Solution of heat equation with variable coefficient using derive

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1 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 Soutio of heat equatio with variabe oeffiiet ug derive RS Lebeo α, I Fedotov ad M Shataov β Departet of Matheatis ad Statistis Tshwae Uiversity of Tehoogy Pretoria, South fria bstrat I this paper, the ethod of approiatig soutios of partia differetia equatios with variabe oeffiiets is studied This is doe by osiderig heat fow through a oe-diesioa ode with variabe ross-setios Two ases are osidered The first oe ivoves quadrati approiatio of the variabe oeffiiet by diret itegratio This ase is studied ug a oi doai The seod ase approiates the variabe oeffiiet quadratiay ad by step futios The soutio of the probe i eah ase is epressed ug Gree's futio, ad the resuts are opared By the suitabe use of a oputer agebra syste (CS) a of these ideas a easiy eough be itrodued at the advaed udergraduate eve INTRODUCTION I this paper the ethod of approiatig soutios of PDEs with variabe oeffiiets is based o the study of the heat fro equatio [], that is, u ρ ( ) k( ) ( ) f(, t) t () I stadard udergraduate ourses of PDEs oe osiders a heat equatio with ostat oeffiiets Equatio () with variabe oeffiiet is usuay soved ug diret ueri ethods I this paper we give a easy to ipeet aaytia soutio of the probe ug the derived Gree s futio at a eve that seior udergraduates wi be abe to uderstad ad ipeet, ug CS The heat equatio uder study is osidered with a variabe ross-setio area ( ) I this ase () π (α β)² is the ross-setioa area of the doai ad f(, t ) The heat apaity is, ρ is the desity of the doai ad k is the thera odutivity of the doai The oeffiiets, ρ ad k are assued to be ostats for this ivestigatio Equatio () is osidered with boudary α ebeors@yahooo β Peraet address: Sesor Siee ad Tehoogy of CSIR Maufaturig ad Materias, PO Bo, Pretoria, South fria - 9 -

2 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 oditios of the first kid desribed by u(, t ) ad ut (,), ad iitia oditio of the for u (,) g ( ) This probe desribes the proess of heat trasfer i a aisyetri body with ross setio () orieted so that the -ais ies aog the ais of the body The β ase where ( ) π ( α β) πα ( γ), is where γ, orrespods to iear α depedee of the boudary equatio This orrespods to the oi shape of the osidered doai It is possibe to fid a eat soutio epressed by Gree s futio The iear futio α β a be approiated by a step futio Physiay this orrespods to a yider osistig of N setios of ostat ross setios j ( j,,, N) The ase for N > was osidered by Fedotov, et a (see [] ad []) The Gree s futio a be obtaied subjet to the soutio satisfyig the boudary oditios at the jutios The otiuity of the soutio at the jutios is desribed as foows: u (, t) u (, t), j j j ad the otiuity heat fow is give by: u (, t) u (, t), ( j, N) j j j j The iitia oditio i our eape is defied by: ( ) ( ) g The resuts a be osidered as a approiatio of the soutio for variabe ross setio ( ), see Figure The resuts of both ethods wi be opared ad the soutios at the ed of the paper show approiate siiarity of resuts Mode of a oe-diesioa doai with variabe ross-setios goverig the PDE ad boudary oditios pproiatig by iear futios The ode of a oi doai of variabe ross-setios of N setios is iustrated by Figure beow The soutio to Equatio () is sought i the for: j u(, t) b X ( ) e t, - -

3 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 where: X( ), (,,,) ( γ ) Figure () is obtaied by appyig the ethod of separatio of variabes to equatio () with appiatio of the boudary oditios of the first kid It is ear that the Syste () is orthogoa with weight equatio () that: ( γ ) It foows fro π ad this forua gives the eigevaues eape is osidered where ad γ The eigevaues obtaied for this eape are show i setio The soutio to the probe is give by the foowig equatio: t u(, t) ( ξ ) X ( ) X ( ξ) g( ξ ) e dξ ϕ () where ϕ ( ξ) X( ξ) dξ ( γ) d, (,,,) ( γ ) is the or squared, ad g( ) π b ( γ ) is the iitia futio The oeffiiets b are desribed as foows: b ( ξ) X ( ) ( ), (,,,) ξ g ξ dξ The soutio give by equatio () a aso be represeted by - -

4 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 where u(, t) G(, ξ, t) g( ξ ) dξ, G(, ξ, t) ( ξ ) X ( ) X ( ξ ) e t is Gree's futio pproiatig by ug step futios The doai for this ase is iustrated i Figure beow Figure The soutio is aso sought i the for: where i this ase: t, ut (, ) bx ( e ), (,,,) N ( ) X ( ) X θ ( ), j θ ( ) H( ) H( ), ( j,,,) j j j is the Heaviside futio differee, ad: j j H ( ) < > The j th eigefutio is epressed as foows: X ( ) os, () ( ) j j j ad is obtaied as desribed uder the oi doai ase, with: N b X( ξ ) g( ξ) d( ξ) ϕ, ad g() is the iitia oditio desribed as foows: - -

