Improved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method

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1 Journal of Applied Mahemaics & Bioinformaics, vol., no., 01, 1-14 ISSN: (prin), (online) Scienpress Ld, 01 Improved Approimae Soluions for Nonlinear Evoluions Equaions in Mahemaical Physics Using he Reduced Differenial Transform Mehod Mahmoud Rawashdeh 1 Absrac In his paper, an improved mehod called he Reduced Differenial Transform Mehod (RDTM) was used o obain approimae numerical and eac soluions for hree differen ypes of nonlinear parial differenial equaions (NLPDEs), such as; Gardner equaion, Varian Nonlinear Waer Wave equaion (VNWW), and he Fifh-Order Koreweg-de Vries (FKdV) equaion. The heoreical analyses of he RDTM are invesigaed for hese equaions and are calculaed in he form of a series wih easily compuable erms. The resuls we obained are compared wih he analyical soluions obained by oher mehods used in he pas. One can conclude ha only few erms of he series epansion are required o obain approimae soluions using he RDTM wih an ecellen accuracy. Mos of he symbolic and numerical compuaions were performed using Mahemaica sofware. 1 Deparmen of Mahemaics and Saisics, Jordan Universiy of Science and Technology, Irbid, 110, Jordan, msalrawashdeh@jus.edu.jo Aricle Info: Received : November 11, 01. Revised : December 9, 01 Published online : June 0, 01

2 Improved Approimae Soluions for Nonlinear Evoluions Equaions Mahemaics Subjec Classificaion: 5J05, 5J10, 5K05, 5L05 Keywords: Reduced Differenial Transform Mehod, Differenial Transform Mehod, Gardner equaion, FKdV equaion 1 Inroducion The Reduced Differenial Transform Mehod [9-11], was firs inroduced by Kesin o solve linear and nonlinear PDEs ha appears in many Mahemaical physics and engineering applicaions. For nonlinear models, he RDTM has shown dependable resuls and gives analyical approimaion ha converges very rapidly and in some cases gives eac soluions. Many numerical mehods were used o solve nonlinear parial differenial equaions, such as, he Adomian Decomposiion Mehod (ADM) [1, ], he Differenial Transform Mehod (DTM) [4], and he Variaional Ieraion Mehod (VIM) [6]. In his paper, we solve he following NLPDEs: Firs, consider he Gardner equaion u = uu + u+ u, (1) 6 6 subjec o he iniial condiion 1 u (,0) = 1 anh. () Second, Nonlinear Varian Waer Wave equaion he: ( ) 0 u + u + u + u + uu = () subjec o he condiions 10 u (,0) = anh 10, Third, he FKdV equaion: 9 = (4) u (,0) sec h u + uu uu + u = 0, (5)

3 Mahmoud Rawashdeh subjec o he condiion u (,0) = e. (6) The goal of he sudy is o use he RDTM o solve hree differen ypes of nonlinear parial differenial equaions (NLPDEs). Efficiency and simple applicabiliy of he mehod for he soluion of complicaed nonlinear parial differenial equaions are he main highlighs of his sudy. Kesin, in his PhD hesis [1], inroduced he reduced form of he differenial ransform mehod (DTM) as reduced differenial ransform mehod (RDTM) and he used he RDTM o solve he Gas Dynamics Equaion and linear and nonlinear Klein Gordon Equaions and more. Also, Kesin and Ouranc (010) used he RDTM o solve linear and nonlinear wave equaions and hey showed he effeciveness, and he accuracy of he mehod. Moreover, hey showed ha he number of ieraions is less han he one used by he DTM. Finally, Alquran [4] used he DTM o solve he Gardner equaion and Kaya and Al-Khaled [9], find a numerical soluion o he Kawahara equaion. Analysis of he RDTM In his secion, we sar wih a funcion of wo variables u(, ) which is analyic and imes coninuously differeniable wih respec o ime and space in he domain of our ineres. Assume we can represen his funcion as a produc of wo single-variable funcions, namely u( ) definiions of he DTM, he funcion can be represened as follows: u(, ) = Fi ( ) G( j ) = U ( ). i= 0 j= 0 = 0 where ( ), = f( ). g ( ). From he i j, (7) U is he ransformed funcion of (, ) u which can be defined as:

4 4 Improved Approimae Soluions for Nonlinear Evoluions Equaions 1 U () = u (,)! = 0 From equaions (7) and (8) we can deduce = 0 = 0. (8) 1 u(,) = u (,)!. (9) Some basic properies of he reduced differenial ransformaion obained from equaions (7) and (8) are given as follows: Theorem.1 If f(,) = αu (,) ± βv (,), hen F ( ) = αu ( ) ± βv ( ), where α and β are consan. Theorem. If f(,) = u (,).(,) v, hen Theorem. If f(,) = u (,).(,). v w (,), hen =. i F( ) U( ) V ( W ) ( ) j i j i i= 0 j= 0 =. F ( ) U ( ) V ( ) i i i= 0 n ( + n)! Theorem.4 If f(,) = u (,), hen F ( ) ( ) n = U+ n. K! n n Theorem.5 If f(,) = u (,), hen F ( ) ( ) n = U n. Theorem.6 If f(,) m u n (,) m =, hen F( ) U ( ) =. n Theorem.7 If f(,) 1, = n whereδ ( n) =. 0, n m n m =, hen F ( ) δ ( n) =,

5 Mahmoud Rawashdeh 5 The proofs of he above heorems and more properies can be found in [1]. To illusrae he RDTM, we wrie he Gardner equaion in sandard form ( ) ( ) ( ) ( ) L u (,) 6 L u (,) L u (,) N u (,) = 0, (10) subjec o iniial condiions where u (,0) = f( ), u(,0) = g ( ), (11) L =, L =, and ( (,)) N u is he nonlinear erm. Now from equaion (10) and (11), we can derive he recursive formulas (according o he heorems menioned above) as: and ( + 1 ) U ( ) = ( U ( ) ) + 6 U ( ) + N( u(,) ) + 1 (1) U ( ) = f( ), U( ) = g ( ). (1) 0 1 To find he res of he ieraions, we firs subsiue equaion (1) ino equaion (1) and hen we find he values of U ( ) s. Finally, we apply he inverse ransformaion o all he values { } 0 U ( ) n = o obain he approimae soluion: n u(,) = U () = 0, (14) where n is he number of ieraions we have used o find he approimae soluion. Hence, he eac soluion of our problem is given by u (,) = lim u(,). (15) n Applicaions In his secion, we es he RDTM on hree numerical eamples and hen compare our approimae soluions o he eac soluions.

6 6 Improved Approimae Soluions for Nonlinear Evoluions Equaions.1 Eamples Now, we presen hree eamples o show he efficiency of he RDTM. Eample.1.1 Consider he Gardner equaion u = uu + u+ u, (16) 6 6 subjec o he iniial condiions 1 u (,0) = 1 anh, 1 u (,0) = sech 4, (17) where he eac soluion is 1 u (, ) = 1 anh. (18) Applying he RDTM o (16) and (17), we obain he recursive relaion i 1 U + 1( ) = ( U( ) ) 6 U( ) 6 U i( ) Ui j( ) ( U j( ) ) ( 1) i= 0 j= 0, (19) where he U ( ), is he ransform funcion of he dimensional specrum. Noe ha 1 U0( ) = 1 anh, U ( ) (0) = sech Now, subsiue Eq. (0) ino Eq. (19) o obain he following: sech U ( = sech sech sech + ( 6sinh( ) 4sinh( ) ), ) 7 7 cosh( ) 108 And so on. So afer he hird ieraion, he differenial inverse ransform of{ U ( )} will provide us wih he following approimae soluion: = 0 (1)

