Linearizability Of Nonlinear Second-Order Ordinary Differential .

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IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19Linearizability of Nonlinear Second-Order OrdinaryDifferential Equations by Using a GeneralizedLinearizing TransformationPrakrong Voraka, Supaporn Suksern, and Nontakan DonjiwpraiAbstract—In this paper, we have proposed the linearizationproblem of second-order ordinary differential equation underthe generalized linearizing transformation. We found the necessary form for reducing the second-order ordinary differentialequation to simple linear equation. We also obtained sufficientcondition for making the above form to be linear. Further,the procedure of linear transformation within the study isdemonstrated in the explicit form. Moreover, we apply theobtained linearization criteria to the interesting problemsof nonlinear ordinary differential equations and nonlinearpartial differential equations, for examples the parachuteequation, the Painlevé - Gambier XI equation, the equationfor the variable frequency oscillator, the one-dimensional nonpolynomial oscillator, the equation that can be linearizable bypoint and Sundman transformations, the modified generalizedVakhnenko equation.Index Terms—linearization problem, generalized linearizing transformation, nonlinear second-order ordinary differential equation.I. IntroductionTHE linearization problem is one of the importantbranches in differential equation field. A numberof mathematicians has been studying this branch continuously until the present time. To discover theory forfinding new knowledge has shown to be a great benefitfor academic world and country development. It is knownthat theories and new knowledge obtained from researchnot only offer benefits to improve existing knowledgewithin the branch itself, but also they can be appliedto other branches or fields and can be key fundamentalto develop basic science which is basic research to buildmany other new knowledge. This would be a fundamentalstep to develop the country.The linearization problem is a branch of study thatcan be applied widely in particular to the study involvingsolving the equations. Most important physical problemsare in the form of nonlinear differential equations whichare normally difficult to solve and there are relativelyfew method to find their exact solutions. Numericalmethod therefore is often used to solve these nonlinearManuscript received June 11, 2020; revised August 17, 2020. Thiswork was financially supported by Naresuan University, Thailandunder Grant no. R2563C017.S. Suksern and N. Donjiwprai (corresponding author e-mail:supapornsu@nu.ac.th) are with the Department of Mathematics,Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.S. Suksern is with the Research center for Academic Excellence inMathematics, Naresuan University, Phitsanulok 65000, Thailand.P. Voraka is with the Faculty of Science and Engineering,Kasetsart University, Chalermphrakiat Sakon Nakhon ProvinceCampus, Sakon Nakhon 47000, Thailand.differential equations but the obtained solutions are justthe approximate solutions. However, the exact solutionis claimed to be more interesting because it can be usedto analyze the properties of the studied equations. Oneof the methods used to determine the exact solutionsis to linearize the interested equation and find solutionsdirectly by fundamental method. The solutions obtainedfrom such linear equation are yet still solutions of initialequation. By mentioned above, we are required to seekfor transformation in order to transform initial equationto be linear