Uttam Ghosh (1), Srijan Sengupta (2a), Susmita Sarkar (2b), Shantanu Das (3)

Transcription

1 Analytical soltion with tanh-method and fractional sb-eqation method for non-linear partial differential eqations and corresponding fractional differential eqation composed with Jmarie fractional derivative Uttam Ghosh (, Srijan Sengpta (a, Ssmita Sarkar (b, Shantan Das (3 (: Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India; (: Department of Applied Mathematics, University of Calctta, Kolkata, India (b: (3Scientist H+, RCSDS, Reactor Control Division BARC Mmbai India Senior Research Professor, Dept. of Physics, Jadavpr University Kolkata Adjnct Professor. DIAT-Pne UGC-Visiting Fellow. Dept of Appl. Mathematics; Univ. of Calctta, Kolkata (3: Abstract The soltion of non-linear differential eqation, non-linear partial differential eqation and nonlinear fractional differential eqation is crrent research in Applied Science. Here tanh-method and Fractional Sb-Eqation methods are sed to solve three non-linear differential eqations and the corresponding fractional differential eqation. The fractional differential eqations here are composed with Jmarie fractional derivative. Both the soltion is obtained in analytical traveling wave soltion form. We have not come across soltions of these eqations reported anywhere earlier. Keywords: Tanh-method, Fractional Sb-Eqation Method, Bossinesq eqation, Non-linear Evoltion eqation, Fractional Differential Eqation, Jmarie Fractional Derivative. Introdction The advanced analytical methods to get soltion to non-linear partial differential eqation is crrent research field in different branches of applied science sch as Applied Mathematics, Physics, Biology and Engineering applications. On the other hand fractional calcls and conseqently soltions of fractional differential eqations is also another growing field of science especially in Mathematics, Physics, Engineering and other scientific fields like controls etc [-5, 5]. The eact soltions of non-linear partial differential eqation and the corresponding fractional differential eqations are physically very important [5-7]. To solve sch type nonlinear differential eqations different methods are sed, sch as Adomain Decomposition Method (ADM [6-7, 5], the Variational Iterative Method (VIM [8-], the Homotopy Pertrbation Method (HPM [-3], Differential Transform Method (DTM []. Another

2 important method to find the traveling wave soltion of the non-linear partial differential eqation is tanh- method was first introdced by Hibin and Kelin [5]. This method was sed directly into a higher-order KdV eqation, to get its soltions. Wazwaz [6, 7] sed tanhmethod to find traveling wave soltions of some non-linear eqations generalized Fisher s eqation, generalized KdV eqation, Zhiber-Shaba eqation and other related eqations. Recently Zhang and Zhang [8] introdce Fractional Sb-Eqation Method to find travelling wave soltion of the non-linear fractional differential eqation. The principle of this method is epressing the soltion of the non-linear differential eqation as polynomials, where the variables of which is one of the soltion of simple and solvable ordinary and partial differential eqation. The method is also based on Homogeneos Balance Principle [9, ] and applied on Jmarie s modified Riemann-Liovelli derivative [, ]. The Fractional Sb-Eqation Method sed by many athors [9,, ] to find the analytic soltion of non-linear fractional differential eqations. Rest of the paper is as follows-in section. we describe tanh-method, in section 3. we describe Fractional Sb-Eqation method, in section 7.-. we describe the soltion of three non-linear eqations in both integer order partial differential and fractional order partial differential eqation types. In this paper we find the soltion of three non-linear eqation sing tanh-method and the corresponding non-linear fractional differential eqation sing Fractional Sb-Eqation method. In all the cases analytical soltions obtained in travelling J wave soltion form. In this paper the symbols for fractional differential operator sed D, means Jmarie fractional derivative.. Tanh-method Let the non-linear partial differential eqation is of the form ( f H (, t,,......( Where (, t is fnction of, t. For instant t is of sch an eqation as H (,,, with H a linear /non-linear operator. Using t travelling wave transformation k + ct, with k constant called wave-nmber (or propagation constant and c a constant; as velocity of propagating wave, eqation ( redce to an ordinary differential eqation in the form H (,,......(

