Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations

Transcription

1 World Applied Sciece Joral 9 (Special Ie of Applied Math): 3-9, ISSN IDOSI Pblicatio, Applicatio of Laplace Decopoitio Method to Sole Noliear Copled Partial Differetial Eqatio Majid Kha, M. Hai, Hoei Jafari ad 3 Yair Kha Departet of Sciece ad Haitie, Natioal Uierit of Copter ad Eergig Sciece, A-K Brohi Road H-/4, Ilaabad, Pakita Departet of Matheatic ad Copter Sciece, Uierit of Mazadara, P.O. Bo , Baboar Ira 3 HITEC Uierit of Teila Catt, Pakita Abtract: I thi article, e deelop a ethod to obtai approiate oltio of oliear copled partial differetial eqatio ith the help of Laplace Decopoitio Method (LDM). The techiqe i baed o the applicatio of Laplace trafor to oliear copled partial differetial eqatio. The oliear ter ca eail be hadled ith the help of Adoia poloial. We illtrate thi techiqe ith the help of three eaple ad relt of the preet techiqe hae cloed agreeet ith approiate oltio obtaied ith the help of Adoia Decopoitio Method (ADM). Ke ord: Approiate oltio Laplace decopoitio ethod oliear copled partial differetial eqatio Adoia decopoitio ethod INTRODUCTION The decopoitio ethod ha bee ho to ole [- efficietl, eail ad accratel a large cla of liear ad oliear ordiar, partial, deteriitic or tochatic differetial eqatio. The ethod i er ell ited to phical proble ice it doe ot reqire ecear liearizatio, pertrbatio ad other retrictie ethod ad aptio hich a chage the proble beig oled, oetie eriol. The Laplace Decopoitio Method (LDM) i a erical algorith to ole oliear ordiar, partial differetial eqatio. Khri [3, 4 ed thi ethod ethod for the approiate oltio of a cla of oliear ordiar differetial eqatio. Agadjao [5 applied thi ethod for the oltio of Dffig eqatio. Elgazer [6 eploit thi ethod to ole Falker-Ska eqatio. Thi erical techiqe baicall illtrate ho the Laplace trafor a be ed to approiate the oltio of the oliear differetial eqatio b aiplatig the decopoitio ethod hich a firt itrodced b Adoia [7. The preet paper ai at offerig a alteratie ethod of oltio to the eitig oe [8 cocerig to the three oliear copled partial differetial eqatio. B ig Laplace trafor algorith baed o decopoitio ethod for olig copled oliear differetial eqatio the eact oltio of iitial ale proble are obtaied. LAPLACE DECOMPOSITION METHOD The ai of thi ectio i to dic the e of Laplace trafor algorith for the oliear partial differetial eqatio. We coider the geeral for of ihoogeeo oliear partial differetial eqatio ith iitial coditio i gie belo L + R + N = h(,t) (.) (,) = f(), (,) t = g() (.) here L i ecod order differetial operator L =, R i the i reaiig liear operator, N repreet a geeral o-liear differetial operator ad h (, t) i orce ter. The ethodolog coit of applig Laplace trafor firt o both ide of Eq. (.) [ (,t) + [ R(,t) + [ N(,t) = [ h(,t) LL L L L (.3) Uig the differetiatio propert of Laplace trafor e get Correpodig Athor: Dr. Majid Kha, Departet of Sciece ad Haitie, Natioal Uierit of Copter ad Eergig Sciece A-K Brohi Road H-/4, Ilaabad, Pakita 3

2 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, [ + L[ + L[ N(,t) [ h(,t) L (,t) f() g() R(,t) L f() g() [ (,t) = + + L[ h(,t) [ R(,t) [ N(,t) (.4) (.5) L[ (,t) = [ R (,t) [ A, + (.4) Applig iere Laplace trafor to Eq. (.)- (.4), So or reqired recrie relatio i gie belo (,t) = K(,t) (.5) The ecod tep i Laplace decopoitio ethod i that e repreet oltio a a ifiite erie gie belo = (,t) (.6) The oliear operator i decopoe a + (,t) = [ R (,t) [ A L + L, (.6) here K(,t) repreet the ter ariig fro orce ter ad precribe iitial coditio. No firt of all e applig Laplace trafor of the ter o the right had ide of Eq. (.6) the applig iere Laplace trafor e get the ale of,,, repectiel. N(,t) = A (.7) here A are Adoia poloial [ of,,,, ad it ca be calclated b forla gie belo d i A = N λ i, =,,,...,!dλ λ= (.8) APPLICATIONS To illtrate thi ethod for copled oliear partial differetial eqatio e take three eaple i thi ectio. Eaple : Coider oliear partial differetial eqatio [8 Pttig Eq. (.6), Eq. (.7) ad Eq. (.8) i Eq. (.5) e ill get (,t) + =, t > (3.) f() g() L (,t) = + + L [ h(,t) [ R(,t) A (.9) ith iitial coditio (, ) = (3.) t (, ) = e (3.3) f() g() L[ (,t) = + + L [ h(,t) [ R(,t) A (.) O coparig both ide of the Eq. (.) e hae f() g() L[ (,t) = + + L [ h(,t) = K(,) (.) L[ (,t) = [ R (,t) [ A (.) L[ (,t) = [ R (,t) [ A (.3) I geeral, the recrie relatio i gie b 4 Applig Laplace trafor algorith e get (,) (,) t(,) (3.4) (,) (,) L (3.5) t (,) = + + Uig gie iitial coditio Eq. (3.5) becoe e (,) = + L (3.6) Applig iere Laplace trafor to Eq. (3.6) e get (,t) = e t + L L (3.7)

