A New Modified Approach for solving sevenorder Sawada-Kotara equations

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seveorder Sawada-Koara eqaos Habbolla Lafzadeh, Shraz

1 Shraz Uversy of Techology From he SelecedWors of Habbolla Lafzadeh A New Modfed Approach for solvg seveorder Sawada-Koara eqaos Habbolla Lafzadeh, Shraz Uversy of Techology Avalable a: hps://wors.bepress.com/habb_lafzadeh//

Joral of mahemacs ad comper Scece 6 ( -7 A New Modfed Approach for solvg seve-order Sawada-Koara eqaos Masod Sarav, Al Nar *, Mar Herma, Javad Vahd, Reza Ahar 5 Deparme of mahemac Islamc Azad Uversy,

2 Joral of mahemacs ad comper Scece 6 ( -7 A New Modfed Approach for solvg seve-order Sawada-Koara eqaos Masod Sarav, Al Nar *, Mar Herma, Javad Vahd, Reza Ahar 5 Deparme of mahemac Islamc Azad Uversy, Nor Brach, Nor, Ira Deparme of Cvl Egeerg, Uversy of Tabrz, Tabrz, Ira Faläfür Mahema d Iformac, Fredrch-Schller-Uversä Jea, Germay Deparme of mahemac Ira Uversy of Scece ad Techology, Tehra, Ira 5 School of Elecrcal ad Comper Egeerg, Uversy of Shraz, Shraz, Ira Arcle hsory: Receved Jaary Acceped Febrary Avalable ole Aprl msarav@yahoo.com, al.ar@yahoo.com Absrac Here, Recosrco of Varaoal Ierao Mehod (RVIM s sed for compg solos of he seveh-order Sawada-Koera eqao (ssk ad a La s seveh order KdV eqaos (LsKdV. The resls are compared wh he Adoma decomposo mehod (ADM ad he ow aalycal solos. Resls obaed epose effecveess ad capably of hs mehod o solve he seve-order Sawada-Koera (ssk ad a La's seve-order KdV (LsKdV eqaos. Keywords: Recosrco of Varaoal Ierao Mehod (RVIM, seve-order Sawada-Koera (ssk, La's seve-order KdV (LsKdV, Adoma decomposo mehod (ADM.. Irodco Esablshg a ew relable algorhm of varaoal erao mehod, Recosrco of varaoal erao mehod s he prpose of hs paper. I varos areas of scece, especally physcs ad egeerg, varaoal mehods were cosdered [, ]. The varaoal erao mehod was proposed by He [ 5] ad was sccessflly appled o aoomos ordary dffereal eqao [6], o Brger s ad copled Brger s eqaos [7], o Bra s eqao [8], o o-lear paral dffereal eqaos wh varable coeffces [9], o lear Helmholz paral dffereal eqao [], ad o Kle Gordo eqaos []. I was show ha hs mehod s more powerfl ha esg echqes sch as he Adoma mehod [, ], homoopy perrbao mehod [] Eergy Balace mehod [5], Homoopy Aalyss Mehod [6], ec. The perrbao mehod sffers from he compaoal worload, especally whe he degree of oleary creases. Moreover, he Adoma mehod sffers from he complcaed algorhms sed o calclae he Adoma polyomals ha are ecessary for olear problems.

3 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 I 9 Hesamedd ad Lafzadeh proposed a ew mehod based o Laplace rasform - Recosrco of varaoal erao mehod (RVIM [7, 8, -6] whch he correcoal fco of he varaoal erao mehod s obaed who sg he Varaoal Theory [-5]. Therefore hs mehod he compley calclag he Lagrage mlpler has bee removed. The RVIM echqe has o specfc reqreme sch as learzao, small parameer ec. I hs paper, RVIM s sed for sdyg he followg seveh order Sawada-Koera eqao ad La s seveh-order KdV eqao. (5 7( 7( =, ( (6 6( ( =. ( Eq. ( s ow as he seveh order Sawada -Koera eqao [9, ] ad Eq. ( s ow as he La s seveh-order KdV eqao [9, ] respecvely. Frher we compare he resl wh gve solos sg ADM [9, ].. Basc Idea of RVIM To clarfy he basc deas of or proposed mehod [7], we cosder he followg dffereal eqao same as VIM based o Lagrage mlpler []: L(,, N(, = f (, ( By sppose ha L( = =,, L( ( Where L s a lear operaor, N a olear operaor ad f,, a homogeeos erm. ( We ca rewre eqao ( dow a correco fcoal as follows: Lj( j = f (, N(, = L( j h((,,, (, (5 herefore L j ( j = h((,, (, (6 Wh arfcal al codos beg zero regardg he depede varable By ag Laplace rasform of boh sdes of he eqao (6 he sal way ad sg he arfcal al codo we oba he resl as follows P( s. U (,,,, = H ((,,,, (7 Where P(s s a polyomal wh he degree of he hghes dervave eqao (7, (he same as he hghes order of he lear operaor L. The followg relaos are possble; j j. [h] = H (8-a