3 Where (, is fnction of one variable. In smmary we are making several variables (, yzt,, to a single variable, with travelling wave transformation sch as + + kz+ ct, with corresponding constants k, k, k, c as described above, and k ky y z is in nits of distance. Ths the operator t y, in terms of with k + ct is following d c t t d z and similarly is d d k k k The transformed eqation, is ths, t which is H (,,. d d k c or k c d d Again, let φ φ( be sch type of fnction which is a soltion of Riccati eqationφ ( σ + φ. Whereσ is a real constant. That is d φ d σ + φ, d Then The above epression is detailed below briefly. σ + φ σ tanh( σ forσ < σ coth( σ σ tan( σ φ ( forσ> σ cot( σ forσ Case-I For σ < then the Riccati eqation can be written eqation, considering integral constant as zero 3

4 d σ + φ Therefore, by inverting above we get the following tanh σ coth σ ( σ ( σ φ ( σ φ ( σ σ tanh φ σ coth Case-II For σ then the Riccati eqation can be written eqation, ignoring integration constant d φ φ φ Case-III For σ > then the Riccati eqation can be written eqation, considering integral constant as zero tan φ ( σ σ d σ + φ cot φ ( σ σ Therefore, inverting above we get the following ( σ ( σ σ tan φ σ cot The soltion of the eqation ( will be epressed in-terms of φ in the form n i ( S aφ...(3 i i Where n is obtained balancing the highest order derivative term and the non-linear terms in eqation (. Then pt the vale of ( from (3 to the eqation ( and comparing the coefficient of i φ Consider the eqation the vales of coefficients can be obtained. t considered eqation redces to. Using the travelling wave transformation k + ct the k c...(

5 Then sing the transformation as defined in (3 in-terms of φ and sing the chain rle of derivative we obtain d d d d sec h ( φ d d d d d d d d d ( φ d ( ( φ φ φ φ d d d d Ptting this transformation in eqation ( we get d S ds ds k ( φ φ( φ cs( φ (5 In the highest order derivative term the highest order ofφ is n + and in the non-linear term it is n +. Eqating them we get n. We have (3 as following Therefore ( S a + aφ ds d S a and Ptting the above vale in eqation (5 we get the following ( φ( φ + φ ( φ a k a c a a Comparing the like powers of φ from both sides of above epression we get a k and a c Hence the corresponding soltion is k k ( S tanh( c φ c that is k t (, tanh( k+ ct c 5

6 3. Fractional sb-eqation method The space and time fractional partial differential eqation is of the form L (, t,,... < <...(6 where (, t and L is fnctions (linear or non-linear operator of the and its derivatives. is order of fractional derivative of Jmarie type, defined as follows J ( f( d, < Γ ( ( d D [ f( ] f ( ( ( f( f( d, < Γ ( d < ( n ( n ( f (, n < n+, n We consider that at < the fnction f( and also f( f( for <. The fractional derivative considered here in the fractional differential eqation are sing Jmarie [] modified Riemann-Lioville (RL derivative which is defined as above. The first epression above is fractional integration of Jmarie type. The modification by Jmarie is to carry RL fractional integration or RL fractional derivation is by forming a new fnction by offsetting the original fnction by sbtraction the fnction vale at the start point; and then operate the RL definition. The above defined fractional derivative has the following properties J γ Γ ( + γ γ D, γ >, Γ ( + γ D f g g D f f D g [ ( ( ] ( ( ( + ( ( ( [ ] ( ( [ ] ( [ ] J J J D f( g ( f g ( D g ( D f( g ( ( g J J J g g The eqation (7 is eample of the fractional differential eqation t.(7 Using the travelling wave transformation k + ct, the eqation (6 redce to (,,,... < <...(8 L Where, ( is fnction of one variable 6

