Boundary Value Problems. Lecture Objectives. Ch. 27

Transcription

1 Boundar Vaue Probes Ch. 7 Lecture Obectves o understand the dfference between an nta vaue and boundar vaue ODE o be abe to understand when and how to app the shootng ethod and FD ethod. o understand what an Egenvaue Probe s. Inta Vaue Probes hese are the tpes of probes we have been sovng wth K ethods d f dt d f dt Subect to: t ( t ) ( t ) Y ( t ) ( t ) Y Y Y t

2 Inta Vaue Probes hese are the tpes of probes we have been sovng wth K ethods d f dt d f dt Subect to: t ( t ) ( t ) Y ( t ) ( t ) Y A condtons are specfed at the sae vaue of the ndependent varabe! Y Y t Boundar Vaue Probes Auar condtons are specfed at the boundares (not ust a one pont ke n nta vaue probes) ( ) wo Methods: Shootng Method Fnte Dfference Method Boundar Vaue Probes Auar condtons are specfed at the boundares (not ust a one pont ke n nta vaue probes) condtons are specfed at dfferent vaues of the ndependent varabe! ( ) wo Methods: Shootng Method Fnte Dfference Method

3 Shootng Method Appcabe to both near & non-near Boundar Vaue (BV) probes. Eas to peent No guarantee of convergence Approach: Convert a BV probe nto an nta vaue probe Sove the resutng probe teratve (tra & error) Lnear ODEs aow a quck near nterpoaton Non-near ODEs w requre an teratve approach sar to our root fndng technques. Shootng Method Coong fn Eape h heat transfer coeffcent k thera conductvt P pereter of fn A cross sectona area of fn abent teperature d hp d ka ( ) ( ) ( L) Anatca Souton hp ka θ ( ) ( ) d θ θ d hp ka Shootng Method Coong fn Eape θ ( ) ( ) ( ) ( L) d θ θ d θ ( ) c e c e Boundar Condtons θ ( ) θ θ ( L) θ θ ( ) c c θ L L θ ( L) ce ce θ θ ( ) ( θ / θ ) snh snh ( L ) θ snh L

4 Shootng Method Basc Method - Coong fn Eape. ewrte as two frst order ODEs d d d hp d ka ( ) d hp d ka. We need an nta vaue for Guess: ( ) ( ) ( ) ( L) ( ) Shootng Method Basc Method - Coong fn Eape. Integrate the two equatons usng K and ; ths w ed a souton at.integrate the two equatons agan usng a nd guess guess for ().. Lnear nterpoate the resuts to obtan the correct nta condton (Note: ths on works for Lnear ODEs. Eape: () act ( ) act Eact Shootng Method Coong fn Eape hp. ka Matab peentaton (rung_fn_uteqn. ) ecast the probe: d d d d d d ( ).. ( ).6e 96.e ( ) ( ) G

5 Non-Lnear BV Probes - Shootng Method Lnear nterpoaton between soutons w not necessar resut n a good estate of the requred boundar condtons ecast the probe as a oot fndng probe he souton of a set of ODEs can be consdered a functon g( o ) where o s the nta condton that s unknown. g( o ) f ( o ) bc Drve g( o ) to get our souton. Iteratve adust our guess. Non-near Shootng ethod Secant Method Consder the foowng ODEs sste d d ( a) d f ( ) ( b) b d. Guess an nta vaue of (.e. (a) ) ust as was done wth the near ethod. Usng K or soe other ODE ethod we w obtan souton at (b). a b. Denote the dfference between the boundar condton and our resut fro the ntegraton as soe functon. ( ) g( ( b) ' ( b) ) true ( b) ( b) guess Fnd the ero of ths functon Non-near Shootng ethod Secant Method. Check to see f s wthn an acceptabe toerance. Have we satsfed the boundar condton (b)? ε ε ε s. If not then use the Secant Method to deterne our net guess. ( ) () oot '( )

6 Non-near Shootng ethod Secant Method ( a) ( b) b BV No ( a) Guess IV ( a) Sove wth K () () (b) true( b) guess ( b) Is sa enough? Yes wrte out souton Fnte Dfference Method Aternatve to the shootng ethod Substtute fnte dfference equatons for dervatves n the orgna ODE. hs w gve us a set of sutaneous agebrac equatons that are soved a nodes. eca usng centra dfferencng: d d d d () - Fnte Dfference Method d ( ) d ewrte n fnte (centra) dfference for: ( ) Mutp b and sove for ( ) ( ) ( ) ()

7 7 Fnte Dfference Method ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) Genera Equaton: Wrte out for a nodes: App boundar condtons: Equatons unknowns. Fnte Dfference Method Put n Matr for: Sove usng one of our Sstes of near agebrac Equatons ethods Fast eas to peent technque for sovng ODEs Fnte Dfference Method Etended to PDEs Consder a spe Eptca Equaton: LaPace s Equaton hs coud descrbe the stead state teperature dstrbuton n D eta pate. Dscrete (wrte n fnte dfference for) our PDE usng Centra Dfference technque: ) (

8 8 Fnte Dfference Method Etended to PDEs Consder a spe Eptca Equaton: LaPace s Equaton - - Fnte Dfference Method Etended to PDEs Sove for If Unfor spacng If Fnte Dfference Method Etended to PDEs Suppose we have a heated pate wth Drchet boundar condtons 8 We can eas use Gauss-Sede to sove our sste of equatons unt: new od new ε ε ε s

9 Fnte Dfference Method Etended to PDEs Heated Pate Matab Eape 9