4 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 B(s = (8-b P(s [b( ] = B(s (8-c Whch ha eqao (8-a he fco H ((,,,,, a h((,,,,, have bee abbrevaed as H, h respecvely. Hece, rewre he eqao (7 as; U (,,,, = H ((,,,,. B( s (9 Now, by applyg he verse Laplace Trasform o boh sdes of eqao (9 ad by sg he (8-a - (8- c, we have; (,, -,,, = h((,, τ,,,. b( τ dτ ( Now, we ms mpose he acal al codos o oba he solo of he eqao (. Th we have he followg erao formlao: (, -,,, = (,,, h((, -,, τ,,,. b( τ dτ ( where s al solo wh or who ow parameers. Assmg s he solo of L, wh al/bodary codos of he ma problem, I case of o ow parameer shold sasfy al/bodary codos. Whe some ow parameers are volved, he ow parameers ca be defed by al/bodary codos afer few erao hs echology s very effecve dealg wh bodary problems. I s worh meog ha, fac, he Lagrage mlpler he He's varaoal erao mehod s λ( τ b( τ as show [7]. = The al vales are sally sed for selecg he zeroh appromao. Wh deermed, he several appromaos >, follow mmedaely. Coseqely, he eac solo may be obaed by sg (,, -,,, = lm (, -,,,. (. Applyg RVIM for ssk ad LsKdV To demosrae he effecveess of he mehod we cosder here Eqs.( ad ( wh gve al codo... Eample : Cosder he ssk eqao ( wh he al codo [9]: A frs rewre eq. ( based o selecve lear operaor as (, = ( ah (, (

5 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 h(,, = = (5 7( 7( ( { ( } Now Laplace rasform s mplemeed wh respec o depede varable o boh sdes of eq. ( ad by sg he ew arfcal al codo (whch all of hem are zero we have { h(,, } { h(,, } s (, = (5 (, = (6 s Ad whereas Laplace verse rasform of /s s as follows [ ] = s (7 Therefore by sg he Laplace verse rasform ad covolo heorem s coclded ha (, = h(, ε, dε (8 Hece, we arrve he followg erave formla for he appromae solo of sbjec o he al codo (. So, echage wh applyg recrsve algorhm, followg relaos are acheved (9 = ( (5 7( 7( dε Now we sar wh a arbrary al appromao (, = ( ah (, ha sasfes he al codo ad by sg he RVIM erao formla (9, we have he followg sccessve appromao 8 (, = ( ah ( sec h ( [ cosh(], 9

6 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 8 (, = ( ah ( ( [76 896cosh(] 9 ( ([ cosh 9( 778cosh( 8cosh( cosh(8] [ cosh( 578cosh( 76cosh(6 5697cosh(8] [ cosh( cosh( 79878cosh( cosh(8] 5 8 [ cosh( cosh(8] Fgre.. The srfaces o boh colms respecvely show he solo (,, for RVIM o he rgh ad ADM o he lef whe =.... Eample : Cosder he LsKdV eqao wh gve al codo [9]: A frs rewre eq. ( based o selecve lear operaor as { ( } (, = (, ( h(,, = = (5 7( 7( ( RVIM's erao formlae -dreco ca be readly obaed. (6 = ( (5 7( 7( dε

7 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 Now we sar wh a arbrary al appromao (, = (, ha sasfes he al codo ad by sg he RVIM erao formla (6, we have he followg sccessve appromao 8 (, = ( 8 (, (, = 6 6(8 6 6 ( 8 ( cosh( cosh( ( ( 8 cosh( (9 8 7 ad so o. Usg he above erm Fg., (, ad eac solo (, = ( [] s draw. Fgre.. The srfaces o boh colms respecvely show he solo (,, for RVIM o he rgh ad ADM o he lef whe =... Coclso I hs paper, we sccessflly apply Recosrco of Varaoal Ierao Mehod (RVIM o appromae he solo of ssk ad LsKdV eqaos. Also, comparsos were made bewee Recosrco of Varaoal Ierao Mehod (RVIM ad Adoma decomposo mehod (ADM for ssk ad LsKdV eqaos. Moreover, he RVIM redces he sze of calclaos by o reqrg he edos Adoma polyomal ad hece he erao s drec ad sraghforward. The resls repored here provde frher evdece of he sefless of RVIM for fdg he aalyc ad merc solos for he lear ad olear dffso eqaos ad, s also a promsg mehod o solve dffere ypes of olear eqaos mahemacal physcs. 5. Refereces [] J. Mosler J, M. Orz, O he mercal mplemeao of varaoal arbrary Lagraga Elera 5

