Application of He's homotopy perturbation method for solving nonlinear wave-like equations with variable coefficients

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1 Inrnaional Jornal of Adancs in Alid Mahmaics and Mchanics Volm Iss : Aailabl onlin a IJAAMM ISSN: 47-9 Alicaion of H's homoo rrbaion mhod for soling nonlinar wa-lik qaions wih ariabl cofficins Smi Ga a Dndra Kmar a and Jagd Singh b a Darmn of Mahmaics Jagan Nah Ga Insi of Enginring and Tchnolog Jair- Rajashan India b Darmn of Mahmaics Jagan Nah Unirsi Jair-9 Rajashan India Rcid 7 Ocobr ; Accd in risd rsion Nombr A B S T R A C T In his ar w al homoo rrbaion mhod HPM for soling nonlinar wa-lik qaions wih ariabl cofficins. HPM ilds solion in raid conrgn sris from asil comabl rms and in som cass ilds ac solions in on iraion. Moror his chniq dos no rqir an discrizaion linarizaion or small rrbaions and hrfor rdcs h nmrical comaions o a gra n. Th solion rocdr b HPM is siml highl accra and comars faorabl wih h ac solions obaind arl in h lirar. Th mhodolog rsnd in h ar is sfl for highl nonlinar roblms consising of mor han on diffrnial qaion. Kwords: Homoo rrbaion mhod Nonlinar wa-lik qaions Aroima solions Eac solions Mahmaica. MSC cods: 9B F C IJAAMM Inrodcion Th homoo rrbaion mhod inrodcd b h Chins rsarchr Dr. Ji Han H in 998 has com o b accd as an lgan ool in h hands of rsarchrs looking for siml highl ffci solions o comlicad roblms in man dirs aras of scinc and chnolog. In a sris of ars H 999 has olind and rfind h HPM showing is sflnss b nonlinar diffrnial qaions. As a rl HPM nds o rodc mch mor lgan solions as comard o h ohr coming chniqs sch as homoo analsis mhod HAM rglar rrbaion mhods c. i is no a h cos of accrac. In gnral h solions rodcd b h HPM ar as accra as h solions Corrsonding ahor addrss: dndra.mahs@gmail.com Dndra Kmar

2 Smi Ga al. 66 gin b h ohr aroima mhods. Rcnl H has roidd a highl rliabl accon of h HPM in a monograh H a. Th HPM is in fac a coling of h radiional rrbaion mhod and homoo in oolog H b. Th mhod ilds a r raid conrgnc of h solion sris in mos cass sall onl fw iraions rodc r accra solions. This mhod was alid o fncional ingral qaions Abbasband 7 cold ssm of raciondiffsion qaion Ganji and Sadighi 6 o Hlmholz qaion and fifh-ordr KdV qaion Rafi and Ganji 6 h idmic modl Rafi al. 7 drmining h frqnc-amlid rlaion of a nonlinar oscillaor wih disconini Ozis and Yildirim 7a ralling wa solion of h KdV qaion Ozis and Yildirim 7b. Momani and Odiba 7 and Odiba and Momani 8 sggsd a modifid HPM and sd i for nonlinar arial diffrnial qaions of fracional ordr and qadraic Riccai diffrnial qaion of fracional ordr. I was alid o h hin film flow of a forh grad flid down an inclind lan Siddiq al. 6 o h nonlinar Volra-Frdholm ingral qaion Ghasmi al. 7 o h singlar VIE of scond ordr Chn and Jiang o h linar rogramming roblm alid in oimizaion chniqs in arios indsris Mhrabinzhad and Sabri-Nadjafi o h amromric nzmaic racion modl alid in nclar chmisr Shanmgarajan al. o coninos olaion modls for singl and inrsing scis in Economics Pamk and Pamk o im-dndn fncional diffrnial qaions Ga al. o fracional gas dnamics qaion Singh al. and o fracional ha and wa-lik qaions Singh and Kmar. As i is said in Blndz al. 9a h sd of nonlinar roblm is of crcial imoranc no onl in all aras of hsics b also in nginring and ohr discilins sinc mos hnomna in or world ar ssniall nonlinar and ar dscribd b h nonlinar qaions. I is r imoran o sol nonlinar roblm and in gnral i is ofn mor difficl o g an analic aroimaion han a nmrical on for h gin nonlinar roblm. Blndz al. 9a 9b did a good insigaion on his mhod. Th ahors of Blndz al. 9b mlod homoo rrbaion mhod o obain highr-ordr analical aroimaions o h riodic solions o a nonlinar oscillaor wih disconiniis for which h lasic rsoring forc is an anismmric and consan forc. Eclln agrmn bwn aroima riods and h ac on has bn dmonsrad and discssd. Th HPM is modifid in Blndz al. 7 b rncaing h infini sris corrsonding o h firs ordr aroimaions bfor inrodcing his solion in h scondordr linar diffrnial qaion and so on. Th fond ha h modifid mhod works wll for h whol rang of iniial amlids and h clln agrmn of h aroimaion frqncis and riodic solions wih h ac ons has bn dmonsrad and discssd. Onl on iraion lads o high accrac of h solion wih a minimal rlai rror. In his ar w considr h following nonlinar wa-lik qaions n F k m n Fij i j X G i X G k m ij i i i j i i i H X S X wih h iniial condiions In. J. of Ad. in Al. Mah and Mch. : 6-79.

