Applying Differential Transformation Method to. the One-Dimensional Planar Bratu Problem

Transcription

1 It J Cotemp Math Siees, Vol, 7, o, Applyig Differetial Trasformatio Method to the Oe-Dimesioal Plaar Brat Problem I H Abdel-Halim Hassa Departmet of Mathematis, Falty of Siee, Zagazig iversity, Zagazig, Egypt ismhalim@hotmailom Vedat Sat Ertür Departmet of Mathematis, Falty of Arts ad Siees Odoz Mayis iversity, Sams,Trey vsertr@omedtr Abstrat This paper is the appliatio of differetial trasformatio method DTM to solve the Brat problem A osiderable researh wors have bee odted reetly i applyig DTM to differet types of partial differetial eqatio of Abdel-Halim Hassa Chaos, Solitos & Fratals [6] ad fratioal differetial eqatios of Arıoğl ad Özol Chaos, Solitos & Fratals [] The oliear eigevale problem Δ + λ e i the it sqare with o the bodary

2 494 I H Abdel-Halim Hassa ad Vedat Sat Ertür is ofte referred to as "the Brat problem" The Brat problem i oe-dimesioal plaar oordiates + λ e with has two ow, bifrated soltios for vales of λ < λ, o soltios for λ > λ ad a iqe soltio whe λ λ The vale of λ is related to the fied poit of hyperboli otaget ftio Two speial ases of the problem are illstrated by sig the tehiqe ad merial reslts ad olsios will be preseted Mathematis Sbjet Classifiatio: 74S, 4B5 eywords: Differetial trasformatio method, Brat problem Itrodtio I this letter, the reslts of applyig differetial trasformatio method [9] to the oe-dimesioal plaar Brat problem [5] will be preseted The Brat problem is a ellipti partial differetial eqatio whih omes from a simplifiatio of the solid fel igitio model i thermal ombstio theory [4] It is also a oliear eigevale problem that is ofte sed as a behmarig tool for merial methods [4], [], [], [8] ad [4] The goal of this paper will be to apply differetial trasformatio method to the - dimesioal plaar Brat problem ad ompare the ow eat soltios to merial soltios proded by differetial trasformatio method The lassial Brat problem is Δ + λ e o Ω : {, y, y }, with o Ω The oe-dimesioal plaar versio of this problem is

3 Differetial trasformatio method λ e, with ad 4 We will orgaize or paper i the followig way I Setio, the differetial trasformatio method will be preseted I Setio, the differetial trasformatio method will be applied to the Brat problem Setio 4, the algorithm is implemeted for two merial eamples Colsio will be preseted i the last setio Desriptio of differetial trasformatio method I order to solve the Brat problem by differetial trasformatio its basi theory is stated i brief i this setio The differetial trasformatio of the th derivative of a ftio is defied as follows: d! d I, is the origial ftio ad is the trasformed ftio As i [] ad [7] the differetial iverse trasformatio of is defied as follows: I fat, from ad, we obtai d! d Eq implies that the oept of differetial trasformatio is derived from the Taylor series epasio From the defiitios of ad, it is easy to prove that the trasformed ftios omply with the followig basi mathematis operatiossee Table, below

4 496 I H Abdel-Halim Hassa ad Vedat Sat Ertür Table The fdametal mathematis operatios Origial ftio Trasformed ftio z ± v Z + V 4 z λ Z λ 5 z d d Z d d z Z z v Z l V l 8 z λ m l, if m, Z λδ m, δ m 9, if m I real appliatios, the ftio is epressed by a fiite series ad a be writte as Eqatio implies that is egligibly small + Nmerial differetial trasformatio algorithm I this setio, the differetial trasformatio method is implemeted for the soltio of the Brat problem whih is give by + λ e ad, where λ > The eat soltio of ad is give [4], [] ad [] preseted here as

5 Differetial trasformatio method osh osh l θ θ, where θ solves 4 osh θ λ θ 4 There are two soltios to 4 for vales of < λ < λ For λ λ >,there are o soltios The soltio is oly iqe for a ritial vale of λ λ whih solves 4 sih 4 θ λ 5 It was evalated i [4], [], [], [], [8] ad [4], that the ritial vale λ is give by 5879 λ 6 The tehiqe osists first of taig differetial trasformatio to both sides of Eq Before doig this, let s epad the o-liear term e as follows:! e 7 Theorem If! z the! Z r Proof By sig Eq8, we obtai differetial trasformatio of as

6 498 I H Abdel-Halim Hassa ad Vedat Sat Ertür From hypothesis, ad by sig Eq5 we obtai! Z r Theorem If ep z the! Z r Proof By sig Theorem, we fid r! ep or! Z r Fially if we apply differetial trasformatio to! r λ 8 The bodary oditios i Eq a be trasformed at as follows:

7 Differetial trasformatio method ad et a Sbstittig 9 ad, ad ito 8, we have λ Sbstittig 9,-, ad ito 8, we have aλ 6 Followig the same rersive proedre, we allate p to the th term The soltio obtaied from Eq has yet to satisfy the seod iitial oditio i Eq4, whih has ot bee maiplated i obtaiig this approimate soltio Applyig this iitial oditio ad the solvig the resltig eqatio for a, will determie the ow ostat a ad evetally the merial soltio 4 Nmerial eamples For prposes of illstratio of differetial trasformatio method for solvig the oe-dimesioal plaar Brat problem, The ompter appliatio program Mathematia was sed to eete the algorithm that was sed with the merial eamples The differetial trasformatio method desribed i Setio is applied to two speial ases of the Brat problem The first oe is whe λ I this ase, the problem - has two loally iqe soltios ad with 549 ad 99 see[4] The soltio of the

8 5 I H Abdel-Halim Hassa ad Vedat Sat Ertür differetial trasformatio method overges to the soltio 549 ad ot to the soltio 99 I Table, the eat soltio for the ase λ derived from Eq is ompared with the merial soltio obtaied by the differetial trasformatio tehiqe Table shows that the absolte errors, ie the differee betwee the approimate soltio ad eat soltio Fig shows the aalyti soltios ad merial soltios obtaied by differetial trasformatio method for λ Table Differetial trasformatio method approimatio for the Brat problem for the ase λ ad r 6 Eat soltio Nmerial soltio Error

9 Differetial trasformatio method 5 Table Differetial trasformatio method approimatio for the Brat problem for the ase λ ad r 6 Eat soltio Nmerial soltio Error The seod ase is whe λ Table shows that the absolte error of the approimatio of the merial tehiqe As λ approahes the ritial vale λ, the error beomes larger, ad the overgee beomes slower as is lear for the ase λ show i Table Also, Fig shows the aalyti soltios ad merial soltios obtaied by differetial trasformatio method for λ Fig The differetial trasformatio soltio dotted rve verss aalyti soltio solid rve for λ

10 5 I H Abdel-Halim Hassa ad Vedat Sat Ertür Fig The differetial trasformatio soltio dotted rve verss aalyti soltio solid rve for λ 5 Colsio Differetial trasformatio method has bee applied to the Brat problem The reslts for two merial eamples showed that the preset method is qite reliable Therefore, this method a be applied to may ompliated o-liear eigevale problem ad does ot reqire liearizatio, disretizatio or pertrbatio Referees [] A Arıoğl, İ Özol, Soltio of fratioal differetial eqatios by sig differetial trasform method, Chaos, Solitos ad Fratals, 4 7,47 48 [] A M Wazwaz, Adomia deompositio method for a reliable treatmet of the Brat-type eqatios, Applied Mathematis ad Comptatio, 66 5, [] C Che, SH Ho, Appliatio of differetial trasformatio to eigevale problems, Applied Mathematis ad Comptatio,79 996,7-88

11 Differetial trasformatio method 5 [4] DA Fra-ameetsi, Diffsio ad Heat Ehage i Chemial ietis, Prieto iversity Press: Prieto, NJ 955 [5] G Brat, Sr les éqatios itégrales oliéaires, Blletis of the Mathematial Soiety of Frae, 4 94, -4 [6] IH Abdel-Halim Hassa, Compariso differetial trasformatio tehiqe with Adomia deompositio method for liear ad oliear iitial vale problems, Chaos, Solitos ad Fratals 6, doi:6/jhaos664 [7] IH Abdel-Halim Hassa, Differet appliatios for the differetial trasformatio i the differetial eqatios, Applied Mathematis ad Comptatio,9,8- [8] J Jaobso, Shmitt, The ioville-brat-gelfad problem for radial operators, Joral of Differetail Eqatios,84, 8-98 [9] J Zho, Differetial Trasformatio ad Its Appliatios for Eletrial Cirits i Chiese, Hazhog ivpress, Wha, Chia, 986 [] JP Abbott, A effiiet algorithm for the determiatio of ertai bifratio poits, Joral of Comptatioal ad Applied Mathematis, 4 978,9-7 [] JP Boyd, Chebyshev polyomial epasios for simltaeos approimatio of two brahes of a ftio with appliatio to the oedimesioal Brat eqatio, Applied Mathematis ad Comptatio, 4, 89- [] JP Boyd, A aalytial ad merial stdy of the two-dimesioal Brat eqatio, Joral of Sietifi Comptig, 986, 8-6 [] R Bmire, Ivestigatios of ostadard Mies-type fiite-differees shemes for siglar bodary vale problems i ylidrial ad spherial oordiates, Nmerial Methods for Partial Differetial Eqatios 9, 8-98

12 54 I H Abdel-Halim Hassa ad Vedat Sat Ertür [4] M Asher, RMM Matheij ad RD Rssell, Nmerial Soltio of Bodary Vale Problems for Ordiary Differetial Eqatios, Soiety for Idstrial ad Applied Mathematis: Philadelphia, PA 995 Reeived: Je, 7