7 d Using the same transformation, the eqation (7 redce to Ths the operator t, in terms of with k + ct is d c c t t d, as described below d d d J ( D and similarly is d d d k k k k D d d d The transformed eqation is ths, k c that is L (,,. The soltion of the eqation (8 is taken in the following series form J ( n i ( bφ...(9 i Where Φ( satisfy the fractional Riccati eqation i Φ σ +Φ < < (, Here σ is a real constant. Say for we obtain the Riccati eqation as dφ we write d, integrating both sides once we get dφ +Φ σ σ symbol, we write as D (Jmarie type the following Φ σ +Φ +Φ d Φ d σ +Φ. From this. In terms of anti-derivative. In similar way we write in terms of fractional integration σ + Φ D Φ Similar to the previos method the coefficients can be obtained. Zhang et al [9] develop the method finding the soltion fractional Riccati eqation in the following form sing generalized hyperbolic and trigonometric fnction 7

8 ( σ ( σ ( σ ( σ σ tanh forσ < σ coth σ tan Φ ( forσ > σ cot Γ ( + forσ + ω Where, the defined higher order transcendental fnctions are as follows sinh ( cosh ( tanh ( coth ( cosh ( sinh ( E( E( E( + E( sinh ( cosh ( sin ( cos ( tan ( cot ( cos ( sin ( E( i E( i E( i + E( i sin ( cos ( Where ( z E z ( k is the one parameter Mittag-Leffler fnction. k Γ +. Nonlinear evoltion eqations (NEEs of shallow water waves (SWW partial and fractional differential eqations The nonlinear evoltion eqation of the shallow water wave eqation is t +...( y + pt + q t Where (, y, t. The Corresponding space and time fractional differential eqation is Where for < <, we have the following p q ( t y t t t y 3 t y t 8

9 5. Kadomtsev-Petviashvili (KP partial and fractional differential eqation The KP eqation was first written in 97 by Soviet physicists Boris B. Kadomtsev and Vladimir I. Petviashvili. It came as a natral generalization of the KdV eqation (derived by Korteweg and De Vries in 895. Whereas in the KdV eqation waves are strictly one-dimensional, in the KP eqation this restriction is relaed. The KP eqation sally written as ( ( yy t Where (, y, t. The Corresponding space and time fractional differential eqation is ( (3 3 y t Where y t 3 3 y, 3 < < 6. Forth order Bossinesq eqations partial and fractional differential eqation In flid dynamics, the Bossinesq approimation for water waves is an approimation valid for weakly non-linear and fairly long waves. The approimation is named after Joseph Bossinesq, who first derived them in response to the observation by John Scott Rssell of the wave of translation (also known as solitary wave or soliton. The 87 paper of Bossinesq introdces the eqations now known as the Bossinesq eqations. The forth order Bossinesq eqation can be written in the following form. Solitons are solitary waves with hge amplitde locally, and zero amplitde at far away, with very high travelling velocity tt + 3( +...( Where (, t.the Corresponding space and time fractional eqation is + 3( +...(5 t Where t t t, < < 9

10 7. Soltion of Nonlinear evoltion eqations (NEEs of shallow water waves (SWW sing tanh method Eqation ( that is t + y + p t + qt is non-linear eqation in (, y, t. Using travelling wave transformation in the form ct + k + my and following sbstittion t ck k k m k c 3 y t t k c in eqation (, is redced to the ordinary differential eqation form + ( + + ( +...(6 km ckp q c k Since the solitary waves are localized in space therefore the soltion and its derivatives at large distance from the plse are known to be etremely small and they vanishes ± [8]. Integrating once w.r.t the above eqation and by assming the condition the derivative terms tend to zero as we get km + ckp ( + q + ( c+ k...(7 Here we consider the soltion in the form n ( φ φ φ... φ...(8 S a + a + a + + a n Where φ ( tanh( Then d d ( φ d d d d ( φ φ( φ d 3 3 d 3d d ( 6 ( ( ( d φ d φ φ d φ φ φ d φ φ