3 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, The Laplace Decopoitio Method (LDM) [3-6 ae a erie oltio of the fctio (, t) i gie b = (,t) (3.8) = Uig Eq. (3.8) ito Eq. (3.7) e get = + = = = (,t) e t L L A () B() (3.9) I aboe Eq. (3.9) A () ad B () are Adoia poloial [ that repreet oliear ter. So Adoia poloial are gie belo A () = (3.) = B () = (3.) = The fe copoet of the Adoia poloial are gie a follo A() = (3.) (,t) [ A () B () L =L L Lte te L [ (, t) = (3.) Therefore the oltio obtaied b LDM i gie belo (,t) = (,t) = et = (3.) Which i ae a oltio obtaied b ADM [8. Eaple : Coider te of oliear copled partial differetial eqatio [8 (,,t) = (,,t) = 5 (3.) (3.3) A() = (3.3) i i (,,t) = 5 A () = ith iitial coditio (3.4) (3.4) B() = (3.5) B() = (3.6) i i B () = (3.7) Fro Eq. (3.9)-(3.) or reqired recrie relatio i gie belo (,t) = et (3.8) (,,) = + (3.5) (,,) = (3.6) (,,) = + (3.7) Applig the Laplace decopoitio ethod (,,) (,,) + (3.8) (,,) (,,) 5 + (3.9) + (,t) A () B(), L (3.9) = = The firt fe copoet of (,t) follo iediatel po ettig (,,) (,,) 5+ (3.3) Uig iitial coditio Eq. (3.8)-(3.3) becoe (,,) = + (+ ) + L (3.3) 5

4 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, 5 (,,) = + ( ) + L (3.3) 5 (,,) = + ( + ) + L (3.33) Applig iere Laplace trafor e get = L L (3.34) (,,t) t () (,,t) = 5t + ( ) + (,,t) = 5t + ( + ) + The recrie relatio are L L (3.35) L L (3.36) (,,t) = t + ( + ) (3.37) + (,,t) L C(,), (3.38) = (,,t) = 5t + ( ) (3.39) L L (3.4) + (,,t) = D (,), = (,,t) = 5t + ( + ) (3.4) + (,,t) L E (,), (3.4) = here C (, ), D (, ) ad E (, ) are Adoia poloial repreetig the oliear ter [ i aboe Eq. (3.37)-(3.4). The fe copoet of Adoia poloial are gie a follo C(,) = (3.43) C(,) = + (3.44) i i C (,) = (3.45) D(,) = (3.46) D(,) = + (3.47) 6 i i D (,) = (3.48) E (,) = (3.49) E(,) = + (3.5) i i E (,) = (3.5) I ie of the recrie relatio (3.37)-(3.4) e obtaied other copoet a follo (,,t) L[ C(,) L L[ ()() = t (3.5) (,,t) L[ D (,) L L[ ( )() = -t (3.53) (,,t) L[ E (,) L = -t (3.54) (,,t) L[ C(,) L + = (3.55) (,,t) L[ D (,) L + = (3.56) (,,t) L[ E(,) L + = (3.57)

5 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, So the oltio of aboe te of oliear partial differetial eqatio are gie belo (3.58) (,,t) = (,t) = + + 3t (,,t) = (,t) = + 3t (3.59) (3.6) (,,t) = (,t) = + + 3t Eaple 3: Coider te of oliear copled partial differetial eqatio [8 (,,t) + = (,,t) + + = (,,t) + + = (3.6) (3.6) (3.63) (,,t) e + = (3.7) J (,) L L (3.7) = + (,,t) =, K (,) = here F (,), G (,), H (,), I (,), J (,), ad K (,) are Adoia poloial [ repreetig oliearitie ariig i aboe te of oliear copled partial differetial eqatio. The fe copoet of aboe Adoia poloial are gie belo F(,) = (3.73) F(,) = + (3.74) i i F(,) = (3.75) G (,) = (3.76) ith iitial coditio (,,) e + (,,) (,,) e + = (3.64) = e (3.65) = (3.66) Applig the ae procedre a applied i preio eaple e arrie at recrie relatio a follo (,,t) e + = (3.67) G(,) = + (3.77) i i G (,) = (3.78) H(,) = (3.79) H(,) = + (3.8) i i H (,) = (3.8) I (,) = (3.8) F(,) L L (3.68) G (,) = = + (,,t) =, I(,) = + (3.83) i i I (,) = (3.84) (,,t) = e (3.69) J (,) = (3.85) H (,) L L (3.7) I (,) = = + (,,t) =, 7 J(,) = + (3.86) i i J (,) = (3.87)