equation.There are a number of interesting transformations.For example, in the case that the transformation consists of derivative, we call it as tangent transformation, in the case that the transformation depends onlyon independent and dependent variables, we call itas point transformation and we will call the tangenttransformation which the independent and dependentvariables can be changed and involves the first derivative as contact transformation. In addition, anothertype of transformation which its transformation set isdifferent from any mentionedabove since there is a nonlocal term T G (t, x) dt, such transformationis called generalized Sundman transformation. In thispaper, we use the generalized linearizing transformationwhich is an extended transformation from generalizedSundman transformation where the selected G functionis G (t, x, x′ ).Up to the present time, all researchers who studythe linearization of second-order ordinary differentialequations via generalized linearizing transformation havenot covered all cases yet. Therefore, in this paper wefocus on the remaining cases that have not yet beenstudied, which we also find that those cases can beapplied to solve several nonlinear equations in real-worldphenomenon.A. Historical ReviewFrom above facts as mentioned, the researcher wouldlike to give a brief background of this study. Since 19thcentury the linearization problem of ordinary differential equation has attracted some interests from variouswell-known mathematicians e.g. S. Lie and E. Cartanetc. The first person who could solve the linearizationproblem of ordinary differential equation is Lie [1]. Liecould discover the standard form of every second-orderordinary differential equation which could be reducedthe form to become linear equation via changing theVolume 50, Issue 4: December 2020

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19independent and dependent variables (or can be calledpoint transformation). Later, Liouville [2] and Tresse[3] used the relative invariants of equivalence groupunder point transformation to solve the equivalence ofsecond-order ordinary differential equations which can bereduced from second-order nonlinear ordinary differentialequations to second-order linear ordinary differentialequations. Moreover, Lie discovered that every secondorder ordinary differential equation can be reduced tosecond-order linear ordinary differential equation without any conditions via contact transformation.Having mentioned some methods above, there are yetstill other methods to solve linearization problem ofsecond-order ordinary differential equation. For example,the method of Cartan [4], the reducing order method, thedifferential substitution method etc.Another transformation that is very interesting andhas not been mentioned yet is the generalized SundmantransformationX F (t, x), dT G(t, x)dt.(1)Duarte, Moreira and Santos [5] used generalized Sundman transformation to determine the conditions forlinearizing the second-order ordinary differential equation to be simple linear equation. In [6] Nakpim andMeleshko demonstrated that the general linear equationin the canonical form of Laguerre was not sufficient forsolving linearization problem via generalized Sundmantransformation. The canonical form of Laguerre couldonly particularly be applied with point and contacttransformations. Therefore, in [6] they found the conditions for linearizing the second-order ordinary differentialequation to be general linear equation.In this paper, we extend the generalized Sundmantransformation which was studied before as shown in[7]-[9], where they called such a transformation in thisform as generalized linearizing transformationX F (t, x), dT G(t, x, x′ )dt.