11 3 d d 3d ( φ φ( φ 3 d d d (36φ 8( φ + φ( φ (8 φ Ptting the vale of and sing the transformation define above eqation (7 redce to the form d S km( 6 ( d S 3 3 φ φ φ 3 ds φ φ + + φ ds + + φ ( ( 3 ck( p q ( ds ( c k(...(9 In the highest order derivative term the highest order ofφ is n + 3 and in the non-linear term it is n +. Eqating them we get n. We have (8 as( S a + aφ (. Ptting the vale of S in (9 we get ( km( φ ( 3 φ a+ ckp ( + q ( φ a + ( c+ k( φ a...( Comparing the powers of φ from both side the following relations obtained for a φ : ( c+ k + ack( p+ q k m φ :( c+ k ack( p+ q 6k m φ : ack( + β k m 3 km Solving the above we get and ( c + k k m. a c( p+ q Therefore the soltion is for all a R. km yt (,, a + tanh ct+ k+ my c( p+ q (

12 Soltion of Nonlinear evoltion eqations Figre : Graphical presentation of for fied y and considering p q m k, c Soltion of Nonlinear fractional evoltion eqations (NEEs of shallow water waves (SWW sing Fractional Sb-eqation method Using the same travelling wave transformation as define in section-7. and with following sbstittions c k t 3 y k t c k k m in eqation (6, that is + + p + q t y t t redce to the form Consider the soltion in the form ( k m + k c ( p+ q ( k + c...( a a a a n n ( + Φ+ Φ Φ...( Where Φ is define in section 3. Comparing the highest powers Φ from the highest order derivative term and the non-linear derivative term from ( and sing from ( we get for n+ 3n+ 3or n. Therefore, a + a Φ(. Ptting the vale of from the

13 above in ( and comparing the like powers of Φ the following relations is obtained in simplest form and sing the relation a Φ : ( c + k σ ac k ( p+ q σ + 6k m σ 3 Φ : ( c + k ac k ( p+ q σ + k m 5 Φ : k m ac k ( p+ q k m Solving the above relation we get a β β and ( c + k + σ k m. Therefore the soltion is following c ( + β k m a + σ tanh ( σ( ct+ k+ my c ( p+ q for σ < k m a + σ coth ( σ( ct+ k+ my c ( p+ q k m Γ ( + yt (,, a + forσ, ω Cons tan t....(3 c ( p+ q ( ct+ k+ my + ω k m a σ tan ( σ( ct+ k+ my c ( p+ q for σ > k m a + σ cot σ( ct k ( + + my c ( p+ q For all a R. Soltion of Nonlinear evoltion eqations for alpha. Soltion of Nonlinear evoltion eqations for alpha.6 Soltion of Nonlinear evoltion eqations for alpha Figre : Graphical presentation of for fied y and considering p q m k, c 3, forσ. 3

14 9. Soltion of Kadomtsev-Petviashvili (KP eqation sing tanh method Using the same travelling wave transformation as define in section-7. and m, k, k & c eqation (, that is 3 yy t ( yy t redces to integration m ( 6...( 3 m kc + k + k Integrating both side two time w.r.t and then sing the condition the solitary waves are localized in space that is,,.. all tends to zero when ζ [8]. We get after first 3 kc ( + 6k + k and after second integration we get ( (5 m kc k k We consider the soltion as defined in eqation (8, that is Then ptting the vales of from (8 in (5 we get n ( S a + aφ+ aφ a n φ. d S ds ( m kc S 3 k S + k ( φ φ( φ...(6 Comparing the highest order derivative and the non-linear term we get for n. Therefore, S a + aφ + a φ, Ptting the vale of in (6 and comparing the like powers ( ( ( of φ the following relations are obtained φ : ( m kc a + 3k a + a k φ : ( m kc a + 6k a a ak φ : ( m kc a 3 k ( a + a a 8a k φ : 6aak + ak 3 φ : 3ak + 6ak Solving the above we get Hence the soltion is 8k + m kc a 6k, a and a k with the relation 9a + k 8k 8 k + m kc yt (,, k tanh ct+ k+ my 6k (