3 67 Alicaion of H s Homoo Prrbaion Mhod X a X X a X. Hr X n and F ij G i ar nonlinar fncion of X and. F ij Gi ar nonlinar fncion of driais of. Whil H S ar nonlinar fncions and i j k m ar ingrs. Ths s of qaions ar of considrabl significanc in arios filds of alid scincs mahmaical hsics nonlinar hdrodnamics nginring hsics biohsics hman momn scincs asrohsics and lasma hsics. Ths qaions dscrib h olion of sochasic ssms. For aml h dscrib h rraic moions of small aricls ha ar immrsd in flids flcaions of h innsi of lasr ligh loci disribions of flid aricls in rbln flows and h sochasic bhaior of chang ras. Ghorishi al. ha sold his of qaion b Adomain Dcomosiion mhod ADM o aoid nralisic assmions in calclaing h Adomain olnomials. ADM is h mos ransarn mhod for solions of h nonlinar roblms; howr his mhod is inold in h calclaion of comlicad Adomain olnomials which narrows down is alicaions. To orcom his disadanag of h ADM w considr h HPM o sol arios nonlinar wa-lik qaions of ariabl cofficins. Homoo rrbaion mhod HPM To illsra h basic idas of h HPM w considr h following diffrnial qaion H 999 A f r r Ω wih h bondar condiion B n r Γ whr A is a gnral diffrnial oraor B is a bondar oraor f r is a nknown analic fncion Γ is h bondar of h domain Ω. Th oraor A can b diidd ino wo ars L and N whr L is linar and N is nonlinar Eq. hrfor can b wrin as follows L N f r. 4 B h homoo chniq w consrc h homoo ν r : Ω [] R which saisf H [ L L ] [ A f r] [] or H L L L [ N f r] [] 6 whr [ ] is an mbdding aramr is an iniial aroimaion of Eq. which saisf h bondar condiions. In. J. of Ad. in Al. Mah and Mch. : 6-79.

4 Smi Ga al. 68 A and h following qaion can b dformd as H L L 7 and H A f r. 8 Now h Eq. can b wrin as owr sris of :. 9 Sing rsls in aroima solion of Eq. :. Th abo solion is conrgs if N L. Alicaions In his scion w al HPM for hr amls and comard or solion wih ADM. Th n rrors bwn ac solion and homoo solion ar dnod b ϕ ac n φ i i whr φ n ar h aroima solions obaind b h HPM. Eaml.. Considr h following wo dimnsional nonlinar wa-lik qaion wih ariabl cofficins Ghorishi al.. wih h iniial condiions Th ac solion is gin b cos sin. According o h HPM l s considr h following homoo. H 4 Now b HPM w ha In. J. of Ad. in Al. Mah and Mch. : 6-79.

5 Alicaion of H s Homoo Prrbaion Mhod In. J. of Ad. in Al. Mah and Mch. : Using Eq. in Eq. 4 and comaring h cofficins of arios owrs of w g : : 6 : and so on. Soling abo diffrnial qaions ndr h iniial condiions m ν m for m w g and so on. Thrfor h aroima solion is gin b which conrgs o h ac solion and sam as obaind b ADM Ghorishi al..

