he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef Cvl Engneerng Department he Islamc Unverst of Gaza
PAR 8 Partal Dfferental Equatons PDEs CH 9; Fnte Dfference Method Ch3; Fnte Element Method Chapter 9
Partal Dfferental Equatons Chapter 9 3
Partal Dfferental Equatons. Laplace equaton. Dffuson equaton 3. Wave equaton A partal dfferental equaton (PDE) nvolves two or more ndependent varables. For eample: t c t f f k 0 f f Chapter 9 4
Fnte Dfference: Ellptc Equatons Smlar to the ODE central dvded dfferences are substtuted for the partal dervatves n the orgnal equaton. hus a partal dfferental equaton s transformed nto a set of smultaneous algebrac equatons that can be solved b the methods descrbed earler. Because of ts smplct and general relevance to most areas of engneerng we wll use a heated plate as an eample for solvng ellptc PDEs. 5
Fnte Dfference: Ellptc Equatons he Central dvded dfferences - + + - 6
Fnte Dfference: Ellptc Equatons 7
Fnte Dfference: Ellptc Equatons 8
he General steps. Choose ntegers n and m to defne step szes h and k n and drecton respectvel.. Partton the nterval [a b] nto m equal parts of wdth h and [c d] nto n equal parts of wdth k n =d b a d c h k m n + - + - o =c o =a m =b 9
he General steps 3. Place a grd b drawng vertcal and horzontal lnes through the ponts of coordnates ( ). 4. For each mesh ponts; h k n =d - + + - o =c o =a m =b Chapter 9 0
he Laplacan Dfference Equatons/ 0 4 0 0 k h Chapter 9 Laplacan dfference equaton. Holds for all nteror ponts Laplace Equaton O[() ] O[() ] -4
Fnte Dfference: Ellptc Equatons he boundar condtons along the edges must be specfed to obtan a unque soluton. he smplest case s where the temperature at the boundar s set at a fed value ths s called; Drchlet boundar condton. Suppose that at the four edges are; 0 o C 50 o C 00 o C and 75 o C he Drchlet (or frst-tpe) boundar condton s a tpe of boundar condton named after Peter Drchlet (805 859). When mposed on an ordnar or a partal dfferental equaton t specfes the values that a soluton needs to take on along the boundar of the doman. In engneerng applcatons a Drchlet boundar condton also referred to as a fed boundar condton.
Fnte Dfference: Ellptc Equatons A balance for node () s: 0 0 75 0 0 4 0 4 75 4 0 Smlar equatons can be developed for other nteror ponts to result a set of smultaneous equatons. 0 Chapter 9 3
Fnte Dfference: Ellptc Equatons he result s a set of nne smultaneous equatons wth nne unknowns: 4 4 3 4 3 3 4 4 3 3 4 3 3 3 4 3 3 3 3 4 3 3 33 33 4 33 75 0 50 75 0 50 75 00 50 Chapter 9 4
he Lebmann Method/ Most numercal solutons of Laplace equaton nvolve sstems that are ver large. For larger sze grds a sgnfcant number of terms wll be zero. For such sparse sstems most commonl emploed approach s Gauss-Sedel whch when appled to PDEs s also referred as Lebmann s method. Chapter 9 5
0 75 0 0 4 4 0 6
Eample 9. Cont. Chapter 9 7
Fg 9.5 Chapter 9 8
Man engneerng problems ehbt rregular boundares. Irregular Boundares Chapter 9 9
0 Frst dervatves n the drecton can be appromated as: ) ( ) ( Irregular Boundares
A smlar equaton can be developed n the drecton. Substtute n Laplace Equaton ) ( ) ( Irregular Boundares 0 0 ) ( ) ( ) ( ) ( Chapter 9