Traveling Waves in Rational Expressions of Exponential Functions to the Conformable Time Fractional Jimbo-Miwa and Zakharov-Kuznetsov Equations
aa r X i v : . [ n li n . S I] M a y Traveling Waves in Rational Expressions of ExponentialFunctions to the Conformable Time Fractional Jimbo-Miwaand Zakharov-Kuznetsov Equations
Alper Korkmaz , ∗ and Ozlem Ersoy Hepson C¸ ankırı Karatekin University, Department of Mathematics, 18200, C¸ ankırı, Turkey. Eski¸sehir Osmangazi University, Department of Mathematics & Computer, 26200, Eski¸sehir, Turkey.
June 26, 2018
Abstract
The conformable time fractional Jimbo-Miwa and Zakharov-Kuznetsov equationsare solved by the generalized form of the Kudryashov method. A simple compatiblewave transformation is employed to reduce the dimension of the equations to one.The predicted solution is of the form of a rational expression of two finite series atboth the numerator and the denominator. The terms of both series are of the pow-ers of some functions having exponential expressions satisfying a particular ODE.The exact solutions are expressed explicitly in terms of powers of some exponentialfunctions in form of rational expressions.
Keywords:
Generalized Kudryashov method; Conformable time fractional Jimbo-MiwaEquation; Conformable time fractional Zakharov-Kuznetsov Equation; ConformableDerivative.
MSC2010:
PACS:
The conformable time fractional Jimbo-Miwa (JM) equation of the form [1] u xxxy + pu y u xx + qu x u xy + rT βt u y − su xz = 0 (1),where u is defined in R , t > p, q, r, s ∈ R − { } , is considered.In this form of the equation, the conformal derivative operator T βt represents the β .thorder derivative ( β ∈ (0 , t . The integer ordered formof the JM equation (1) is a member of the KP-hierarchy and is not capable of passing ∗ [email protected] G ′ /G )-expansion and theseparation of variables [18, 19]. Some fractional forms of the JM equation (in modifiedRiemann-Liouville and conformable derivative forms) have also been solved exactly byvarious methods such as modified and generalized Kudryashov, Riccati equation methods[1, 20, 21].The Zakharov-Kuznetsov (ZK) equation in conformable time fractional form in threespace dimension given in [1] as T βt u + puu x + qu zzz + ru xxz + su yyz = 0 (2)is considered as the second equation to be solved exactly. The equation was derived tomodel stable 3D ion-sound solitons with weak non linearity in a low pressure magnetizedplasma [22]. When the plasma contains κ -distributed hot and cold electrons in varioustemperatures, the ZK equation can also be obtained by using reductive perturbationtechniques [23]. Electron acoustic waves with weak non linearity were examined in amagnetic field for both magnetized and unmagnetized ions [24].A general form ZK equation with arbitrarily chosen power non linearity was integratedby analysis of Lie symmetry, the tanh-function, ( G ′ /G )-expansion and simple ansatzmethods to construct cnoidal wave, singular and non-singular periodic, solitary waveand non topological soliton type exact solutions [25]. Some traveling wave type exactsolutions were suggested by using exp-function and ( G ′ /G )- and F -expansion approaches[26]. Some more kink, antikink, solitary and periodic type traveling wave solutions were2btained the planar bifurcation theory [27]. The β th order derivative in conformable sense is defined as T βt ( u ( t )) = lim τ → u ( t + τ t − β ) − u ( t ) τ , β ∈ (0 , . (3)in the positive half space for a function u : [0 , ∞ ) → R [30]. The β th conformablederivative has the properties given below. Theorem 1
Let β ∈ (0 , , and assume that u and v are β -differentiable in the positivehalf plane (interval) t > . Then, • T βt ( au + bv ) = aT βt ( u ) + bT βt ( v ) • T βt ( t p ) = pt p − β , ∀ p ∈ R • T βt ( λ ) = 0 , for all constant function u ( t ) = λ • T βt ( uv ) = uT βt ( v ) + vT βt ( u ) • T βt ( uv ) = vT βt ( u ) − uT βt ( v ) v • T βt ( u )( t ) = t − β dudt for ∀ a, b ∈ R [31, 32]. The conformable derivative supports many significant properties like Laplace transform,exponential function, chain rule, Gronwall’s inequality, various integration rules andTaylor series expansion [33].