(2)Notice that for the case G1 0, the generalized linearizing transformation becomes a generalized Sundmantransformation, so that they assumed G1 ̸ 0.The authors of [7] obtained that any second-orderlinearizable ordinary differential equation which can be′′mapped into the equation X 0 via a generalizedlimearizing transformation has to be of the form′′x A3 (t, x)x′3 A2 (t, x)x′2 A1 (t, x)x′ A0 (t, x) 0,and the functions Ai ’s (i 0, 1, 2, 3) are connected tothe transform functions F and G through the relationsA3 (G1 Fxx Fx G1x )/M,A2 (G2 Fxx 2G1 Fxt Fx G2x Ft G1x Fx G1t )/M,A1 (2G2 Fxt G1 Ftt Fx G2t Ft G2x Ft G1t )/M,A0 (G2 Ftt Ft G2t )/MX F (t, x) , dT (G1 (t, x) x′ G2 (t, x)) dt.(4)with M Fx G2 Ft G1 ̸ 0.They have analyzed a particular case of equation (3),namely, A3 0 and A2 0 in equation (4). Completeanalysis of the compatibility of arising equations is givenfor the case Fx ̸ 0.Therefore, in this paper we will apply the generalizedlinearizing transformation with second-order ordinarydifferential equation to complete the remaining cases(Fx 0) which are different from the work by Chandrasekar and Lakshmanan [7].II. Formulation of the Linearization TheoremsA. Obtaining Necessary Condition of LinearizationWe begin with investigating the necessary conditionsfor linearization. We consider the second-order ordinarydifferential equation′′x F (t, x, x′ )(5)which can be transformed to a simplest linear equation′′They demonstrated that this transformation can beused to linearize a more extensive class of nonlinearstandard differential equations including some equationsthat can’t be linearized by the non-point and invertiblepoint transformations. In the case that the functionG in (2) does not depend on the variable x′ , thenit can be turned into a non-point transformation. IfG is a differentiable function, then it turns into aninvertible point transformation. In this way, (2) is aunified transformation as it incorporates non-point andinvertible point transformations as extraordinary cases.A case of an equation that can be linearized by a changeof the structure (2) is given in [8].In [7], Chandrasekar, Senthilvelan and Lakshmananapplied a particular class of transformations (2), wherethe function G(t, x, x′ ) is linear with respect to x′ .They payed attention to the case where G is apolynomial function in x′ and in particular where it islinear in x′ with coefficients which are arbitrary functionsof t and x. To be specific, they focused here on the case(3)X 0(6)under the generalized linearizing transformationX F (t, x) ,dT [G1 (t, x) x′ G2 (t, x)] dt,(7)where G1 ̸ 0. So, we arrive at the following theorem.Theorem 2.1: Any second-order ordinary differentialequations (5) obtained from a linear equation (6) by ageneralized linearizing transformation (7) has to be theform′′x A3 (t, x)x′3 A2 (t, x)x′2 A1 (t, x)x′ A0 (t, x) 0,(8)whereA3 ( Fxx G1 Fx G1x )/(Ft G1 Fx G2 ),A2 ( 2Ftx G1 Ft G1x Fxx G2 Fx G1t Fx G2x )/(Ft G1 Fx G2 ),A1 ( 2Ftx G2 Ftt G1 Ft G1t Ft G2x Fx G2t )/(Ft G1 Fx G2 ),A0 ( Ftt G2 Ft G2t )/(Ft G1 Fx G2 ).Volume 50, Issue 4: December 2020(9)(10)(11)(12)