5 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 g( ) bx( ) ppiatio of the boudary oditios at the edpoits ad at the jutios to equatio (), resuts i a (N N) bok atri, whih is hepfu i deteriig the eigevaues The soutio of the probe is give by the foowig equatio: N t ut (, ) ( ξ ) X( X ) ( ξ) g( ξ) e d( ξ), ϕ () where the weight futio is: ( ), N N N N N ad the or squared is desribed as foows: The soutio is aso give i the for: N ϕ ( ξ) X ( ξ) dξ N u(, t) G(, ξ, t) g( ξ ) dξ, where: is Gree's futio G(, ξ, t) ϕ ( ξ ) X ( ) X ( ξ ) e t Resuts The oi doai ase The first five eigevaues of the sooth futios i the oi ase, as a eape, where, are give i the Tabe - -

6 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 The first five eigefutios orrespodig to the eigevaues i Tabe are show i Tabe Graphs of the sooth eigefutios i Tabe are represeted i f Figure, where derive was used Figure Figure, whih was obtaied ug derive, represets the aaytia soutio of the probe Figure The stepped doai ase - -

7 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber To obtai the eigevaues for the stepped ase, a eape with N is aso osidered For this eape, a (8 8) atri was obtaied fro the foowig syste of equatios [ ] [ ] [ ] [ ] [ ] [ ] os os os os os os os os os os os os os os (6) The variabe area for the doai with variabe ross setios of four setios is desribed as foows: 7 6 ) ( The foowig tabe shows the first five eigevaues obtaied fro soutio of the trasedeta Syste (6), ug the ethod of fidig roots as used by Fedotov, et a [] The otiuous eigefutios, with disotiuous derivatives due to approiatio, ug o-sooth futios ad orrespodig to the eigevaues i Tabe, are give i Tabe

8 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 Tabe shows the vaues of the oeffiiets,,, 8 Mathad was used to obtai graphs of the eigefutios i Tabe, as depited i Figure Figure Figure 6 gives the soutio of the probe (USING Mathad) Notie agai the jup of the derivatives at the poits where jutios our This is a property of eigefutios Cousio Figure 6 I this paper approiatios of soutios of partia differetia equatios i a doai of varyig area were studied, ug CS ad tehiques that are easy - 6 -

9 Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, 6 Septeber 8 eough for seior udergraduate studets to uderstad Two ases were osidered The first oe ivoved a oi doai where the equatio uder study was approiated by sooth eigefutios The seod ase ivoved a stepped doai ad here the equatio was approiated by o-sooth eigefutios The use of tehoogy suh as derive ad Matab yieded siiar aaytia soutios, as show i Figures ad 6 (whih are osey reated)this shows that the ethods of approiatig soutios of PDEs give siiar resuts Differet kids of boudary oditios ay be osidered, for eape, boudary oditios of the seod ad third kids [] I this paper these boudary oditios were ot preseted for the sake of sipiity This oept a be eteded to doais of ore opiated shapes eape of suh a shape that a be approiated either by iear or step futios is iustrated by Figures 7(a) ad 7(b) Figure 7(a) Figure 7(b) These figures represet the doai of a arbitrary shape approiated first by ug iear futios (Figure 7(a)) ad seod by step futios (Figure 7(b)) Referees [] SJ Farow (99) Partia Differetia Equatios for Sietists ad Egieers, New York: Dover Pubiatios pp 7- [] I Fedotov, S Joubert, J Marais ad M Shataov (6) other approah to vibratio aaysis of stepped strutures Eetroi Trasatios o Nueria, pp 66-7 [] I Fedotov, M Shataov, K Nioaides ad J Marais () uified approah to vibratio aaysis of stepped strutures I Proeedigs of the Europea Cogress o Coputatioa Methods i ppied Siees ad Egieerig, -8 Juy,, Uiversity of Jyväskyä, Jyväskyä, Fiad vaiabe at: (essed: Septeber 8) [] I Fedotov, M Shataov ad JN Mwabakaa (8) O the probe of roots braketig ad hoog guess vaues for roots of agebrai ad trasedeta equatios Buffespoort TIME8 Peer-reviewed Coferee Proeedigs, -6 Septeber, South fria, ISBN , pp 6 7 [] RS Lebeo (8) pproiatig soutios of partia differetia equatios with variabe oeffiiets, MTeh ii thesis, Tshwae Uiversity of Tehoogy, pp 7-,