15 Soltion of Kadomtsev-Petviashvili eqation Figre 3: Graphical presentation of for fied y and considering m k, c Soltion of Kadomtsev-Petviashvili (KP eqation sing Fractional Sbeqation method The eqation (3, that is ( 6 3 y t + + can be written in the following form + 6( (7 t yy Using the same travelling wave transformation ct + k + my and with following sbstittions kc t ( k ( k m yy eqation (7 redce to ( ck m + 6 k ( + 6k + k... (8 We want to find the soltion in the form as define in eqation (, that is n ( a + aφ+ a Φ a n Φ, and then comparing the highest order derivative term and the non-linear term we get for n. Therefore ( a (7 and comparing the coefficients of powers of Φ( we get 5 + a Φ( + a Φ (. Ptting in eqation

16 Φ : ( ck m aσ + 6 k ( aaσ + aσ + 6k aσ 3 Φ : ( ck m aσ + 6 k (aaσ + 6 aaσ + 6k aσ Φ : 8( ck m aσ + 6 k (8aaσ + a σ + aσ + 36k aσ Φ : ( ck m aσ + 6 k (aa+ 8 aaσ + k a 3 Φ : 6( ck m a+ 6 k (6aa+ 6aσ + 3 a + k aσ Φ : 7k aa + ak 5 Φ : 6k a + a k 6 Solving we get a, a and a m 8σ k k c 6k m k k c m k k c k. Hence the soltion is following ( σ ( σ 8σ + k σ tanh ( ct+ k+ my 6k for σ < 8σ + k σ coth ( ct+ k+ my 6k ( Γ ( + m 8σ k k c ( k for σ, ω Cons tan t. ( ω 6 k ( ct k my m 8σ k k c k σ tan ( c ( σ t+ k+ my 6 k for σ > m 8σ k k c k σ cot ( σ( ct k my + + 6k 6

17 Soltion of Kadomtsev-Petviashvili eqation for alpha. Soltion of Kadomtsev-Petviashvili eqation for alpha.6 Soltion of Kadomtsev-Petviashvili eqation for alpha Soltion of Kadomtsev-Petviashvili eqation for alpha.8 Soltion of Kadomtsev-Petviashvili eqation for alpha.85 Soltion of Kadomtsev-Petviashvili eqation for alpha Figre : Graphical presentation of for fied y and considering m k, c 3.68, forσ.. Soltion of Forth order Bossinesq eqation sing tanh Method Eqation (, that is non-linear eqation in (, t. Using travelling wave transformation in the form k + ct and with following sbstittion c tt k ( k ( k in eqation ( the eqation redced to the form ( c k 3 k ( + k...(9 7

18 Here we consider the soltion as defined in eqation (8, that is n ( S a + aφ+ a φ a n φ. Ptting the vales of from (9 in eqation (9 and sing the transformation as define in section 7. we get d S ds ( c k ( φ φ( φ d S ds ds 3 k {( φ φ( φ } S ( φ + + k 3 d S 3d S ( φ φ( φ 3 d S ds + ( φ (36φ 8 + φ( φ (8 φ......(3 Comparing the highest order derivative term and the non-linear term we obtain n. Therefore ( S a + a φ( + a φ (. Ptting this vale of S in eqation (3 and comparing the powers of φ we get φ : ( c k a 3 k (a a + a 6a k φ : ( c k a 3 k ( aa + aa + 6ak φ : 8( c k a 3 k ( 6a a + a + 36a k φ : ( c k a 3 k (aa 36 aa ak 3 φ : 6( c k a 3 k ( 6a + a a a k 5 φ : a k + 7aa k φ : 6 ak ( a + k 6 Solving the above set of eqation we get is 8k + c k a 6k 8k + c k 6k, a and a k. Hence the soltion yt (,, k tanh ct+ k ( 8