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19Proof. Applying a generalized linearizing transformation(7), one obtains the following transformationsDt FX ′ (T ) Dt [G1 x′ G2 ]dtFt x′ Fx G1 x ′ G 2 P (t, x, x′ ),Dt PX ′′ (T ) Dt [G1 x′ G2 ]dtPt Px x′ Px′ x′′ ,G1 x ′ G2and G3t , one finds(18)Fxx (Fx G1x A3 K)/G1 ,Fx G3x G21Kx ( Fx G1t G1 Fx G1x G1 G3 3G1x K A2 G1 K 3A3 G1 G3 K)/(2G1 ), (19)Kt ( Fx G1t G1 G3 Fx G1x G1 G23 Fx G3x G21 G3 4G1t K G1x G3 K 2G3x G1 K 2A1 G1 K 3A2 G1 G3 K 3A3 G1 G23 K)/(2G1 ),(20)G3t G3x G3 A0 A1 G3 A2 G23 A3 G33 .(21)Comparing the mixed derivative (Kx )t (Kt )x , oneobtainswhereG3xx (2A0x Fx G31 2A1x Fx G31 G3 4A1x G21 KFtt (G1 x′ G2 ) Ft (G1 x′′ G1t x′ G2t ),(G1 x′ G2 )2Ftx (G1 x′ G2 ) Ft (G1x x′ G2x ),Px (G1 x′ G2 )2Ft G1 x′′Px′ ,(G1 x′ G2 )2Pt and Dt t x′ x x′′ x′ . is a total derivative.Substituting the resulting expression into the linearequation (6) we arrive at the necessary form (8), whereA0 , A1 , A2 and A3 are some functions of t and x asdefined in system of equations (9)-(12). 2A2t G21 K 2A2x Fx G31 G23 6A2x G21 G3 K 6A3t G21 G3 K 2A3x Fx G31 G33 6A3x G21 G23 K 4Fx G1tx G21 G3 2Fx G1tt G21 3Fx G21t G1 6Fx G1t G1x G1 G3 2Fx G1t G3x G21 2Fx G1t A1 G21 4Fx G1t A2 G21 G3 6Fx G1t A3 G21 G23 2Fx G1xx G21 G23 3Fx G21x G1 G23 2Fx G1x G3x G21 G3 2Fx G1x A0 G21 2Fx G1x A2 G21 G23 4Fx G1x A3 G21 G33 Fx G23x G31 2G1tx G1 K 3G1t G1x K G1t A2 G1 K 3G1t A3 G1 G3 K 2G1xx G1 G3 K 3G21x G3 K G1x G3x G1 KB. Obtaining Sufficient Conditions of Linearization andLinearizing TransformationFor obtaining sufficient conditions of linearizability ofequation (8), one has to solve the compatibility problemof the system of equations (9)-(12), considering it asoverdetermined system of partial differential equationsfor the functions F, G1 and G2 with given coefficients Aiof equation (8).For convenience of calculations, we setG3 G2.G1 G1x A2 G1 G3 K 3G1x A3 G1 G23 K 5G3x A2 G21 K 15G3x A3 G21 G3 K 6A0 A3 G21 K 6A1 A3 G21 G3 K 6A2 A3 G21 G23 K 6A23 G21 G33 K)/(4G21 K). (22)The compatibility analysis depends on the value ofFx . A complete study of all cases is cumbersome. Herea complete solution is given for the case where Fx 0.Case Fx 0Since Fx 0, then substituting it into Fxx in equation(18), one gets the conditionSo that system of equations (9)-(12) becomeA3 ( Fxx G1 Fx G1x )/(G1 (Ft Fx G3 )),A2 ( 2Ftx G1 Ft G1x Fxx G1 G3 Fx G1t Fx G1x G3 Fx G3x G1 )/(G1 (Ft Fx G3 )),A1 ( 2Ftx G1 G3 Ftt G1 Ft G1t Ft G1x G3 Ft G3x G1 Fx G1t G3 Fx G3t G1 )/(G1 (Ft Fx G3 )),A0 ( Ftt G1 G3 Ft G1t G3 Ft G3t G1 )/(G1 (Ft Fx G3 )).A3 0.(13)Comparing the mixed derivative (Ft )x (Fx )t , oneobtains the derivative(14)G1x A2 G1 3A3 G1 G3(24)and this satisfies equation (Fxx )t (Ft )xx . Setting(15)(16)According to the notation K G1 (Fx G3 Ft ), we definethe derivative Ft asFt (Fx G1 G3 K)/G1 .(23)(17)λ1 A1x 2A2t ,λ2 A0xx A0x A2 A2tt A2t A1 A2x A0 λ1t A1 λ1then, equation (G3xx )t (G3t )xx becomesG3x λ1 G3 A2 λ1 λ2 0.(25)The compatibility analysis depends on the value of λ1 .A complete study of all cases is given here.Solving equations (13)-(16) with respect to Fxx , Kx , KtVolume 50, Issue 4: December 2020

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 193.3.1. Case λ1 0From equation (25), one finds the condition(26)λ2 0.3.3.2. Case λ1 ̸ 0Equation (25) provides the derivativeG3x (G3 A2 λ1 λ2 )/λ1 .(27)Subtituting G3x into G3xx in equation (22), one arrivesat the conditionλ2x ( A2t λ21 λ1x λ2 λ31 )/λ1 .(28)Comparing the mixed derivaties (G3x )t (G3t )x , onegets the conditionλ2t (A0x λ21 λ1t λ2 A0 A2 λ21 A1 λ1 λ2 λ22 )/λ1 .(29)Combining all derived results in the case Fx 0 thefollowing theorems are proven.Theorem 2.2: Sufficient conditions for equation (8) tobe equivalent to a linear equation (6) via generalizedlinearizing transformation (7) with the function F F (t) is the equation (23) and the additional conditionsare as follows.(a) If λ1 0, then the condition is equation (26).