Theorem 2
Let u be an β -differentiable function in conformable sense. Also supposethat v is differentiable in classical sense and is defined in the range of u . Then, T βt ( u ◦ v )( t ) = t − β v ′ ( t ) u ′ ( v ( t )) (4) A general non-linear PDE given as F ( u, T βt u, u x , u y , u z , T βt u, u xx , ... ) = 0 (5)where u = u ( x, y, z, t ), β ∈ (0 ,
1] can be reduced to F ( U, U ′ , U ′′ , . . . ) = 0 (6)3here ( ′ ) indicates classical derivative of U wrt ω by a simple transform of travelingwave u ( x, y, z, t ) = U ( ω ) , ω = ax + by + cz − νβ t β (7)The compatible forms of the traveling wave transform were used in some recent studies[1, 29, 34]. Implementing the classical balance procedure in the related terms gives therelation the numbers M and N required for the construction of the solution U ( ω ) = M P i =0 a i P i ( ω ) N P j =0 b j P j ( ω ) , a M = 0 , b N = 0 (8)where P ( ω ) satisfies the ODE dPdω = (cid:0) P − P (cid:1) ln ( A ) (9)with the positive real A = 1. It should be noted that the solution of this ODE is P ( ω ) = 11 + ˜ dA ω (10)for a nonzero ˜ d . The solution procedure follows by substituting a more clear form of thesolution (8) determined by using suitable values of M and N into (6). The coefficients ofpowers of P in the resultant polynomial is forced to be zero. Thus, an algebraic systemof equations is constructed. The solution of this system gives the relation between theparameters used in the target equation and the wave transform. Once the relationsbetween the parameters are determined the solution to (6) can be expressed explicitly.The final step of the procedure is to express the solutions of the target fractional PDEin terms of the original variables. The wave transform (7) reduces the JM equation (after integrating the resultant ODEonce) to a b d d ω U ( ω ) + 12 a b ( p + q ) (cid:18) dd ω U ( ω ) (cid:19) + ( − sac + ν br ) dd ω U ( ω ) = K (11)where K is the constant of integration. Balancing U ′′′ and ( U ′ ) gives the relation M = N + 1. Choose M = 2 and N = 1. Then, the solution (8) is expressed as U ( ω ) = P i =0 a i P i ( ω ) P j =0 b j P j ( ω ) = a + a P + a P b + b P (12)4ith a = 0 and b = 0. Substituting this solution into (11) gives a b a d ω P + 6 a (cid:16) dd ω P (cid:17) d d ω P + 2 a P d d ω Pb + b P − (cid:18) a d ω P + 2 a (cid:16) dd ω P (cid:17) + 2 a P d d ω P (cid:19) b ω P ( b + b P ) + 6 (cid:16) a ω P + 2 a P dd ω P (cid:17) b (cid:16) dd ω P (cid:17) ( b + b P ) − (cid:16) a ω P + 2 a P dd ω P (cid:17) b d ω P ( b + b P ) − (cid:0) a + a P + a P (cid:1) b (cid:16) dd ω P (cid:17) ( b + b P ) +6 (cid:0) a + a P + a P (cid:1) b (cid:16) dd ω P (cid:17) d d ω P ( b + b P ) − (cid:0) a + a P + a P (cid:1) b d ω P ( b + b P ) + 1 / a b ( p + q ) a ω P + 2 a P dd ω Pb + b P − (cid:16) a + a P + a ( P ( ω )) (cid:17) b ω P ( b + b P ) + ( − acs + bν r ) a ω P + 2 a P dd ω Pb + b P − (cid:16) a + a P + a ( P ) (cid:17) b ω P ( b + b P ) = K (13) P isequalized to zero to give an algebraic