Chapter Introduction to Partial Differential Equations

Transcription

1 hpte Intodtion to Ptil Diffeentil Eqtions Afte eding this hpte o shold be ble to: 1. identif the diffeene between odin nd ptil diffeentil eqtions.. identif diffeent tpes of ptil diffeentil eqtions. Wht is Ptil Diffeentil Eqtion (PDE) A diffeentil eqtion with one independent vible is lled n odin diffeentil eqtion. An emple of sh n eqtion wold be d 3 5 3e (0) 5 d whee is the dependent vible nd is the independent vible. Wht if thee is moe thn one independent vible? Then the diffeentil eqtion is lled ptil diffeentil eqtion. An emple of sh n eqtion wold be 3 sbjet to etin onditions: whee is the dependent vible nd nd e the independent vibles. Fom Odin to Ptil Diffeentil Eqtion Assme we pt spheil steel bll tht is t oom tempete in hot wte. The tempete of the bll is going to inese with time. Wht if we wish to find wht this tempete vs. time pofile wold loo lie fo the bll? We wold develop mthemtil model fo this bsed on the lw of onsevtion of het eneg. Fom n eneg blne Het gined - Het lost Het stoed (1) The eneg stoed in the mss is given b Het stoed in the bll m () whee m mss of bll g speifi het of the bll J /( g K) tempete of the bll t given time K

2 hpte Hot Wte Spheil Bll The te of het gined b the bll de to onvetion is Rte of het gined de to onvetion ha( ) (3) whee h the onvetive ooling oeffiient W /( m K ). A sfe e of bll m mbient tempete of the hot wte K As o n see we hve the epession fo the te t whih het is gined (not the het gined) so we ewite the het eneg blne s Rte t whih het is gined - Rte t whih het is lost Rte t whih het is stoed (4) This gives s d ha( ) m (5) dt Eqtion (5) is fist ode odin diffeentil eqtion tht when solved with the initil ondition ( 0) 0 wold give s the tempete of the spheil bll s fntion of time. Howeve we mde lge ssmption in deiving Eqtion (5) - we ssmed tht the sstem is lmped. Wht does lmped sstem men? It implies tht the intenl ondtion in the sphee is lge enogh tht the tempete thoghot the bll is nifom. This llows s to me the ssmption tht the tempete is onl fntion of time nd not of the lotion in the spheil bll. The sstem being onsideed lmped fo this se depends on: mteil of the bll geomet nd het ehnge fto (onvetion oeffiient) of the bll with its sondings.

3 Intodtion to Ptil Diffeentil Eqtions Wht hppens if the sstem nnot be teted s lmped sstem? In tht se the tempete of the bll will now be fntion not onl of time bt lso the lotion. In spheil o-odintes the lotion is given b φ o-odintes. Fige 1 Spheil oodinte Sstem. The diffeentil eqtion wold now be ptil diffeentil eqtion nd is given s t t ρ φ (0) 0 ( ) 0 h t the sfe (6) whee theml ondtivit of mteil ) /( K m W ρ densit of mteil 3 / m g As n intodtion to solve PDEs most tetboos onentte on line seond ode PDEs with two independent vibles nd one dependent vible. The genel fom of sh n eqtion is 0 D B A (7) Whee B A nd e fntions of nd nd D is fntion of nd. z φ P ρ

4 hpte Depending on the vle of B 4A nd ode line PDE n be lssified into thee tegoies. 1. if B 4A < 0 it is lled ellipti. if B 4A 0 it is lled pboli 3. if B 4A > 0 it is lled hpeboli Ellipti Eqtion The Lple eqtion fo sted stte tempete in plte is n emple of n ellipti seond ode line ptil diffeentil eqtion. The Lple eqtion fo sted stte tempete in plte is given b T T 0 (8) Ug the genel fom of seond ode line PDEs with one dependent vible nd two independent vibles A B D 0 A 1 B 0 1 D 0 gives B 4A 0 4(1)(1) 4 4 < 0 This lssifies Eqtion (8) s ellipti. Pboli Eqtion The het ondtion eqtion is n emple of pboli seond ode line ptil diffeentil eqtion. The het ondtion eqtion is given b T T (9) t Ug the genel fom of seond ode line PDEs with one dependent vible nd two independent vibles A B D 0 A B 0 0 D 1 gives B 4A 0 4(0)( ) 0 This lssifies Eqtion (9) s pboli.

5 Intodtion to Ptil Diffeentil Eqtions Hpeboli Eqtion The wve eqtion is n emple of hpeboli seond ode line ptil diffeentil eqtion. The wve eqtion is given b 1 (10) t Ug the genel fom of seond ode line PDEs with one dependent vible nd two independent vibles A B D 0 1 A 1 B 0 D 0 gives 1 B 4A 0 4(1)( ) 4 4 > 0 This lssifies Eqtion (10) s hpeboli. PARTIAL DIFFERENTIAL EQUATIONS Topi Intodtion to Ptil Diffeentil Eqtions Smm Tetboo notes fo the intodtion of ptil diffeentil eqtions Mjo All engineeing mjos Athos At Kw Si Hsh Gpti Dte Feb Web Site