(b) If λ1 ̸ 0, then the conditions are equations (28)and (29).Theorem 2.3: Provided that the sufficient conditionsin Theorem 2.2 are satisfied, the transformation (7) withthe function F F (t) mapping equation (8) to a linearequation (6) is obtained by solving the compatible systemof equations :(a) (17), (19), (20), (21), (22), and (24).(b) (17), (19), (20), (21), (24), and (27).III. Some ApplicationsIn this section we focus on finding some applicationswhich satisfy Theorem 2.1, Theorem 2.2 and Theorem2.3. The obtained results are as follows.A. Parachute EquationAn application to this equation can be applied to amodel of motion for a parachutist by using Newton’s lawII which isF ma. The movement of skydiver whenthe coefficient of air opposition changes between free-falland the last consistent state drop with the parachute isslowly conveyed.Consider the parachute equation [10], in the formx′′ kx′2 g 0,(30)′with initial conditions x(0) 0 and x (0) 0.2dDHere k πρC8m , where m is the mass of the body and parachute, ρ is the density of the fluid in which the body moves, Cd is the drag coefficient for the parachute(1.5 for parabolic profile and 0.75 for flat), D is the effective diameter of the parachute.Equation (30) is an equation of the form (8) in Theorem2.1 with the coefficientsA3 0, A2 k, A1 0, A0 g, λ1 0, λ2 0.One can check that these coefficients obey the conditions in Theorem 2.2. case (a). Thus, equation (30) islinearizable via a generalized linearizing transformation.For finding the functions F , G1 and G2 we have to solveequations in Theorem 2.3 case (a), which become1 G3 k),Ft GK1 , Kx kK, Kt K(2G1tG G1G3t g G23 k, G3xx 0, G1x G1 k.(31)One can find the particular solution for equations in (31)as G1 ekx , G3 kg i, G2 kg iekx ,K ekx kgit , F ikg e kgit .So that, one obtains the linearizing transformation i kgitg kxkx ′ X e, dT (e x (32)ie )dt.kkgHence, equation (30) is mapped by the transformation(32) into the linear equation′′(33)X 0.The general solution of equation (33) isX c1 T c2 ,(34)where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (32) to equation(34), we obtain that the general solution of equation(30) isi e kgit c1 ϕ(t) c2 ,kgwhere the function T ϕ(t) is a solution of the equation dTg kx′ (x i)e .dtkB. Painlevé - Gambier XI EquationIn [11], Koudahoun, Akande, Adjai, Kpomahou andMonsia considered the Painlevé - Gambier XI equationx′2 0.(35)xTo investigate the exact classical and quantum mechanical solutions, they offered a generalized singulardifferential equation of quadratic Lienard type.By using our obtained theorems, we get the results asfollow. Equation (35) is an equation of the form (8) inTheorem 2.1 with the coefficients1A3 0, A2 , A1 0, A0 0, λ1 0, λ2 0.xOne can check that these coefficients obey the conditions in Theorem 2.2. case (a). Thus, equation (35) islinearizable via a generalized linearizing transformation.For finding the functions F , G1 and G2 we have to solveequations in Theorem 2.3 case (a), which becomex′′ Ft GK1 , Kx G3t G23xKx,Kt , G3xx G3x2 ,K(2G1t x G1 G3 ),G1 xG1G1x x .(36)One can find the particular solution for equations in (36)asG1 x, G3 0, G2 0, K x, F t.Volume 50, Issue 4: December 2020

