Partial Differential Equations

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1 EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates). oe dimesioal wave eqatio c two dimesioal wave eqatio c oe dimesioal eat eqatio c two dimesioal eat eqatio c two dimesioal aplace eqatio two dimesioal Poisso eqatio f ( y) Te epressio ad its iger dimesioal aalogs are ow as te aplacia of ad deoted by Δ or. Te order of a partial differetial eqatio (PDE) is te igest order of derivative wic appears. A PDE is called liear if te ow fctio ad te partial derivatives are of te first degree ad at most oe of tese appears i ay give term. Oterwise te eqatio is called oliear. I may cases sbscripts are sed to deote partial derivatives. For eample y ad y. A geeral liear PDE of order two i two variables ca be epressed as A B y C yy D E y F G were A B C D E F ad G are fctios of ad y. Te eqatio is called omogeeos if G. PDEs are classified ito tree grops: Elliptic PDE: if B 4AC < Parabolic PDE: if B 4AC Hyperbolic: B 4AC > Page of

2 EE 84 Matematical Metods i Egieerig Te modelig of a pysical problem is ofte doe wit a PDE ad a set of oter eqatios tat describe te beavior of te soltio o te bodary of te regio der cosideratio. Te eqatios are called bodary coditios. Iitial coditios may also be sed to describe te pysical peomeo. A PDE alog wit te bodary ad iitial coditios is called ad iitial-bodary vale problem or simply a bodary vale problem. Teorem : Sperpositio Priciple If ad are soltios of a liear omogeeos partial differetial eqatio te ay liear combiatio c c were c ad c are costats is also a soltio. If i additio ad satisfy a liear omogeeos bodary coditio te so will c c. Eample: Cosider te two dimesioal aplace eqatio. It is liear secod order ad omogeeos. Te fctios y y y y l( y ) e si y are all soltios to te aplace y eqatio. Ay liear combiatio of te above fctios will also satisfy te aplace eqatio. Before sperposig soltios cec tat te PDE i qestio is liear ad omogeeos. Soltio of PDEs: Metod of Separatio of Variables Oe dimesioal wave eqatio et s cosider a oe dimesioal wave eqatio of a vibratig strig. Cosider a strig is fasteed at ad at. Te strig is free to vibrate i a fied plae. ( t) deotes te positio of te poit o te strig at time t. satisfies te oe dimesioal wave eqatio c () Te bodary coditios are: ( t) ad ( t) Te iitial coditios are: ( ) f ( ) ad ( ) g() for all t >. for < <. Te bodary coditios state tat te eds of te strig are eld fied at all time. Te iitial coditios give te iitial positio (sape) of te strig f ( ) ad its iitial velocity g. ( ) Page of

3 EE 84 Matematical Metods i Egieerig Figre : Iitial sape of a stretced strig. et s assme tat ( t) ca be epressed as a prodct of two fctios. ( t) X ( ) T (t) () were X ( ) is a fctio of aloe ad T ( t) is a fctio of t aloe. Te problem terefore becomes fidig X ( ) ad T ( t). Differetiatig ( t) X ( ) T (t ) wit respect to ad t we get X T ad XT Sbstittig tese to te wave eqatio we get X T c X T. Dividig by c XT we get T X (3) c T X Te variables are separated i te sese tat te left side of Eq. (3) is a fctio of t aloe ad te rigt side is a fctio of aloe. Sice te variables t ad are idepedet of eac oter te oly way to get eqality is to ave te fctios o bot sides of Eq. (3) are costat ad eqal. T X ad were is a arbitrary costat ad called te separatio c T X costat. T c T X X We ave to separate te variables i te bodary coditios sig Eq. (). X ( ) T ( t) ad X ( ) T ( t) for all t >. If X ( ) or X ( ) te T ( t) mst be zero for all t ad terefore is zero. To avoid tis trivial soltio we set X ( ) ad X ( ). Te bodary vale problem i X becomes X X X ad X ( ) ( ) Solve te bodary vale problem i X First assme is positive say μ wit μ >. Terefore X μ X ad its μ geeral soltio is: X c e c e ( ) μ Page 3 of 3