system of equations − Kb = 0 a ba b b (ln ( A )) − a ba b (ln ( A )) − saca b b ln ( A )+ saca b ln ( A ) + ν bra b b ln ( A ) − ν bra b ln ( A ) − Kb b = 0 − a ba b b (ln ( A )) − a ba b b (ln ( A )) + 7 a ba b (ln ( A )) +4 a ba b b (ln ( A )) − a ba b (ln ( A )) + 1 / a bpa b (ln ( A )) − a bpa a b b (ln ( A )) +1 / a bpa b (ln ( A )) + 1 / a bqa b (ln ( A )) − a bqa a b b (ln ( A )) + 1 / a bqa b (ln ( A )) + saca b b ln ( A ) − saca b b ln ( A ) − saca b ln ( A ) + 2 saca b b ln ( A ) + 2 saca b ln ( A ) − ν bra b b ln ( A ) + 2 ν bra b b ln ( A ) + ν bra b ln ( A ) − ν bra b b ln ( A ) − ν bra b ln ( A ) − Kb b = 0 saca b b ln ( A ) + 5 saca b b ln ( A ) − ν bra b b ln ( A ) − ν bra b b ln ( A )+2 a bqa a b b (ln ( A )) + 2 a bpa a b b (ln ( A )) − Kb b − saca b ln ( A )+2 ν bra b ln ( A ) − a bpa b (ln ( A )) − a bpa b (ln ( A )) − a bqa b (ln ( A )) − a bqa b (ln ( A )) + 12 a ba b b (ln ( A )) + 10 a ba b b (ln ( A )) − a ba b b (ln ( A )) + a ba b (ln ( A )) + 38 a ba b (ln ( A )) − a ba b (ln ( A )) − ν bra b b ln ( A ) + 2 ν bra b b ln ( A ) + 2 saca b b ln ( A ) − saca b b ln ( A )+2 a bpa a b (ln ( A )) + 2 a bqa a b (ln ( A )) − saca b ln ( A ) + ν bra b ln ( A ) − a ba b b (ln ( A )) − a ba b b (ln ( A )) − a bqa a b b (ln ( A )) − a bpa a b b (ln ( A )) = 04 saca b b ln ( A ) − ν bra b b ln ( A ) − saca b b ln ( A ) − saca b b ln ( A )+ ν bra b b ln ( A ) + 5 ν bra b b ln ( A ) − a bqa a b b (ln ( A )) − a bpa a b b (ln ( A )) − Kb + 1 / a bpa b (ln ( A )) + 1 / a bpa b (ln ( A )) + 1 / a bqa b (ln ( A )) +1 / a bqa b (ln ( A )) − a ba b b (ln ( A )) − a ba b b (ln ( A )) + 6 a ba b b (ln ( A )) − a ba b (ln ( A )) − a ba b (ln ( A )) + 6 a ba b (ln ( A )) − a bpa a b (ln ( A )) − a bqa a b (ln ( A )) − a bpa a b (ln ( A )) − a bqa a b (ln ( A )) + saca b ln ( A ) − ν bra b ln ( A ) + a ba b b (ln ( A )) + 41 a ba b b (ln ( A )) − a ba b b (ln ( A )) + 2 a bpa b (ln ( A )) + 2 a bqa b (ln ( A )) + a bqa a b b (ln ( A )) + a bpa a b b (ln ( A )) + 4 a bqa a b b (ln ( A )) + 4 a bpa a b b (ln ( A )) = 024 a ba b (ln ( A )) − a ba b b (ln ( A )) + 28 a ba b b (ln ( A )) − a ba b (ln ( A )) − a bpa a b b (ln ( A )) + 2 a bpa a b (ln ( A )) + 2 a bpa a b (ln ( A )) − a bpa a b b (ln ( A )) − a bpa b (ln ( A )) + 2 a bpa b b (ln ( A )) − a bqa a b b (ln ( A )) +2 a bqa a b (ln ( A )) + 2 a bqa a b (ln ( A )) − a bqa a b b (ln ( A )) − a bqa b (ln ( A )) +2 a bqa b b (ln ( A )) − saca b b ln ( A ) + saca b ln ( A ) + 4 ν bra b b ln ( A ) − ν bra b ln ( A ) = 036 a ba b b (ln ( A )) − a ba b b (ln ( A )) + 7 a ba b (ln ( A )) − a bpa a b (ln ( A )) + a bpa a b b (ln ( A )) + 2 a bpa b (ln ( A )) − a bpa b b (ln ( A )) + 1 / a bpa b (ln ( A )) − a bqa a b (ln ( A )) + a bqa a b b (ln ( A )) + 2 a bqa b (ln ( A )) − a bqa b b (ln ( A )) +1 / a bqa b (ln ( A )) − saca b ln ( A ) + ν bra b ln ( A ) = 024 a ba b b (ln ( A )) − a ba b (ln ( A )) + 2 a bpa b b (ln ( A )) − a bpa b (ln ( A )) + 2 a bqa b b (ln ( A )) − a bqa b (ln ( A )) = 06 a ba b (ln ( A )) + 1 / a bpa b (ln ( A )) + 1 / a bqa b (ln ( A )) = 0(14) a , a , a , b , b , ν and K gives a = − a a = −