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19So that, one obtains the linearizing transformation′X t, dT xx dt.D. The One-Dimensional Non-Polynomial Oscillator(37)Hence, equation (35) is mapped by the transformation(37) into the linear equation′′(38)X 0.The general solution of equation (38) is(39)X c1 T c2 ,where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (37) to equation(39), we obtain that the general solution of equation(35) is t c1 ϕ(t) c2 ,where the function T ϕ(t) is a solution of the equationdT xx′ .dtC. Equation for the Variable Frequency OscillatorIn 2013, Mastafa, Al-Dueik and Mara’beh [12] considered the ordinary differential for the variable frequencyoscillatorx′′ xx′2 0.(40)They showed that this equation can be linearizable bygeneralized Sundman transformation.By using our obtained theorems, we get the results asfollow. Equation (40) is an equation of the form (8) inTheorem 2.1 with the coefficientsA3 0, A2 x, A1 0, A0 0, λ1 0, λ2 0.One can check that these coefficients obey the conditions in Theorem 2.2. case (a). Thus, equation (40) islinearizable via a generalized linearizing transformation.For finding the functions F , G1 and G2 we have to solveequations in Theorem 2.3 case (a), which becomeK(2G1t G1 G3 x),G1 GK1 ,Ft Kx xK, Kt G3t G23 x, G3xx G3 , G1x G1 x.(41)One can find the particular solution for equations in (41)asG1 ex22, G3 0, G2 0, K ex22, F t.So that, one obtains the linearizing transformationX t, dT ex22x′ dt.(42)Hence, equation (40) is mapped by the transformation(42) into the linear equation′′(43)X 0.In the note [13], Mathew and Lakshmanan presenteda remarkable nonlinear system that all its boundedperiodic motions are simple harmonic. The system is aparticle obeying the highly nonlinear equation of motion(1 λx2 )x′′ (α λx′2 )x 0,where λ and α are arbitrary parameters.By using our obtained theorems, we get the results asfollow. Equation (45) is an equation of the form (8) inTheorem 2.1 with the coefficientsλxαxA3 0, A2 , A1 0, A0 ,(λx2 1)(λx2 1)λ1 0, λ2 αλx( λx2 2).One can check that the condition (23) in Theorem 2.2.case (a) are satisfied. Now, the condition (26) is satisfiedwhen the following condition holds, that is,αλx( λx2 2) 0.Two cases arise, that are α 0 and λ x22 . (Note thatfor λ 0 equation (45) is linear equation.)Here we consider only case α 0. In this case, theequation (45) takes the form(1 λx2 )x′′ λxx′2 0.λxKFt GK1 , Kx (1 λx2) ,K(2G1t λx2 2G1t λxG1 G3 ),G1 (1 λx2 )λx2 G23λG3 ( λx2 1) (1 λx2 ) , G3xx (1 λx2 )2 ,λxG1G1x (1 λx2) .Kt G3t where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (42) to equation(44), we obtain that the general solution of equation(40) is t c1 ϕ(t) c2 ,where the function T ϕ(t) is a solution of the equationx2dT e 2 x′ .dt(47)One can find the particular solution for equations in (47)as1G1 1 , G3 0, G2 0,2(1 λx ) 21K 1(1 λx2 ) 2, F t.So that, one obtains the linearizing transformationX t, dT 11(1 λx2 ) 2x′ dt.(48)Hence, equation (46) is mapped by the transformation(48) into the linear equation′′X 0.(49)The general solution of equation (49) isX c1 T c2 ,(44)(46)The linearizing transformation is found by solving equations in Theorem 2.3 case (a), which becomeThe general solution of equation (43) isX c1 T c2 ,(45)(50)where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (48) to equation(50), we obtain that the general solution of equation(46) is t c1 ϕ(t) c2 ,where the function T ϕ(t) is a solution of the equation1dT′ 1 x .dt(1 λx2 ) 2Volume 50, Issue 4: December 2020

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19E. Equation That Can Be Linearizable by Point andSundman TransformationsConsider the nonlinear second-order ordinary differential equationx′′ µ3 xk3 x′2 µ2 xk2 x′ µ1 xk1 0,(56), we obtain that the general solution of equation(52) is t c1 ϕ(t) c2 ,where the function T ϕ(t) is a solution of the equation(51)where k3 , k2 , k1 , µ1 , µ2 and µ3 ̸ 0 are arbitrary constants. The Lie criteria [1], showed that the nonlinearequation (51) is linearizable by a point transformation ifand only if µ1 0 and µ2 0. In [6], Nakpim andMeleshko showed that the nonlinear equation (51) islinearizable by a generalized Sundman transformationif and only if µ2 ̸ 0 and µ1 0.By using our obtained theorems, we get the results asfollow. Equation (51) is an equation of the form (8) inTheorem 2.1 with the coefficientsµ3 xk3 1dT e k3 1 x′ ,dtwhere k3 ̸ 1.For k3 1, one can find the particular solution forequations in (53) asG1 xµ3 , G3 0, G2 0, K xµ3 , F t.So that, one obtains the linearizing transformationX t, dT xµ3 x′ dt.