4 EE 84 Matematical Metods i Egieerig It is owever easy to verify tat te oly way to satisfy te coditio X( ) is to tae c c. Terefore discard μ. X ( ) ad Now cec if μ. Te eqatio for X becomes X μ X wit te geeral soltio X ( ) c cosμ c siμ. Te coditio X ( ) implies tat c ad ece X ( ) siμ. Te coditio X( ) implies tat si μ. π si μ i tr implies tat μ were is ay iteger. π Terefore X ( ) X si 3 Wit π μ we ow ave to solve for T. π T c T Te geeral soltio of tis eqatio is π π T( t) acosc t bsi c t were T a cos λ t b si λ t were 3 π λ c ad 3 Combiig te soltios of X ad T we obtai a ifiite set of prodct soltios of () all satisfyig te correspodig bodary coditios: π t a t b t ( ) si ( cos λ si λ ) 3 Ay liear combiatio of te above will satisfy () ad te bodary coditios. Terefore by sperpositio priciple try a ifiite liear combiatio as a possible soltio tat will satisfy () te bodary coditios ad te iitial coditios. π t a t b t ( ) si ( cos λ si λ ) We mst ow determie te ow coefficiets a ad b sc tat te fctio ( t) satisfies te iitial coditios. At t we get π ( ) f ( ) a si < < (4) Epressio (4) is te Forier series of ( ) f valid betwee < < (called alf-rage epasio). Terefore te coefficiets a are te sie coefficiets ad ca be determied by Page 4 of 4

5 EE 84 Matematical Metods i Egieerig π a f ( ) si d 3 Te secod iitial coditio ca be sed to determie ( t) term by term wit respect to t b. Differetiate te series for π si a si t b cos t Now set t ( λ λ λ λt) π g ( ) λ b si t t π b g si d Sice tis is te alf-rage epasio of g ( ) λ ( ) 3 π b g si λ d Terefore ( ) π b g ( ) si λ d were π λ c as determied before. π b g ( ) si d 3 cπ Oe dimesioal eat eqatio Cosider temperatre distribtio i a iform bar of legt wit islated lateral srface ad o iteral sorces of eat sbect to certai bodary ad iitial coditios. ( t) o o Figre : Islated bar wit eds ept at zero degree. et ( t) represet te temperatre at poit of te bar at time t. Te eds of te bar are eld at costat temperatre. Te iitial temperatre distribtio of te bar is ( ) f (). Fid ( t) for < < t > were satisfies te oe dimesioal eat eqatio Page 5 of 5

6 EE 84 Matematical Metods i Egieerig c (5) Te bodary coditios are: ( t) ad ( t) Te iitial coditios are: ( ) f ( ) for < <. for all t >. et s assme tat ( t) ca be epressed as a prodct of two fctios. ( t) X ( ) T (t) were X ( ) is a fctio of aloe ad T ( t) is a fctio of t aloe. Sbstitte ( t) X ( ) T ( t) i te eat eqatio. X T c X T T X c T X Terefore to old te eqality costat. T c T ad X X Separatig variables i te bodary coditios we get ( ) ( ) ( ) ( ) X T t ad X T t for all t >. T X ad were is te separatio c T X X ( ) ad ( ) I order to avoid trivial soltios we eed X problem i X is: X X X ( ) ad X ( ) Tis part was solved i te oe dimesioal wave eqatio. π μ 3 π X si ad ( ) X 3 Sbstittig te vale of i te differetial eqatio for T we get π T c T (6) Te geeral soltio of (6) is T ( t) b e λ t 3. Te bodary vale π were λ c 3 λ ( ) t π Terefore te prodct soltio is t be si 3 Page 6 of 6

7 EE 84 Matematical Metods i Egieerig Eac is a soltio of te eat eqatio ad te give bodary coditios. Usig sperpositio priciple we ca write λ ( t) b e t π si Te coefficiet b ca be obtaied tilizig te iitial coditio. Set t. π f ( ) ( ) b si π b f si d 3 Terefore ( ) Fiite Differece Metod for te Heat Eqatio Or oe dimesioal eat eqatio ad te bodary ad iitial coditios for te islated bar wit eds eld at temperatre zero are agai c < < t > Te bodary coditios are: ( t) ad ( t) Te iitial coditios are: ( ) f ( ) for < <. for all t >. Fiite differece metod ca be tilized to approimate te vales of te ow fctio at discrete itervals. Te process begis by dividig te idepedet variables ito a discrete grid strctre. et s divide (positio) ito parts wit a step size of. Similarly divide t (time) ito m parts wit a step size of. i i i 3 t i 3 m Or obective is to approimate te vales i ( i ) were < i < ad represets te vale of at te grid poit ( i ). i Page 7 of 7