19 Soltion of Forth order Bossinesq eqation Figre 5: Graphical presentation of for fied y and considering c k..: Soltion of Forth order fractional Bossinesq eqation sing Fractional Sb-eqation method Using the same type of travelling wave transformation ct + k and with following sbstittion c t k ( k ( k,, the fractional Bossinesq eqation (5 which is following form + 3( + redce to the t ( c k 3 k ( k +...(3 We want to find the soltion in the form as define in eqation (, comparing the highest order derivative term and the non-linear term we get n. Therefore, ( a + aφ( + aφ (. Comparing the like powers of Φ ( we get the following 9

20 Φ : ( c k a σ 3 k σ (a a + a + 6a k σ 3 Φ : σ( c k a k ( a aσ + 3 aa σ + 6aσ k Φ : 8 σ ( c k a k ( a a + a σ + 36σ a k + 36a k σ Φ : ( c k a k a a + 8aa σk + aσk Φ 3 : 6( c k a 8 k ( a + a a + 96k a σ + σa k Φ : ak + 7aak 5 Φ : 6 ak ( a + k 6 Solving the above system of eqation we get a soltion is, a and a c 8σ k k 6k ( σ ( σ c 8σ k k + k σ tanh ( ct+ k 6k for σ < c 8σ k k + k σ coth ( ct k + 6k ( Γ ( + ( + + ω k. Hence the c 8σ k k (, k forσ, ω Cons tan t. y 6 k ( ct k c 8σ k k k σ tan ( σ( ct+ k 6k for σ > c 8σ k k k σ cot ( σ( ct k + 6k Soltion of Forth order Bossinesq eqation for alpha.6 Soltion of Forth order Bossinesq eqation for alpha.8 Soltion of Forth order Bossinesq eqation for alpha Figre 6: Graphical presentation of for fied y and considering c k, forσ. From figre -6 it is clear that soltions of the fractional differential eqations depend on order of fractional derivative.

21 3. Conclsion In all the three cases the soltion obtained by tanh-method coincides with one soltion of obtained by Fractional Sb-Eqation method, when fractional order tends to one. Ths the soltion of the non-linear partial differential eqation and the non-linear fractional differential eqation is obtained in analytic form. However, the soltion obtained by fractional sb-eqation method is more general compare to the tanh-method, as it encompasses the fractional generalization of the non-linear partial differential eqations. Reference [] Kilbas A, Srivastava HM, Trjillo JJ. Theory and Applications of Fractional Differential Eqations. North- Holland Mathematics Stdies, Elsevier Science, Amsterdam, the Netherlands. 6;:-53. [] Miller KS, Ross B. An Introdction to the Fractional Calcls and Fractional Differential Eqations.John Wiley & Sons, New York, NY, USA; 993. [3] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon, Switzerland; 993. [] Podlbny I. Fractional Differential Eqations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. 999;98. [5] Oldham KB, Spanier J. The Fractional Calcls, Academic Press, New York, NY, USA; 97. [6] El-Sayed AMA, Gaber M. The adomian decomposition method for solving partial differential eqations of fractal order in finite domains. Phys. Lett. A. 6;359(:75-8. [7] A El-Sayed AMA, Behiry SH, Raslan WE. Adomian s decomposition method for solving an intermediate fractional advection dispersion eqation. Compt. Math. Appl. ;59(5: [8] J.H. He, Commn. Nonlinear Sci. Nmer. Siml., 3-35 (997. [9] W GC, Lee EWM. Fractional variational iteration method and its application. Phys. Lett. A. ;37(5: [] S. Go and L. Mei, Phys. Lett. A 375, ( 39. [] He J-H, Homotopy pertrbation techniqe, Compt Methods. Appl. Mech. Eng.999;78(3-:57-6. [] J.H. He, Inter. J. Non-Linear Mech. 35, 37-3 (. [3] H. Jafari, S. Momani, Solving fractional diffsion and wave eqations by modified homotopy pertrbation method, Phys. Lett. 7 A [] Z. Odibat and S. Momani, Appl. Math. Lett., 9-99(8. [5] L. Hibin, W. Kelin, Eact soltions for two nonlinear eqations: I, J. Phys. A: Math. Gen. 3 (

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