12 ln ( A ) ab p + qb = − b ν = − a (cid:16) A )) a b − cs (cid:17) brK = 0 (15)for arbitrarily chosen a , b , c , a and b . Thus, the solution to (11) is determined as U ( ω ) = (cid:18) − / a + a dA ω −
12 ln ( A ) ab ( p + q ) (1 + dA ω ) (cid:19) (cid:18) − / b + b dA ω (cid:19) − (16)Returning the original variables ( x, y, z, t ) gives the solution to the conformable timefractional JM equation (1) as u ( x, y, z, t ) = − a + a
1+ ˜ dA ax + by + cz + a ( A ))2 a b − cs ) br tββ − ln( A ) ab ( p + q )
1+ ˜ dA ax + by + cz + a ( A ))2 a b − cs ) br tββ − / b + b
1+ ˜ dA ax + by + cz + a ( A ))2 a b − cs ) br tββ ! (17)The algebraic system (14) has one more solution as a = b (12 ln ( A ) ab + pa + qa ) b ( p + q ) a = −
12 ln ( A ) ab p + qν = − a (cid:16) (ln ( A )) a b − cs (cid:17) brK = 0 (18)for arbitrary a , b , b , a , b and c . Then, the solution to (11) is expressed as U ( ω ) = (cid:18) b (12 ln( A ) ab + pa + qa ) b ( p + q ) + a
1+ ˜ dA ω − ln( A ) ab ( p + q ) (
1+ ˜ dA ω ) (cid:19)(cid:16) b + b
1+ ˜ dA ω (cid:17) (19)7hus, the solution to the conformable time fractional JM equation (1) is constructed as u ( x, y, z, t ) = b (12 ln( A ) ab + pa + qa ) b ( p + q ) + a
1+ ˜ dA ax + by + cz + a ( (ln( A ))2 a b − cs ) br tββ − ln( A ) ab ( p + q )
1+ ˜ dA ax + by + cz + a ( (ln( A ))2 a b − cs ) br tββ b + b
1+ ˜ dA ax + by + cz + a ( (ln( A ))2 a b − cs ) br tββ (20) The traveling wave transform (7) reduces the conformable time fractional ZK equation(2) to − ν U ( ω ) + 12 ap ( U ( ω )) + (cid:0) a br + b cs + c q (cid:1) d d ω U ( ω ) = K, K constant (21)after integrating the resultant ODE once. Balancing U and U ′′ gives the relation M = N + 2. Assuming N = 1 and M = 3 gives the predicted solution of the form U ( ω ) = a + a P + a P + a P b + b P (22)where a = 0 and b = 0. Substituting this solution into (21) and arranging the resultantequation gives 8 A )) a bra b + 6 (ln ( A )) b csa b + 1 / apa b + 6 (ln ( A )) c qa b (cid:17) P ( ω )+ (cid:16) A )) a bra b −
10 (ln ( A )) a bra b + 2 (ln ( A )) b csa b −
10 (ln ( A )) b csa b + 16 (ln ( A )) c qa b b +1 / apa b + apa a b + 2 (ln ( A )) c qa b −
10 (ln ( A )) c qa b + 16 (ln ( A )) a bra b b + 16 (ln ( A )) b csa b b (cid:17) P ( ω )+ (cid:16) − ν a b − A )) a bra b + 12 (ln ( A )) a bra b + 4 (ln ( A )) a bra b − A )) b csa b +12 (ln ( A )) b csa b + 4 (ln ( A )) b csa b + 6 (ln ( A )) c qa b b −
27 (ln ( A )) c qa b b + 1 / apa b + apa a b + apa a b − A )) c qa b + 12 (ln ( A )) c qa b +4 (ln ( A )) c qa b + 6 (ln ( A )) a bra b b −
27 (ln ( A )) a bra b b + 6 (ln ( A )) b csa b b −
27 (ln ( A )) b csa b b (cid:17) P ( ω )+ (cid:16) − ν a b + 6 (ln ( A )) b csa b −
21 (ln ( A )) b csa b − A )) c qa b b + 11 (ln ( A )) c qa b b + (ln ( A )) b csa b + 6 (ln ( A )) a bra b −
21 (ln ( A )) a bra b + (ln ( A )) a bra b + 1 / apa b − ν a b b + (ln ( A )) c qa b + apa a b + apa a b + apa a b +6 (ln ( A )) c qa b −