8 EE 84 Matematical Metods i Egieerig t i t i- i i i - t t i lim lim ( t ) ( t) Figre 3: Grid poits i te t-plae. ( t ) ad ( t) ( t) ( t) ( t) If ad are sfficietly small we ca write t ( t) ( t) ( t ) ( t) ad ( t) ( t) ( t) Note: Te approimate epressio for te secod derivative ca be obtaied sig Taylor s epasio. At te grid poit ( i ) we ave te followig approimatios wit respect to te first derivative ad te secod derivative. ( ) ( i ) ( i ) t i ( i i ) Page 8 of 8

9 EE 84 Matematical Metods i Egieerig ( i ) ( ) i i i Sbstittig tese discrete approimatios ito te eat eqatio we get c ( ) ( ) i i i i c i ( ) i i i i i c were s. ( s) s( ) i i i i Te bodary ad iitial coditios ave to be discretized as well. ( t) ( ) ad t for all t > ( ) ad ( ) ( ) f ( ) for < < ( i ) f (i i ) i > To compte we eed te tree vales ad as sow i Figre 4. i i i i i i- i i Figre 4: Movig forward i time step. et s cosider a merical eample. A ti bar of it legt is placed i boilig water. After reacig C trogot te bar is removed from te boilig water. Wit te lateral sides ept islated sddely at time t te eds are immersed i a medim wit costat temperatre C. Use te fiite differece metod to approimate te soltio to tis problem at t. sec. Tae c /. Divide te legt ito eqal parts ad ece.. Tae a time step of.. c s ( / ) (. ) (. ) 4 Page 9 of 9

10 ( s) i s( i i ) ( ( 4) ) i ( )( i i ) ( ) 4 i i 4 i i i i / Bodary coditios: (. ) ad (. ) Iitial coditios: (. i ) f (. ) i > i i ( )/ 4 ( * ) 4 ( )/ 4 ( * ) 4 / 3 / EE 84 Matematical Metods i Egieerig 7 ( )/ 4 ( * ) 4 75 ( )/ 4 ( * ) 4 / 3 / ( )/ 4 ( * 75 ) / ( )/ 4 ( * 75) / Fiite Differece Metod for te Wave Eqatio et s tilize te oe dimesioal wave eqatio already cosidered i te earlier sectio. c < < t > tt Page of

11 EE 84 Matematical Metods i Egieerig Te bodary coditios are: ( t) ad ( t) Te iitial coditios are: ( ) f ( ) ad t ( ) g( ) for all t >. for. As before let deote step size i ad deote step size i t. Or obective is to approimate te vales i ( i ) were i ad represets te vale of at te grid poit ( ) i i. i i- i i i - i t Figre 5: Grid poits i te t plae. i i ( i ) ( ) i i i yy ( i ) ( ) i i i Sbstittig tese ito te wave eqatio we get Page of

12 EE 84 Matematical Metods i Egieerig ( s) i s( i i ) i i () c were s. Figre 6: Steppig forward i time. Now we ave to discretize te bodary ad iitial coditios. ( t) ( ) ad t for all t > ( ) ad ( ) ( ) f () ad t ( ) g() ( i ) f (i > for. i ) i () We ow se te iitial data f ad g to obtai te discrete form of te secod iitial coditio. Usig te cetered first differece approimatio ( t ) ( t ) t ( t) Settig t we get ( ) ( ) g( ) t ( ) ( ) ( ) g() ( i ) ( i ) g(i) ( i) g () i i Set i Eq. (). ( s) i s( i i ) i ( s) s( ) i i i i i i Page of

13 i ( s) f ( i) s[ f {( i ) } f {( i ) } ] i ( i) s[ f {( i ) } f ( i) f {( i ) } ] i i f (3) Add () ad (3) ad divide by. s f ( i) g( i) [ f {( i ) } f ( i) f {( i ) } ] (4) i Eample: 6 c f ( ) if < < 3 g ( ) oterwise Use ad. Terefore s ad 6. Wit s epressio () becomes i i i i (5) ( i ) f (i i ) i () s f ( i) g( i) [ f {( i ) } f ( i) f {( i ) } ] (4) i EE 84 Matematical Metods i Egieerig Usig () ad (4) related to te bodary coditios we get te followig Now we ca se (5) to geerate reslts for te remaiig grid poits Page 3 of 3

Chapter 4. Fourier Series

Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,