8 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 (VALE formlao Ieraoal Joral for Nmercal Mehods Egeerg, 67(9, 7 89, (6. [] P. Thoreddy, M. Orz, Varaoal r-adapo ad shape-opmzao mehod for fedeformao elascy, Ieraoal Joral for Nmercal Mehods Egeerg, 6(,, (. [] J. H. He, Appromae aalycal solo for seepage flow wh fracoal dervaves poros meda, Comper Mehods Appled Mechacs ad Egeerg, 67, 57 68, (998. [] R. Seaada, M. Dehgha, S.M. Vaezpor, M. Sarav., The covergece of He s varaoal erao mehod for followg egral eqao Compers & Mahemacs wh Applcao 58, 67 7, (9. [5] A. Nar, M. Mgha, Applcao of He s varaoal erao mehod for solvg seveh-order dffereal eqao Amerca Joral of Compaoal ad Appled Mahemac (, 7-, (. [6] J. H. He, Appromae solo of olear dffereal eqaos wh covolo prodc oleares. Comper Mehods Appled Mechacs ad Egeerg, 67, 69 7, (998. [7] M.A. Abdo, A.A. Solma, Varaoal erao mehod for solvg Brgers ad copled Brgers eqao, J. Comp. Appl. Mah., 8, 5 5, (5. [8] M. Sarav, M. Herma, D. Kaser, Solo of Bra s Eqao by He's Varaoal Ierao Mehod, Amerca Joral of Compaoal ad Appled Mahemac (, 6-8, (. [9] J. H. He Varaoal prcple for some olear paral dffereal eqaos wh varable coeffce Chao Solos ad Fracal 9, 87 85, (. [] S. Moma, S. Absaad, Applcao of He s varaoal-erao mehod o Helmholz eqao, Chaos Solos Fracal 7(5, 9, (5. [] S. Abbasbady, Nmercal solo of olear Kle Gordo eqaos by varaoal erao mehod, Ieraoal Joral for Nmercal Mehods Egeerg, 7, , (7. [] G. Adoma, Solvg Froer Problems of Physcs: The Decomposo Mehod, Klwer, Boso, (99. [] G. Adoma, A revew of he decomposo mehod appled mahemac Joral of Mah. Aal. Appl., 5, 5 5, (988. [] M. Sarav, M. Herma ad H. Ebrahm Khah, The comparso of homoopy perrbao mehod wh fe dfferece mehod for deermao of mamm beam defleco, Joral of Theorecal ad Appled Physc 7:8, (, do:.86/

9 M. Sarav, A. Nar, M. Hermaa, J. Vahd, R. Ahar / J. Mah. Comper Sc. 6 ( -7 [5] A. Nar, S. Esmaelzade Tolo, K. Rashed ad H. R. Khalaj Hedaya, Applcao of eergy balace mehod for a coservave X/ force olear oscllaor ad he Doffg eqao Ieraoal Joral of Nmercal Mehods ad Applcao 5(, 57-66, (. [6] Y. Kha, R. Taghpor, M. Falaha, A. Nar, A ew approach o modfed reglarzed log wave eqao, Neral Comp & Applc, (, DOI.7/s [7] E. Hesamedd, H. Lafzadeh, Recosrco of Varaoal Ierao Algorhms sg Laplace Trasform, Iera. J. Nolear Sc. Nmer. Smlao, (, 65-7, (9. [8] A. Nar, Z. Mgha, S.M. Sagheba, S.B. Nojabae ad M. Dae. Developme ad Valdao of a Aalycal Mehod o he Solo of Modellg he Pollo of a Sysem of Lae Research Joral of Appled Scece Egeerg ad Techology, 5(, 96-, (. [9] S. M. El-Sayed ad D. Kaya, A applcao of he ADM o seve order Sawada-Koera eqao Appl. Mah. Comp., 57, 9, (. [] W. Herema, P.P. Baerjee, A. Korpel, G. Assao, A. va Immerzeele ad A. Meerpoel, Eac solary wave solos of olear evolo ad wave eqaos sg a drec algebrac mehod, J. Phys. A: Mah. Ge., 9, 67-68, (986. [] E. J. Pares ad B.R. Dffy, A aomaed ah-fco mehod for fdg solary wave solos o o-lear evolo eqao Comp. Phys. Comm., 98, 88, (996. [] G. Adoma, Solvg Froer Problems of Physcs: The Decomposo Mehod, Klwer, (99. [] A.M. Wazwaz, The varaoal erao mehod: A powerfl scheme for hadlg Lear ad olear dffso eqao Compers ad Mahemacs wh Applcao 5(7, 9-99, (7. [] A. Nar, A ew approach for solvg Gas dyamcs eqao. Aca Techca Corvess Blle of Egeerg,, -6, (. [5] A. Nar, J. Vahd, M. Jafarejad Ghom, M. Mgha, Recosrco of varao Ierao Mehod for Solvg Ffh Order Cadrey-Dodd-Gbbo (CDG Eqao, Ieraoal joral of Scece ad Egeerg Ivesgao (6, 8-, (. [6] S. Ghasempoor, J. Vahd, A. Nar, M. Mgha, Aalycal approach o some hghly olear eqaos by meas of he RVIM, Res. J. Appl. Sc. Eg. Techol., 5(, 96-, (. 7

Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations

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