21 (ln ( A )) c qa b − A )) a bra b b + 11 (ln ( A )) a bra b b − A )) b csa b b + 11 (ln ( A )) b csa b b (cid:17) P ( ω )+ (cid:16) apa a b + apa a b + apa a b + 9 (ln ( A )) c qa b −
10 (ln ( A )) c qa b − (ln ( A )) c qa b + 2 (ln ( A )) c qa b +3 (ln ( A )) a bra b b + 3 (ln ( A )) b csa b b + (ln ( A )) a bra b b + (ln ( A )) b csa b b − A )) a bra b b − A )) b csa b b + 1 / apa b − ν a b b − Kb + 9 (ln ( A )) a bra b +9 (ln ( A )) b csa b + 3 (ln ( A )) c qa b b −
10 (ln ( A )) b csa b −
10 (ln ( A )) a bra b + 2 (ln ( A )) a bra b +2 (ln ( A )) b csa b − (ln ( A )) a bra b − (ln ( A )) b csa b + (ln ( A )) c qa b b − A )) c qa b b − ν a b − ν a b (cid:17) P ( ω )+ (cid:16) − ν a b − ν a b − A )) a bra b + 4 (ln ( A )) a bra b − A )) b csa b + (ln ( A )) a bra b + (ln ( A )) b csa b + 4 (ln ( A )) b csa b + 3 (ln ( A )) c qa b b − (ln ( A )) c qa b b − Kb b + 1 / apa b − ν a b b + (ln ( A )) c qa b + apa a b + apa a b − A )) c qa b + 4 (ln ( A )) c qa b + 3 (ln ( A )) a bra b b − (ln ( A )) a bra b b + 3 (ln ( A )) b csa b b − (ln ( A )) b csa b b (cid:17) P ( ω )+ (cid:16) − ν a b − (ln ( A )) c qa b b + (ln ( A )) a bra b + (ln ( A )) b csa b − Kb b + 1 / apa b − ν a b b + (ln ( A )) c qa b + apa a b − (ln ( A )) a bra b b − (ln ( A )) b csa b b (cid:17) P ( ω )+ 1 / apa b − Kb − ν a b = 0 (23) P ( ω )in the previous equation for a , a , a , a , b , b and ν gives a = 0 a = 1 ap × (cid:16) b (cid:16) − a br (ln ( A )) − b cs (ln ( A )) − c q (ln ( A )) ± q a b r (ln ( A )) + 2 a b crs (ln ( A )) + 2 a bc qr (ln ( A )) + b c s (ln ( A )) + 2 b c qs (ln ( A )) + c q (ln ( A )) − Kap (cid:19)(cid:19) a = 12 (ln ( A )) b (cid:0) a br + b cs + c q (cid:1) apa = −
12 (ln ( A )) b (cid:0) a br + b cs + c q (cid:1) apb = 0 ν = ± q a b r (ln ( A )) + 2 a b crs (ln ( A )) + 2 a bc qr (ln ( A )) + b c s (ln ( A )) + 2 b c qs (ln ( A )) + c q (ln ( A )) − Kap (24) where ap = 0 for arbitrarily chosen a , b , c , K and b . Thus, the solutions to (21) areexpressed as U , ( ω ) = 1 b (cid:16)
11+ ˜ dAω (cid:17) × (cid:20) ap × (cid:16) b (cid:16) − a br (ln ( A )) − b cs (ln ( A )) − c q (ln ( A )) + ν (cid:17)(cid:17) × (cid:18)
11 + ˜ dA ω (cid:19) +12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) ap × (cid:18)
11 + ˜ dA ω (cid:19) −
12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) ap × (cid:18)
11 + ˜ dA ω (cid:19) = (cid:20) ap × (cid:16) − a br (ln ( A )) − b cs (ln ( A )) − c q (ln ( A )) + ν (cid:17) +12 (ln ( A )) (cid:16) a br + b cs + c q (cid:17) ap × (cid:18)
11 + ˜ dA ω (cid:19) −
12 (ln ( A )) (cid:16) a br + b cs + c q (cid:17) ap × (cid:18)
11 + ˜ dA ω (cid:19) (25) for arbitrarily chosen b , K , a , b and c . The return to the original variables gives thesolutions as u , ( x, y, z, t ) = (cid:20) ap × (cid:16) − a br (ln ( A )) − b cs (ln ( A )) − c q (ln ( A )) + ν (cid:17) +12 (ln ( A )) (cid:16) a br + b cs + c q (cid:17) ap ×
11 + ˜ dA ax + by + cz − ν tββ −