7 Mahmoud Rawashdeh 7 u, = U( ) = U( ) + U( ) + U( ) + U( ) +... ( ) = = 1 anh sech 4 sech + sech sech cosh( ) 6sinh( ) 4sinh( ) sech + 471cosh( ) sech sech sech (79sinh( ) 46sinh( ) + 6(84 + sinh( ))) + 1+ cosh( ) Eample.1. We consider he Nonlinear Varian Waer Wave equaion ( ) 0 u + u + u + u + uu =, () subjec o he condiions (, 0) anh, (, 0) sech 5 u u anh 5 = 10 = , () where he eac soluion 10 9 u (, ) = anh (4) 10 5 Similar o he previous eample, by he heorems above applied o Eq. () and Eq. () we ge 1 ) ) ) ( ) ( ( U ) ( )) (5) 5 U + 1( = ( U( ) + U( + 5 ( U ) + Ui i ( + 1) i= 0 and

8 8 Improved Approimae Soluions for Nonlinear Evoluions Equaions U0( ) = anh, U1( ) = sech anh, (6) where, he U ( ), is he ransform funcion of he dimensional specrum. Now, subsiue Eq. (6) ino Eq. (5) o obain he following: ( 6+ 5 ) sech sech 88cosh( ) 5cosh( ) cosh( ) 41cosh 88cosh( ) 8cosh 8cosh U( ) = sech 5 sech cosh cosh sech sech 5 + 0sinh So afer he hird ieraion, he differenial inverse ransform of{ U ( )} will give = 0 he following approimae soluion: u, = U ( ) ( ) = 0 = U( ) + U( ) + U( ) + U( ) +... = sech anh anh... Eample.1. We consider he FKdV equaion u + uu uu + u = 0, (7) subjec o he iniial condiion u (,0) = e, (8) where he eac soluion u (,) = e. (9) Applying he RDTM o (8) and (7), we obain he recursive relaion

9 Mahmoud Rawashdeh 9 U 1 ( U ( U ( U ( U ( ) U ( ) 1 ) = i ) ) ) + i i i + 1 i 0 = i= 0 (0) So for = 0, we obain U ( ) 1 = e. Now for 1 we obain ( ) e e U =, U( ) =, 4 6 ( ) e e U =, U5( ) =,.. Thus e e e e u (,) = e e = e O [] = e. This is he eac soluion of Eq. (7). 4 Tables and Figures In his secion, we shall illusrae he accuracy and efficiency of he RDTM. For his purpose, we consider he same values for and, specifically, = { 0.5, 0.,0.,0.5} and = {0.000,0.0004,0.0006,0.001}. Also we can do he same for he oher eample. Table 1: Comparison of absolue errors of he soluion for Gardner equaion, by RDTM and he DTM for differen values of and Eac DTM RDTM Error(DTM)(n=8) Error(RDTM)(n=) E E E E E E E E

10 10 Improved Approimae Soluions for Nonlinear Evoluions Equaions E E E E E E E E-7 Table : Comparison of absolue errors of he soluion for nonlinear varian waer wave equaion, by RDTM and he DTM for differen values of and Eac DTM RDTM Error(DTM)(n=11) Error(RDTM)() E E E E E E E E E E E E E -8

11 Mahmoud Rawashdeh E E E -7 Figure 1: The approimae, eac soluions and absolue error, respecively for eample.1.1 when -0.5< <0.5 and 0< < Noe ha; Figure 1 shows he eac soluion, approimae soluion and he absolue error, respecively. approimae eac Figure : The approimae and eac soluions for eample.1.1 when -0.5< <0.5 and =0.0, 0.04, 0.06, 0.08, 0.1.