6 Smi Ga al. In. J. of Ad. in Al. Mah and Mch. : Eaml.. Considr h following nonlinar wa-lik qaion wih ariabl cofficins Ghorishi al wih h iniial condiions. Th ac solion is gin b. According o h HPM considr h following homoo H. 8 Now b HPM w ha. Using Eq. in Eq. and comaring h cofficins of arios owrs of w finds : 8 : : ]. 8 Soling abo diffrnial qaions w g

7 7 Alicaion of H s Homoo Prrbaion Mhod and so on. Thrfor h aroima solion is gin b which is h ac solion and is sam as obaind b ADM Ghorishi al.. Eaml.. Considr h following nonlinar wa-lik qaion wih ariabl cofficins Ghorishi al.. < < > 6 wih h iniial condiions. 7 Th ac solion of Eq. 6 is gin b sin. According o h HPM firs w consrc h homoo in h following form H. 8 Now b HPM w ha. 9 In. J. of Ad. in Al. Mah and Mch. : 6-79.

8 Smi Ga al. 7 Using Eq. 9 in Eq. 8 and comaring h cofficins of arios owrs of w finds ha h following Eq. 8 is dcomosd in o infini no of linar arial diffrnial qaions soling hs qaions ndr h iniial condiions w g!! and so on. 7 7! Thrfor h aroima solion is gin b 7 sin!! 7! which is an ac solion and is sam as obaind b ADM Ghorishi al.. 4 Tabls Tabl : Th following abl shows absol rrorϕ 7 for ariabls and aris from. o. for Eaml E- 9.6E-.E-8.89E-7.686E-6.6E-.9866E-.8666E E-7.966E E-.66E-.689E E-7.86E E 9.6E-.E-8.44E-7.86E-6 In. J. of Ad. in Al. Mah and Mch. : 6-79.

9 7 Alicaion of H s Homoo Prrbaion Mhod Tabl : Th following abl shows absol rror ϕ 8 for ariabl and aris from. o. for Eaml E-.764E-.647E-8.689E-7.984E E-.6E E E E E E-.689E E-7.96E-6.986E- 9.86E- 8.E E E-6 Tabl : Th following abl shows absol rror ϕ for ariabl and aris from. o. for Eaml E E E-7.686E-....9E E E-7.6E-7 Figrs HPM Eac Figr : Comarison of ac solion and HPM solion of Eaml. a. In. J. of Ad. in Al. Mah and Mch. : 6-79.

10 Smi Ga al HPM Eac. 4 6 Figr : Comarison of ac solion and HPM solion of Eaml HPM Eac Figr : Comarison of ac solion and HPM solion of Eaml...E- 9.E-6 8.E-6 7.E-6 6.E-6.E-6 4.E-6.E-6.E-6.E-6.E..7.9 Figr 4: Errors of Eaml. form arios als of o a. In. J. of Ad. in Al. Mah and Mch. : 6-79.

11 7 Alicaion of H s Homoo Prrbaion Mhod.E- 8.E-6 6.E-6 4.E-6.E-6.E Figr : Errors of Eaml. from arios als of o Figr 6: Eac solion grah of Eaml. for o Figr 7: Aroima solion grah of Eaml. o fifh aroimaion for o. In. J. of Ad. in Al. Mah and Mch. : 6-79.

12 Smi Ga al Figr 8: Eac solion grah of Eaml. for o Figr 9: Aroima solion grah of Eaml. o fifh aroimaion for o Figr : Aroima solion grah of Eaml. o fifh aroimaion for o. In. J. of Ad. in Al. Mah and Mch. : 6-79.

13 77 Alicaion of H s Homoo Prrbaion Mhod -.. Figr : Aroima solion grah of Eaml. o fifh aroimaion for o. 6 Conclsions In his ar h HPM has bn sccssfll mlod o obain h aroima analical solions of h nonlinar wa-lik qaions wih ariabl cofficins. In Eamls - w obsr ha HPM solions ar mor accra han h ADM. Figs. o show ha h solions obaind b HPM ar mch closr o h ac solion for arios als of im aramr. Tabl o dnos h lss rror for h solion obaind b HPM and ac solion. HPM aoids h difficlis arising in finding h Adomain olnomials. In addiion h calclaions inold in HPM ar r siml and sraigh forward. I is shown ha h HPM is a romising ool for boh linar and nonlinar arial diffrnial qaions. Mahmaica. has bn sd for nmrical comaions in his ar. Rfrncs Abbasband S. 7: Alicaion of H's homoo rrbaion mhod o fncional ingral qaions. Chaos Solions Fracals Blndz A. Pascal C. Orno M. Gallgo S. and Ni C. 7: Alicaion of modifid H's homoo rrbaion mhod o obain highr ordr aroimaions of a forc nonlinar oscillaor. Phsics Lrs A Blndz A. Pascal C. Blndz T. and Hrnandz A. 9a: Solion for an anismmric qadraic nonlinar oscillaor b a modifid H's homoo rrbaion mhod. Nonlinar Analsis RWA Blndz A. Pascal C. Orno M. Blndz T. and Gallgo S. 9b: Alicaion of modifid H's homoo rrbaion mhod o obain highr ordr aroimaions o a nonlinar oscillaor wih disconiniis. Nonlinar Analsis RWA Chn Z. and Jiang W. : Picwis homoo rrbaion mhod for soling linar and nonlinar wakl singlar VIE of scond ordr. Alid Mahmaics and Comaion In. J. of Ad. in Al. Mah and Mch. : 6-79.