(57)Hence, equation (51) is mapped by the transformation(57) into the linear equationA3 0, A2 µ3 xk3 , A1 µ2 xk2 ,A0 µ1 xk1 , λ1 k2 µ2 xk2 ,′′(58)X 0.(k1 k3 )λ2 xµ1 µ3 k1 x x(k1 k3 )µ1 µ3 k3 xThe general solution of equation (58) is xk1 µ1 k12 xk1 µ1 k1 x2k2 µ22 k2 x.Now, the conditions in Theorem 2.2. case (a) is satisfiedwhen the following conditions holds, that are,k2 µ2 xk2 0,x(k1 k3 ) µ1 µ3 k1 x x(k1 k3 ) µ1 µ3 k3 x xk1 µ1 k12 xk1 µ1 k1 x2k2 µ22 k2 x 0.(59)X c1 T c2 ,where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (57) to equation(59), we obtain that the general solution of equation(51) is t c1 ϕ(t) c2 ,Two cases arise.where the funtion T ϕ(t) is a solution of the equationCase 1: µ2 0 and µ1 0In this case, the equation (51) takes the formdT xµ3 x′ .dtCase 2: k2 0 and µ1 0In this case, the equation (51) takes the formx′′ µ3 xk3 x′2 0.(52)The linearizing transformation is found by solving equations in Theorem 2.3 case (a), which becomeFt GK1 , Kx µ3 xk3 K,K(2G1t µ3 xk3 G1 G3 ), G3tG1µ3 k3 xk3 G3G3xx , G1x xKt µ3 xk3 G23 ,µ3 x 3k3 1K eK ex′ dt.(54)Hence, equation (52) is mapped by the transformation(54) into the linear equation′′(55)X 0.The general solution of equation (55) isX c1 T c2 ,G3t (61),One can find the particular solution for equations in (61)asµ3 xk3 1G1 e k3 1 , G3 0, G2 0,So that, one obtains the linearizing transformationµ3 xk3 1k3 1K(2G1t µ3 xk3 G1 G3 µ2 G1 ),G1G3 µ3 k3 xk3k3G3 (x G3 µ3 µ2 ), G3xx xG1x G1 µ3 xk3 .Kt , F t.X t, dT eFt GK1 , Kx Kµ3 xk3 ,µ3 xk3 G1 ., G3 0, G2 0,µ3 xk3k3 1(60)The linearizing transformation is found by solving equations in Theorem 2.3 case (a), which become(53)One can find the particular solution for equations in (53)ask 1G1 ex′′ µ3 xk3 x′2 µ2 x′ 0.µ3 xk3k3 1 µ2 t, F eµ 2 tµ2 .So that, one obtains the linearizing transformationX µ3 xk3 1eµ2 t, dT e k3 1 x′ dt.µ2Hence, equation (60) is mapped by the transformation(62) into the linear equation′′X 0.(56)where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (54) to equation(62)(63)The general solution of equation (63) isVolume 50, Issue 4: December 2020X c1 T c2 ,(64)

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (62) to equation(64), we obtain that the general solution of equation(60) iseµ2 t c1 ϕ(t) c2 ,µ2where the function T ϕ(t) is a solution of the equationwhere k3 ̸ 1.For k3 1, one can find the particular solution forequations in (61) asG1 xµ3 , G3 0, G2 0,µ2 tK xµ3 e µ2 t , F eµ2 .So that, one obtains the linearizing transformationeµ2 t, dT xµ3 x′ dt.µ2(65)Hence, equation (60) is mapped by the transformation(65) into the linear equation′′(66)X 0.2ut utx 2[uux utx ut (uuxx u2x )] 2u2 uxx 2uu3x ρuux βux q(ut uux ) 0.(69)Of particular interest among solutions of equation (69)are travelling wave solutions:u(t, x) H(x Dt),µ3 xk3 1dT e k3 1 x′ ,dtX Consider a modified generalized Vakhnenko equation(68), we can rewrite it in the formwhere D is a constant phase velocity and the argumentx Dt is a phase of the wave. Substituting the representation of a solution into equation (69), one finds2D2 H ′ H ′′ 2DH ′ (2HH ′′ H ′2 ) 2H 2 H ′ H ′′ 2HH ′3 ρHH ′ βH ′ q( DH ′ HH ′ ) 0.