fractional differential equation; The Burger-like equation;The Sawada-Kotera EquationMSC 2010 No.: 34A08, 35R11, 83C151. IntroductionObvious, fractional differential equations (FDEs) have been the focus of many studies due to theirfrequent appearance in several applications in physics, biology, engineering, signal processing,systems identification, control theory, finance, fractional dynamics, and other different sciences(see Kilbas et al. (2006), Mille and Ross (1993), and Podlubny (1999)). The exact solution ofthe fractional differential equations have received tremendous attention of great, several inves-tigators have successfully used reliable and algebraic methods for this purpose (see Taghizadehet al. (2015, 2016)). However, there are several methods to obtain exact solutions of fractional9701Eskandari and Taghizadeh: Exact Solutions of Nonlinear Space-time Fractional DifferentialPublished by Digital Commons @PVAMU, 2020AAM: Intern. J., Vol. 15, Issue 2 (December 2020) 971differential Burger-like equation and fractional differential Sawada-Kotera equation; for example:the Sub-equation method, see Wang et al. (2017), the (G′/G) method, see Cerdid et al. (2019),the F-expansion method, see Inan et al.(2017), the exp(−φ(ξ)) method see Ali et al. (2016) andother methods, see Bulut et al. (2013), but we apply the exp-function method to nonlinear FDES’s,space-time fractional Burger-like equation, see Cerdid et al. (2019) and Inan et al. (2017) and time-fractional fifth-order Sawada-Kotera equation, see Ali et al. (2016) and Guner et al. (2017).The organization of the paper follows in the second and third sections of the paper we define theRiemann-Jumarie Derivative and we describe the exp-function method respectively. In Section 4and Section 5 application of the exp-function method for Burger-like equation and Sawada-Koteraequation respectively. In the end, the general conclusion is given in Section 6.2. The Riemann-Jumarie DerivativeAssume that f : R −→ R, x −→ f(x) denotes a continuous (but not nesseciary differentiable)function, then Riemann-Jumarie derivative (see Jumarie(2006, 2009)),Dαt f(t) ={1Γ(1−α)ddt∫ t0 (t− ξ)−α(f(ξ)− f(0))dξ, if 0 < α ≤ 1,(f (n)(t))(α−n), if n ≤ α < n+ 1, n ≥ 1.(1)Some important properties for the Riemann-Jumarie derivative of order α are listed below,Dαt tβ =Γ(1 + β)Γ(1 + β − α)tβ−α, (2)Dαt (f(t)g(t)) = g(t)Dαt f(t) + f(t)Dαt g(t), (3)Dαt f [g(t)] = f′g[g(t)]Dαt g(t) = Dαt f [g(t)](g′(t))α. (4)We consider the fractional partial differential equation, with independent variable t, x1, x2, ..., xnand dependent variable u :P (u,Dαt u,Dαx1u, ...,D2αt U, ...) = 0 0 < α ≤ 1, t > 0. (5)By using the fractional variable transformationu(t, x1, ..., xn) = u(ξ) and ξ =ctαΓ(1 + α)+k1xα1Γ(1 + α)+ ...