6 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, K (,) = (3.88) K(,) = + (3.89) i i K (,) = (3.9) Therefore other copoet of the oltio are gie belo (,,t) L[ F(,) G (,) L Le e e e e L e + + = e L + = e t (3.9) (,,t) [ H(,) I(,) L L L e = Le + e e e e Le = e L = e t (3.9) (,,t) L[ J (,) K (,) L Le + e e e e Le + + = e L + = e t (3.93) (,,t) [ F(,) G(,) L + + L( ) ( ) Lte + + = e L 3 + e = t (3.94)! 8 (,,t) [ H(,) I(,) L + + L ( ) ( ) Le t e = t (3.95)! (,,t) L[ J (,) K (,) + + L ( ) ( ) Le + + t e = t (3.96)! So or reqired oltio are gie belo (,,t) = (,,t) = t = e + e t + e +...! + t t = e + t = e + + (3.97)! (,,t) = (,,t) = t = e + e t + e +...! t = e + t ! (,,t) = (,,t) = t = e + e t+ e +...! t = e + (3.98) + t t = e + t = e + + (3.99)! Fro (3.), (3.58)-(3.6) ad (3.98)-(3.99), approiate oltio obtaied b Laplace decopoitio ethod i iilar to the oltio obtaied b Adoia decopoitio ethod [8. CONCLUSION I thi article, Laplace dcopoitio ethod (LDM) i applied to ole oliear copled partial differetial eqatio ith iitial coditio. The relt of three eaple are copared ith ADM [8. The relt of thee three eaple tell that both ethod

7 World Appl. Sci. J., 9 (Special Ie of Applied Math): 3-9, ca be ed alteratiel for the oltio of high-order iitial ale proble. REFERENCES. Adoia, G., 994. Frotier proble of phic: The decopoitio ethod. Boto: Kler Acadeic Pbliher.. Adoia, G., 995. Serrao SE, J. Appl. Math. Lett., : Gejji, V.D. ad H. Jafari, 5. Adoia decopoitio: A tool for olig a te of fractioal differetial eqatio. J. Math. Aal. Appl., 3: Jafari, H. ad V.D. Gejji, 6. Reied Adoia decopoitio ethod for olig a te of oliear eqatio. Appl. Math. Copt., 75: Ch, C., H. Jafari ad Y. Ki, 9. Nerical ethod for the ae ad oliear diffio eqatio ith the hootop pertrbatio ethod. Copt. Math. Appl., 57: Jafari, H. ad V.D. Gejji, 6. Solig liear ad oliear fractioal diffio ad ae eqatio b Adoia decopoitio. Appl. Math. Copt., 8: Jafari, H. ad V.D. Gejji, 6. Reied Adoia decopoitio ethod for olig te of ordiar ad fractioal differetial eqatio. Appl. Math. Copt., 8: Jafari, H. ad S. Seifi, 9. Hootop aali ethod for olig liear ad oliear fractioal diffio-ae eqatio. Co. Noliear Sci. Ner. Silat., 4: Jafari, H., A. Golbabai, S. Seifi ad K. Saead, Hootop aali ethod for olig lti-ter liear ad oliear diffio ae eqatio of fractioal order. Copt. Math. Appl, (Article i Pre).. Wazaz, A.M., 6. The odified decopoitio ethod ad Padé approiat for a bodar laer eqatio i boded Adoia. Appl. Math. Copt., 77: Wazaz, A.M., 8. A td o liear ad oliear Schrodiger eqatio b the ariatioal iteratio ethod. Chao Solito ad Fractal, 37: Wazaz, A.M.,. A e techiqe for calclatig Adoia poloial for oliear poloial. Appl. Math. Copt., : Khri, S.A.,. A Laplace decopoitio algorith applied to cla of oliear differetial eqatio. J. Math. Appl., 4: Khri, S.A., 4. A e approach to Brat' proble. Appl. Math. Copt., 47: Elci Yfogl (Agadjao), 6. Nerical oltio of Dffig eqatio b the Laplace decopoitio algorith. Appl. Math. Copt., 77: Naer, S. Elgazer, 8. Nerical oltio for the Falker--Ska eqatio. Chao Solito ad Fractal, 35: Adoia, G., 994. The decopoitio ethod. Boto: Kler Acad. Pbl. 8. Wazaz, A.M.,. Partial differetial eqatio ethod ad applicatio. Netherlad Balkea Pbliher. 9