A Direct Method for Solving Nonlinear PDEs and. New Exact Solutions for Some Examples

Transcription

1 In. J. Conemp. Mah. Sciences, Vol. 6, 011, no. 46, A Direc Mehod for Solving Nonlinear PDEs and New Eac Solions for Some Eamples Ameina S. Nseir Jordan Universiy of Science and Technology Deparmen of Mahemaics and Saisics P. O. Bo 3030, Irbid 110, Jordan anseir@js.ed.jo Absrac A generalizaion for he raional eponenial mehod is presened. The Vakhnenko eqaion or he redced Osrovsky eqaion, and he modified Camassa Holm and Degasperis Procesi were discssed. Eac solions for hese eqaions were fond. The mehod was also applied on some oher eamples of nonlinear PDEs and new eac solions were obained. All calclaions involved were performed sing Mahemaica sofware version 6.0. Keywords: Direc mehod; Eac solions for nonlinear PDEs; Vakhnenko eqaion; modified Camassa Holm and Degasperis Procesi eqaion 1. Inrodcion The pas decades have winessed significan ineres and progress in finding solions o nonlinear parial differenial eqaions (NPDEs ha resemble physical phenomena. Boh mahemaicians and physiciss have performed pleny of research regarding his maer. A glance a he lierare reveals a lo of effecive mehods ha solve his ype of NPDEs. Eamples are he inverse scaering ransform mehod [7], Hiroa s mehod [13], Painlevé analysis [ 4], ep-fncion mehod [8,11,1], homoopy perrbaion mehod [9], F-epansion mehod [1], variaional ieraion mehod [10], homogeneos balance mehod [14,1], ailiary eqaion mehod [18], anh-fncion mehod [15], improved anh-fncion mehod [3,19], raional eponenial mehod [], and ohers.

2 84 A. S. Nseir. The mehod To inrodce he generalizaion of raional eponenial (GRE mehod for solving nonlinear PDEs le G(; ; ; ; ;... = 0, (1 where G is a polynomial of and all parial derivaives of. Sppose ha is a fncion of wo variables and. Now le =, ( m k ( c (, k ( c where A, a 0, a 1, k, and c are consan o be deermined, and m is an ineger. To find he vale of m we sbsie Eq ( ino Eq (1 and balance he mos nonlinear erms in he eqaion. Afer ha we se he coefficien of all powers of he eponenial fncion o zero and find he vales of he consans A, a 0, a 1, k, and c. 3. Applicaions Eample 1. Vakhnenko eqaion or he redced Osrovsky eqaion In 1978, Osrovsky derived an eqaion for weakly nonlinear srface and inernal waves in a roaing ocean [17]: ( + c0 + p + q = γ (3 where c 0 is he velociy of dispersionless linear waves, p is he nonlinear coefficien; q and c are he dispersion coefficiens. Since hen his eqaion is known as Osrovsky eqaion [,6]. When q = 0, he eqaion redces o ( + c0 + p = γ (4 This eqaion was considered for he firs ime in he original paper by Osrovsky [17]. Laer, he same eqaion was derived for differen physical siaions by many ahors. An ineresing and imporan discover has been made by Vakhnenko and Parkes [0], who have demonsraed ha he redced Osrovsky eqaion (4 can be ransformed o a new inegrable eqaion as follows: + = 0 (5

3 Direc mehod for solving nonlinear PDEs 85 I is worh menions ha Vakhnenko and coworkers fond ha his eqaion is compleely inegrable by inverse scaering mehod [19]. This eqaion was sdied by several researchers and research grops who fond eac solion sing differen echniqes. Ysfoğl and Bekir solved his eqaion sing hyperbolic angen mehod [7]. Yasar, employed he improved anh fncion mehod o find differen ypes of eac solions [6]. Now we apply he GRE mehod o find a new form of eac solion for Eq (5. Le =, m k ( c (, k ( c and sbsie (, ino Eq (5 hen eqaing he powers of eponenial fncions of nonlinear erms yields o m =. Then sbsiing he resling eqaion ino Eq (5 o find he consans A, k, a 0, a 1, and c. This implies ha, =, (6 k ( c (, k ( c where A, a 0, a 1, and c are arbirary consans, and A k = m. 6a a 0 1 Eample. The modified Camassa Holm and Degasperis Procesi (CH-DP eqaion Eqaions of he form + ( b + 1 = b + (7 have been invesigaed by Msafa [16]. When b =, Eq. (7 redces o he Camassa Holm (CH eqaion + 3 = + However, when b = 3, Eq. (7 redces o he Degasperis Procesi (DP eqaion + 4 = 3 + Boh he CH and he DP eqaions are bi-hamilonian and have an associaed isospecral problem. These eqaions are formally inegrable by means of he inverse scaering mehod [16]. A modified CH DP eqaion was sggesed by Wazwaz [3].

4 86 A. S. Nseir + ( b + 1 = b + (8 where b is any real nmber. Fan and coworkers sggesed he following general form of modified CH DP eqaion + α β = (9 where α and β are parameers [5]. They sed nmerical simlaions o discss a loop solion for special cases. Applying GRE mehod o he above eqaion, resls in m =. Tha is, =. (10 k ( c (, k ( c Afer sbsiing Eq (10 ino (9, collecing he coefficiens of like powers of he eponenial fncions, hen solving he resling sysem we find k = m1, + 3β + β c =, α a 0, and a 1 are arbirary, so ha he eac solion is 1a a ce m( c 0 1 (, = m( c (1 + β. (11 Eample 3. The combined KdV-mKdV eqaion p + q + = 0 (1 + where p, q (q 0 are real consans. Eq. (1 is sed o describe a variey of wave phenomena in plasma, solid sae, and qanm physics [5]. Applying he GRE mehod, one ges m = 1, i.e. k ( c (, = k ( c a0 (13 Now sbsie Eq (13 ino (1 o find he vales of he nknowns. The final form of he solion is

5 Direc mehod for solving nonlinear PDEs 87 (, = p( a 6a ce 0 1 m c ( c + a e 1 m c ( c (14 where p c =, a 0, and a 1 are arbirary. 6q Eample 4. The KdV Brgers Kramoo eqaion + p + q + r = 0, (15 + where p, q and r are real consans. This eqaion occpies a prominen posiion in describing physical processes in moion of rblence and oher nsable sysems [4]. To solve he eqaion, assme ha =. (16 m k ( c (, k ( c Then sbsie Eq (16 ino Eq (15 and balance he powers of eponenial fncion of nonlinear erms wih linear ones, we ge m =. And he final solion will be =, (17 k ( c (, k ( c where, 3 A = 4( a a kp + 3a a k q + 7a a k, r 3 c = kp + k q + k r, k, a 0, and a 1 are arbirary consans. Conclsion The above eamples illsrae ha he mehod presened in his work is efficien, easy o se, and can be applied o many nonlinear PDEs. The mehod can be

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8 90 A. S. Nseir [6] E. Yasar, New ravelling wave solions o he Osrovsky eqaion, Appl. Mah. and Comp., 16 (010, [7] E. Ysfoğl, A. Bekir, A ravelling wave solion o he Osrovsky eqaion, Appl. Mah. and Comp., 186 (007, Received: May, 011