12 1 Improved Approimae Soluions for Nonlinear Evoluions Equaions Figure : The approimae, eac soluions and absolue error, respecively for eample.1. when -0.5< <0.5 and 0< < Noe ha; Figure shows he eac soluion, approimae soluion and he absolue error, respecively. approimae.00 eac Figure 4: The approimae and eac soluions for eample.1.1 when -0.5< <0.5 and =0.0, 0.04, 0.06, 0.08, Conclusion In his paper, he Reduced Differenial Transform Mehod (RDTM) was proposed for solving he Gardner equaion, Nonlinear Varian Waer Wave equaion, and he Fifh-Order Koreweg-de Vries (FKdV) equaion. We

13 Mahmoud Rawashdeh 1 successfully found approimae soluions for he firs wo nonlinear PDEs by firs applying he RDTM o all hree physical models. Also I was being able o find eac soluion for eample (.1.). The resuls we obained were in ecellen agreemen wih he eac soluions. The RDTM inroduces a significan improvemen in he fields over eising echniques. Also a comparaive sudy has been conduced beween he DTM and he RDTM. My goal in he fuure is o apply he RDTM o oher nonlinear PDEs which arises in oher areas of science. Compuaions of his paper have been carried ou using compuer pacage Mahemaica 7. Acnowledgemens. This wor was suppored in par by he deanship of research gran from Jordan Universiy of Science and Technology. The auhor would lie o han he Edior and he anonymous referees for heir commens and suggesions on his paper. References [1] G. Adomian, Solving fronier problems of physics: he decomposiion mehod, Kluwer Acad. Publ, [] G. Adomian, A new approach o nonlinear parial differenial equaions, J. Mah. Anal. Appl., 10, (1984), [] A. Ali and A. Soliman, New Eac Soluions of Some Nonlinear Parial Differenial Equaions, Inernaional Journal of Nonlinear Science., 5, (008), [4] M. Alquran, Applying Differenial Transform Mehod o Nonlinear Parial Differenial Equaions: A Modified approach, Applicaions and Applied Mahemaics: An Inernaional Journal., 7, (01),

14 14 Improved Approimae Soluions for Nonlinear Evoluions Equaions [5] S. Haq, A. Hussain, S. Islam, Soluions of Coupled Burger s, Fifh-Order KdV and Kawahara Equaions Using Differenial Transform Mehod wih Padé Approiman, Selcu J. Appl. Mah., 11, (010), 4-6. [6] J. H. He, Variaional ieraion mehod-a ind of non-linear analyical echnique: some eamples, In. J. Nonlinear Mech., 4(4), (1999), [7] B. Ibis and M. Bayram, Approimae Soluions for Some Nonlinear Evoluions Equaions By Using The Reduced Differenial Transform Mehod, Inernaional Journal of Applied Mahemaical Research,, (01), [8] D. Kaya: Eac and numerical solion soluions of some nonlinear physical models. Appl. Mah. Comp., 15, (004), [9] D. Kaya, K. Al-Khaled, A numerical comparison of a Kawahara equaion, Phys. Le. A, 6, (007), [10] T. Kawahara, Oscillaory soliary waves in dispersive media, J. phys. Soc. Japan,, (197), [11] Y. Kesin, G. Ouranc, Reduced Differenial Transform Mehod for fracional parial differenial equaions, Nonlinear Science Leers A, 1(), (010), [1] Y. Kesin and G. Ouranc, Reduced Differenial Transform Mehod for Parial Differenial Equaions, Inernaional Journal of Nonlinear Sciences and Numerical Simulaion, 10(6), (009), [1] Y. Kesin, Ph.D. Thesis, Selcu Universiy, (in Turish), 010. [14] B. Solanalizadeh, Applicaion of Differenial Transformaion Mehod for Numerical Analysis of Kawahara Equaion, Ausralian Journal of Basic and Applied Sciences, 1, (011), [15] A.M. Wazwaz, Parial Differenial Equaions and Soliary Waves Theory, Springer-Verlag, Heidelberg, 009. [16] A.M.Wazwaz, A sine cosine mehod for handling nonlinear wave equaions, Mah. Compu. Modeling, 40, (004),