12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) ap ×
11 + ˜ dA ax + by + cz − ν tββ (26) where ν = ± q a b r (ln ( A )) + 2 a b crs (ln ( A )) + 2 a bc qr (ln ( A )) + b c s (ln ( A )) + 2 b c qs (ln ( A )) + c q (ln ( A )) − Kap (27) a = (cid:16) − (ln ( A )) a brb − (ln ( A )) b csb − (ln ( A )) c qb + b ν (cid:17) b apb a = 12 (ln ( A )) a brb − (ln ( A )) a brb + 12 (ln ( A )) b csb − (ln ( A )) b csb + 12 (ln ( A )) c qb − (ln ( A )) c qb + b νapa = − (cid:16) a brb − a brb + b csb − b csb + c qb − c qb (cid:17) (ln ( A )) apa = −
12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) apν = ± q (ln ( A )) a b r + 2 (ln ( A )) a b crs + 2 (ln ( A )) a bc qr + (ln ( A )) b c s + 2 (ln ( A )) b c qs + (ln ( A )) c q − Kap (28) where b , b , K , a , b and c are arbitrary constants. Thus, the solutions to (21) areobtained as U , ( ω ) = b + b × (cid:16)
11+ ˜ dAω (cid:17) × (cid:16) − (ln ( A )) a br − (ln ( A )) b cs − (ln ( A )) c q + ν (cid:17) b ap +
12 (ln ( A )) a brb − (ln ( A )) a brb + 12 (ln ( A )) b csb − (ln ( A )) b csb + 12 (ln ( A )) c qb − (ln ( A )) c qb + b νap ! × (cid:18)
11 + ˜ dA ω (cid:19) + − (cid:16) a brb − a brb + b csb − b csb + c qb − c qb (cid:17) (ln ( A )) ap × (cid:18)
11 + ˜ dA ω (cid:19) + −
12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) ap × (cid:18)
11 + ˜ dA ω (cid:19) (29) for arbitrarily chosen b , b , K , a , b and c and ν = ± q (ln ( A )) a b r + 2 (ln ( A )) a b crs + 2 (ln ( A )) a bc qr + (ln ( A )) b c s + 2 (ln ( A )) b c qs + (ln ( A )) c q − Kap (30)
Returning the original variables gives the solution to the comformable time fractionalZK equation (2) as u , ( x, y, z, t ) = b + b ×
11+ ˜ dAax + by + cz − ν tββ × (cid:16) − (ln ( A )) a br − (ln ( A )) b cs − (ln ( A )) c q + ν (cid:17) b ap +
12 (ln ( A )) a brb − (ln ( A )) a brb + 12 (ln ( A )) b csb − (ln ( A )) b csb + 12 (ln ( A )) c qb − (ln ( A )) c qb + b νap ! ×
11 + ˜ dA ax + by + cz − ν tββ + − (cid:16) a brb − a brb + b csb − b csb + c qb − c qb (cid:17) (ln ( A )) ap ×
11 + ˜ dA ax + by + cz − ν tββ + −
12 (ln ( A )) b (cid:16) a br + b cs + c q (cid:17) ap ×
11 + ˜ dA ax + by + cz − ν tββ (31) where ν = ± q (ln ( A )) a b r + 2 (ln ( A )) a b crs + 2 (ln ( A )) a bc qr + (ln ( A )) b c s + 2 (ln ( A )) b c qs + (ln ( A )) c q − Kap (32) Conclusion
In the study, the generalized Kudryashov method is implemented to some conformabletime fractional PDEs defined in three space dimensions, name the conformable timefractional JM and ZK equations. The compatible wave transform has a significant rolein the solutions steps. Reducing both equations to some ODEs and implementationof the generalized form of the Kudryashov method derive some explicit exact solutionsto them. These explicit solutions can be represented in rational forms of some finiteexponential function series.
Acknowledgement:
A part of this study was presented orally in International Congresson Fundamental and Applied Sciences 2017, Sarajevo, Bosnia and Herzegovina.