(70)By using the obtained theorems, we get the results asfollow. Equation (70) is an equation of the form inTheorem 2.1 with the coefficients1, A1 0,A3 0, A2 (D H)ρH β qD qHA0 2(D2 2DH H 2 ) , λ1 0, λ2 ρD β.From Theorem 2.2. case (a), equation (70) is linearizableif only if ρD β 0.The general solution of equation (66) is(67)X c1 T c2 ,where c1 and c2 are arbitrary constants. Applying thegeneralized linearizing transformation (65) to equation(67), we obtain that the general solution of equation(60) iseµ2 t c1 ϕ(t) c2 ,µ2where the function T ϕ(t) is a solution of the equationdT xµ3 x′ ,dtRemark 3.1: The conditions in Theorem 2.2. case (b)are satisfied if only if µ1 0.F. Modified Generalized Vakhnenko EquationIn 2009, Ma, Li and Wang [14] focus on a modifiedgeneralized Vakhnenko equation (mGVE), 1 (L2 u pu2 βu) qLu 0, L u , (68) x2 t xwhere ρ, q, β are arbitrary non-zero constants.To develop the specific solutions for mGVE is exceedingly significant. For models, when ρ β 0 and q 1,equation (68) is reduced to notable Vakhnenko equation(VE), which oversees the nonlinear engendering of highrecurrence wave in a loosening up medium [15]-[17]. TheVE has solition solutions [17]. When ρ q 1 and βan arbitrary non-zero constant, equation (68) is becomeas the generalized VE (GVE), in [18] it was indicatedthat GVE has N-soliton solution. When ρ 2q and βis an arbitrary non-zero constant, equation (68) has aloop-like, hump-like and cusp-like soliton solutions [19].In [20], it was appeared that equation (68) has travellingwave solution and single-soliton solution.G. Burgers’ EquationBurgers’ equation is acquired because of the relationship between nonlinear wave movement and lineardiffusion. It is the model for the investigation of consolidated impact of nonlinear advection and diffusion. Thepresence of the viscous term covers the wave-breaking,smooth out stun discontinuities, and thus we wish to geta tide and smooth solution. Also, as the dispersion termturns out to be vanishingly small, the smooth viscoussolutions converge non-uniformly to the appropriatediscontinuous shock wave, causing to another system forexamining traditionalist nonlinear dynamical processes.Consider the nonlinear convection-diffusion equation u u 2u u v 2 0, v 0,(71) t x xwhich is known as Burgers’ equation. This equationbalances between time advancement, nonlinearity, anddissemination. This is the nonlinear model equation fordiffusive waves in fluid dynamics. Burgers (1948) firstbuilt up this equation basically to illuminate disturbancedepicted by the collaboration of two inverse impacts ofconvection and dissemination.The term uux will have a stunning up impact that willmake waves break and the term vuxx is a diffusion termlike the one appearing in the heat equation.Of particular interest among solutions of equation (71)are travelling wave solutions:u(t, x) H(x Dt),where D is a constant phase velocity and the argumentx Dt is a phase of the wave. Substituting the representation of a solution into equation (71), one finds DH ′ HH ′ vH ′′ 0.Volume 50, Issue 4: December 2020(72)

IAENG International Journal of Applied Mathematics, 50:4, IJAM 50 4 19By using the obtained theorems, we get the results asfollow. Equation (72) is an equation of the form inTheorem 2.1 with the coefficientsA3 0, A2 0, A1 D Hv ,

equations to second-order linear ordinary differential equations. Moreover, Lie discovered that every second-order ordinary differential equation can be reduced to second-order linear ordinary differential equation with-out any conditions via contact transformation. Having mentioned some methods above, there are yet still other methods to solve .

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