+knxαnΓ(1 + α), (6)where c and ki, (i = 1, ..., n) are nonzero arbitrary constants. The fractional differential equation(5) reduced to a nonlinear ordinary differential equation (ODE)F (u, u′, u′′, ...) = 0, (7)where the prime denotes the derivation with respect to ξ and u′ = dudξ .2Applications and Applied Mathematics: An International Journal (AAM), Vol. 15 [2020], Iss. 2, Art. 15https://digitalcommons.pvamu.edu/aam/vol15/iss2/15972 E. M. Eskandari et al.3. The Exp-function methodIn the exp-function method, we assume that the solution is a function,u(ξ) =∑m2p=−m1 ap exp[pξ]∑n2q=−n1 aq exp[qξ], (8)where m1,m2, n1 and n2 are unknown positive integers that will be determined by using balancemethod, for constructing the relations between m2,m1, n2 and n1, first we consider the highestdegree and the highest derivative order of function u(ξ) in the nonlinear (ODE) (7). For relationbetween m2 and n2 by creating a balance between the first sentence of the highest degree and thefirst sentence of the highest derivative order. Similarly, the relation between m1 and n1 is obtainedby balancing the last sentence of the highest degree and the last sentence of the highest derivativeorder. Ebaid (2012) proved in Theorem 1 and Theorem 2 that m2 = n2 and m1 = n1 are the onlyrelations that can be obtained by applying the balancing method in the nonlinear (ODE) (7). In theexp-function method, additional calculations of balancing the highest order linear term with thehighest order nonlinear term are not longer required in future because we only use their first andlast sentences. Hence, the method becomes more straightforward.4. The space-time fractional Burger-like equationBurger’s equation is related to applications in acoustic phenomena and has been utilized to modelturbulence and certain steady-state viscous flows. We consider conformable space-time fractionalBurger-like equation (see Cerdid et al. (2019) and Inan et al. (2017)),Dαt u+Dαxu+ uDαxu+12D2αx u = 0, t > 0, 0 < α ≤ 1, (9)where α is a parameter that describes the order of the fractional time derivative. By using variabletransformationsu(x, t) = u(ξ), and ξ =ctαΓ(1 + α)+kxαΓ(1 + α), (10)where c and k are constants, Equation (9) is reduced to an ordinary defferential equation(c+ k)u′ + kuu′+k22u′′ = 0, (11)where u′ = dudξ .Integrating equation (11 ) with respect to ξ yields(c+ k)u+12ku2 +k22u′ = 0. (12)By using Equation (8) and Equation (12), we haveu′(ξ) =A exp[(n2 +m2)ξ] + ...+B exp[−(n1 +m1)ξ]C exp[2n2ξ] + ...+D exp[−2n1ξ], (13)u2(ξ) =E exp[2m2ξ] + ...+ F exp[−2m1ξ]G exp[2n2ξ] + ...+H exp[−2n1ξ], (14)3Eskandari and Taghizadeh: Exact Solutions of Nonlinear Space-time Fractional DifferentialPublished by Digital Commons @PVAMU, 2020AAM: Intern. J., Vol. 15, Issue 2 (December 2020) 973where A,B,C,D,E, F,G and H are coefficients determined by am2 , a−m1 , bn2 and b−n1 .By balancing the highest order Exp-function in u′ and u2 in Equation (12), we have n2 +m2 = 2m2,which implies m2 = n2. Similarly, balancing the lowest order exp-function in u′ and u2, we haven1 + m1 = 2m1, which implies m1 = n1. For simplicity, we set m2 = n2 = 1 and m1 = n1 = 1 soEquation (8) reduces tou(ξ) =a1 exp[ξ] + a0 + a−1 