14 Smi Ga al. 78 Ganji D. D. and Sadighi A. 6: Alicaion of H's homoo rrbaion mhod o nonlinar cold ssm of racion-diffsion qaions. Inrnaional Jornal of Nonlinar Scincs and Nmrical Simlaion Ghasmi M. Taassoli M. and Babolian E. 7: Nmrical solions of h nonlinar Volra-Frdholm ingral qaions b sing homoo rrbaion mhod. Alid Mahmaics and Comaion Ghorishi M. Ismail A. I. B. and Ali N. H. M. : Adomain dcomosiion mhod for nonlinar wa-lik qaion wih ariabl cofficins. Alid Mahmaical Scincs Ga S. Singh J. and Kmar D. : Alicaions of homoo rrbaion ransform mhod for soling im-dndn fncional diffrnial qaions. Inrnaional Jornal of Nonlinar Scinc H J. H. 999: Homoo rrbaion chniqs. Comr Mhods in Alid Mchanics and Enginring H J. H. a: A riw on som nw rcnl dlod nonlinar analical chniqs. Inrnaional Jornal of Nonlinar Scincs and Nmrical Simlaion. 7. H J. H. b: A coling of homoo chniq and rrbaion chniq for nonlinar roblms. Inrnaional Jornal of Nonlinar Mchanics H J. H. : Homoo rrbaion mhod: a nw nonlinar analical chniq. Alid Mahmaics and Comaion Mhrabinzhad M. and Sabri-Nadjafi J. : Alicaion of H's homoo rrbaion mhod o linar rogramming roblm Inrnaional Jornal of Comr Mahmaics Momani S. and Odiba Z. 7: Homoo rrbaion mhod for nonlinar arial diffrnial qaions of fracional ordr. Phsics Lrs A Odiba Z. and Momani S. 8: Modifid homoo rrbaion mhod: Alicaion o Riccai diffrnial qaion. Chaos Solions Fracals Ozis T. and Yildirim A. 7a: A comarai sd of H's homoo rrbaion mhod for drmining frqnc-amlid rlaion of a nonlinar oscillaors wih disconini. Inrnaional Jornal of Nonlinar Scinc and Nmrical Simlaion Ozis T. and Yildirim A. 7b: Tralling wa solion of KdV qaion sing H' homoo rrbaion mhod. Inrnaional Jornal of Nonlinar Scinc and Nmrical Simlaion Pamk S. and Pamk N. : H's homoo rrbaion mhod for coninos olaion modl for singl and inracing scis. Comrs and Mahmaics wih Alicaions In. J. of Ad. in Al. Mah and Mch. : 6-79.

15 79 Alicaion of H s Homoo Prrbaion Mhod Rafi M. and Ganji D. D. 6: Elici solions of Hlmholz qaion and fifh-ordr KdV qaion sing homoo rrbaion mhod. Inrnaional Jornal of Nonlinar Scincs and Nmrical Simlaion Rafi M. Ganji D. D. and Daniali D. 7: Solion of idmic modl b homoo rrbaion mhod. Alid Mahmaics and Comaion Shanmgarajan A. Alwaraan S. Somasndaram S. and Lakshmanan R. : Analic solion of amromric nzmaic racion basd on homoo rrbaion mhod. Elcrochimica Aca Siddiq A. M. Mahmood R. and Ghori Q. K. 6: homoo rrbaion mhod for hin film flow of a forh grad flid down an inclind lan. Phsics Lrs A Singh J. Kmar D. and Kilicman A. : Homoo rrbaion mhod for fracional gas dnamics qaion sing smd ransform. Absrac and Alid Analsis Aricl ID ags. Singh J. and Kmar D. : An alicaion of homoo rrbaion ransform mhod o fracional ha and wa-lik qaions. Jornal of Fracional Calcls and Alicaions In. J. of Ad. in Al. Mah and Mch. : 6-79.