exp[−ξ]b1 exp[ξ] + b0 + b−1 exp[−ξ]. (15)Substitating Equation (15) into Equation (12) and the help of Maple, we have1A(R1 exp[2ξ] +R2 exp[ξ] +R3 +R4 exp[−ξ] +R5 exp[−2ξ]) = 0, (16)whereA = (b1 exp[ξ] + b0 + b−1 exp[−ξ])2,R1 = (c+ k)a1b1 +12ka21,R2 = (c+ k)[a0b1 + a1b0] + ka1a0 +12k2a1b0 −12k2a0b1,R3 = (c+ k)[a0b0 + 2a1b−1 + a−1b1] + k[a1a−1 + a−1b1 +12a20] + k2[a1b−1 − a−1b1],R4 = (c+ k)[a0b−1 + a−1b0] + kb−1a0 −12k2a−1b0 +12k2a0b−1,R5 = (c+ k)a−1b−1 +12ka2−1. (17)Solving this system of algebraic equation by using symbolic computation, we obtain the followingresults.If a1 = 0, a0 = 0 , a−1 = −2kb−1, b1 = b1, b0 = 0, b−1 = b−1, c = k2 − k and k = k.If we set b1 = 12 , b−1 = −12 and k = 2, then we have a−1 = −2 and c = 2, we obtain the followinghyperbolic function solitary solution,u(x, t) =2 exp[−2( xαΓ(1+α) +tαΓ(1+α))]sinh(2( −xαΓ(1+α) +tαΓ(1+α))). (18)5. The space-time fractional fifth-order Sawada-Kotera equationWe consider the space-time fractional fifth-order Sawada-Kotera equation (see Ali et al. (2016)and Guner et al. (2017)),Dαt u+D5αx u+ 45U2Dαxu+ 15(DαxuD2αx u+ uD3αx u) = 0, t > 0, 0 < α ≤ 1, (19)4Applications and Applied Mathematics: An International Journal (AAM), Vol. 15 [2020], Iss. 2, Art. 15https://digitalcommons.pvamu.edu/aam/vol15/iss2/15974 E. M. Eskandari et al.where α is a parameter describing the order of the fractional time-derivative. The function u(x, t) isassumed to be a causal function of time. As in the previous section, we use variable transformationsu(x, t) = u(ξ), and ξ =ctαΓ(1 + α)+kxαΓ(1 + α), (20)where c and k are constants, Equation (19) is reduced to an ordinary defferential equationcu′ + k5u(5) + 45ku2u′ + 15(k3u′u′′ + k3uu(3)) = 0, (21)where u′ = dudξ . Integrating Equation (21) with respect to ξ yields.cu+ k5u(4) + 15ku3 + 15k3(uu′′) = 0. (22)We have(u(ξ))(4) =I exp[(15n2 +m2)ξ] + ...+ J exp[−(15n1 +m1)ξ]K exp[16n2ξ] + ...+ L exp[−16n1ξ], (23)uu′′(ξ) =M exp[(3n2 + 2m2)ξ] + ...+N exp[−(3n1 + 2m1)ξ]P exp[5n2ξ] + ...+Q exp[−5n1ξ], (24)where I, J,K,L,M,N, P and Q are coefficients determined by am2 , a−m1 , bn2 and b−n1 .By the same approach as in before this section, balancing the terms u(4) and uu′′ in Equation (22)we have m2 = n2,m1 = n1. To simplicity, we set m2 = n2 = 1 and m1 = n1 = 1 so equation (8)degrades tou(ξ) =a1 exp[ξ] + a0 + a−1 exp[−ξ]b1 exp[ξ] + b0 + b−1 exp[−ξ]. (25)Substiuting Equation (25) into Equation (22), and by the help of Maple, we have1A(R1 exp[5ξ] +R2 exp[4ξ] +R3 exp[3ξ] +R4 exp[2ξ] +R5 exp[ξ] +R6+R7 exp[−ξ] +R8 exp[−2ξ] +R9 exp[−3ξ] +R10 exp[−4ξ] +R11 exp[−5ξ]) = 0,(26)where5Eskandari and Taghizadeh: Exact Solutions of Nonlinear Space-time Fractional DifferentialPublished by Digital Commons @PVAMU, 2020AAM: Intern. J., Vol. 15, Issue 2 (December 2020) 975A = (b1 exp[ξ] + b0 + b−1 exp[−ξ])5,R1 = ca1b41 + 15ka31b21,R2 = 15k3a1a0b31 + ca0b41 + 4ca1b31b0 − 15k3a21b21b0 + 45ka21a0b21 + k5a0b41 − k5a1b31b0 + 30ka31b1b0,R3 = 30ka31b1b−1 + 15ka31b20 + ca−1b41 + 45ka1a20b21 − 16k5a1b31b−1 + 15k3a20b31 − 60k3a21b21b−1+ 60k3a1a−1b31 + 16k5a−1b41 − 11k5a0b31b0 + 4ca0b31b0 + 90ka21a0b1b0 + 6ca1b21b20 + 45ka21a−1b21+ 4ca1b31b−1 − 15k3a1a0b21b0 + 11k5a1b21b20,R4 = −135k3a1a0b21b−1 + 77k5a1b21b0b−1 + 30ka31b0b−1 − 76k5a0b31b−1 + 4ca1b1b30 + 45ka21a0b20− 11k5a1b1b30 + 15k3a21b30 + 75k3a0a−1b31 + 6ca0b21b20 − k5a−1b31b0 − 30k3a21b1b−1b0+ 90k3a1a−1b21b0 + 12ca1b21b0b−1 + 11k5a0b21b20 + 15ka30b21 − 15k3a1a0b1b20 + 4ca−1b31b0+ 90ka21a−1b1b0 + 4ca0b31b−1 + 90ka21a0b1b−1 + 90ka1a0a−1b21 + 90ka1a20b1b0,R5 = 4ca0b1b30 + k5a1b40 − 60k3a1a−1b21b−1 + 60k3a1a−1b1b20 + 105k3a0a−1b21b0 − 58k5a1b1b20b−1+ 12ca0b21b−1b0 + 12ca1b1b20b−1 + 47k5a0b21b−1b0 + 90ka21a0b0b−1 − k5a0b1b30 + 45ka1a20b20+ 30ka30b1b0 − 15k3a20b1b20 + 15k3a0a1b30 + ca1b40 + 15ka31b2−1 + 60k3a2−1b31+ 180ka1a0a−1b1b0 − 150k3a1a0b1b−1b0 + 6ca1b21b2−1 + 4ca−1b31b−1 + 6ca−1b21b20+ 176k5a1b21b2−1 − 176k5a−1b31b−1 + 11k5a−1b21b20 + 45ka21a−1b20 + 45ka1a2−1b21 + 45ka20a−1b21+ 60k3a21b20b−1 − 75k3a20b21b−1 + 90ka21a−1b1b−1 + 90ka1a20b1b−1,R6 = 30k3a1a−1b30 + 230k5a0b21b2−1 + 45ka0a2−1b21 + 105k3a21b0b2−1 + 30ka30b1b−1 + 90ka21a−1b0b−1+ 90ka1a20b0b−1 + 90ka1a0a−1b20 + 90ka1a2−1b1b0 + 90ka20a−1b1b0 − 75k3a1a0b1b2−1+ 180ka1a0a−1b1b−1 − 60k3a1a−1b1b−1b0 − 115k5a1b1b0b2−1 − 115k5a−1b21b0b−1 + 12ca−1b21b0b−1+ 12ca1b1b0b2−1 − 10k5a0b1b−1b20 + 4ca1b30b−1 − 120k3a20b1b−1b0 + 12ca0b1b−1b20 + 45ka21a0b2−1+ 4ca−1b1b30 + 5k5a−1b1b30 + 5k5a1b30b−1 + 6ca0b21b2−1 + ca0b40 + 15k3a30b20 + 45k3a1a0b−1b20− 75k3a0a−1b21b−1 + 45k3a0a−1b1b20 + 105k3a2−1b21b0,R7 = −150k3a0a−1b1b−1b0 + 180ka1a0a−1b0b−1 + 4ca0b−1b30 − k5a0b−1b30 + 30ka30b0b−1 + 45ka20a−1b20+ 15k3a0a−1b30 − 15k3a20b−1b20 + 4ca1b1b3−1 + 6ca1b20b2−1 + 6ca−1b21b2−1 − 176k5a1b1b3−1 + 11k5a1b20b2−1+ 176k5a−1b21b2−1 + 45ka21a−1b2−1 + 45ka1a20b2−1 + 45ka1a2−1b20 − 75k3a20b1b2−1 + 60k3a2−1b1b20+ ca−1b40 + k5a−1b40 + 15ka3−1b21 + 60k3a21b3−1 − 58k5a−1b1b20b−1 + 12ca0b1b2−1b0 + 12ca−1b1b20b−1+ 47k5a0b1b2−1b0 + 90ka1a2−1b1b−1 + 90ka20a−1b1b−1 + 90ka0a2−1b1b0 − 60k3a1a−1b1b2−1+ 105k3a1a0b2−1b0 + 60k3a1a−1b20b−1,R8 = 4ca0b1b3−1 − 76k5a0b1b3−1 + 90k3a1a−1b0b2−1 + 90ka0a2−1b1b−1 − k5a1b0b3−1 + 12ca−1b1b0b2−1+ 90ka20a−1b0b−1 + 77k5a−1b1b0b2−1 − 15k3a0a−1b20b−1 + 6ca0b2−1b20 + 30ka3−1b1b0 − 30k3a2−1b1b−1b0+ 45ka0a2−1b20 − 11k5a−1b30b−1 + 15k3a2−1b30 + 4ca1b0b3−1 + 75k3a1a0b3−1 − 135k3a0a−1b1b2−1+ 90ka1a0a−1b2−1 + 11k5a0b2−1b20 + 90ka1a2−1b0b−1 + 15ka30b2−1 + 4ca−1b30b−1,R9 = 4ca−1b1b3−1 + 30ka3−1b1b−1 + 6ca−1b20b2−1 − 15k3a0a−1b0b2−1 + 90ka0a2−1b0b−1 + 15ka3−1b20+ 60k3a−1a1b3−1 + 45ka20a−1b2−1 − 16k5a−1b1b3−1 + 15k3a20b3−1 − 60k3a2−1b1b2−1 + 11k5a−1b20b2−1+ ca1b4−1 + 4ca0b3−1b0 − 11k5a0b3−1b0 + 16k5a1b4−1 + 45ka1a2−1b2−1,R10 = k5a0b4−1 + 30ka3−1b0b−1 − k5a−1b0b3−1 + ca0b4−1 + 45ka0a2−1b2−1 + 4ca−1b0b3−1,+ 15k3a−1a0b3−1 − 15k3a2−1b0b2−1R11 = ca−1b4−1 + 15ka3−1b2−1. (27)Solving the above algebraic equations by using Maple for a1, a0, a−1, b1, b0, b−1, c and k, we have6Applications and Applied Mathematics: An International Journal (AAM), Vol. 15 [2020], Iss. 2, Art. 15https://digitalcommons.pvamu.edu/aam/vol15/iss2/15976 E. M. Eskandari et al.the solution if a1 = 0, a0 = b0, a−1 = 0, b1 = b1, b0 = b0 and b−1 = 14b20b1, k = −1 and c = −1.We take b1 = 12 and b0 = 1,then a0 = 1 and b−1 =12 sou(x, t) =12sech2[12(−xαΓ(1 + α)+tαΓ(1 + α))]. (28)(a) u(x, t) of equation(18) (b) u(x, t) of equation(28)Figure 1. a. solution of Burger-like equation for α = 12 b. solution of Sawada-Kotera equation for α =126. ConclusionIn the present paper, the exact solution of some fractional differential equations is obtained by theexp-function method. Also, by balancing the first sentence of the highest degree and first sentencethe highest derivative order of nonlinear (ODE) and similarly, balancing the last sentence of thehighest degree and last sentence the highest derivative order of nonlinear (ODE), leads m2 = n2and m1 = n1. Consequently, the simplest choice m2 = n2 = 1 and m1 = n1 = 1 has to beonly considered. Hence, this method can be useful to solve other nonlinear FDEs in mathemati-cal physics, because it is an easy, direct, concise, basic and powerful method to implement, alsothe exp-function method generalized solitary solutions and periodic solutions of nonlinear equa-tions. The principal merits of this method over the other methods are that it provides more generalsolutions with some arbitrary parameters. It should be noticed that the solutions are accurate.Acknowledgment:Authors thank reviewers for their comments that help improve the paper.7Eskandari and Taghizadeh: Exact Solutions of Nonlinear Space-time Fractional DifferentialPublished by Digital Commons @PVAMU, 2020AAM: Intern. J., Vol. 15, Issue 2 (December 2020) 977REFERENCESAli, M.Y., Hafez, M.G., Chowdary, M.K.H. and Akter, M.T. (2016). Analytical and traveling wavesolutions to the fifth order stender Sawada-Kotera Equation via the generalized exp(−φ(ξ))-expansion method, Journal of Applied Mathematics and Physics, Vol. 4, No. 2, pp. 262-271.Bulut, H., Tuz, M. and Aktürk, T. (2013). New multiple solution to the Boussinesq equation andthe Burgers-like Equation, Journal of Applied Mathematics, Vol. 2012, Artick ID952614, pp.1-6.Cerdik, Y.H. and Girgin, A. (2019). 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Complex solutions of the time fractional Gross-Pitaevskii (FGP) equation with external potential by using a reliable method, Applicatiionsand Applied Mathematics, Vol. 11, pp. 943-956.Wang, X.M., Bilige, S.D. and Bai, Y.X. (2017). A general sub-equation method to the Burgers-likeEquation, Thermal Science, Vol. 21, pp. 1681-1687.8Applications and Applied Mathematics: An International Journal (AAM), Vol. 15 [2020], Iss. 2, Art. 15https://digitalcommons.pvamu.edu/aam/vol15/iss2/15

Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method

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Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method

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