Substituting Fourier law of conduction into the First law of thermodynamics for solid in differential cartesian element

Transcription

1 EA ODUIO FORMUAIO I ARESIA OORDIAE SYSEM Foue w of ouco eco ; Q q A F lw of heoc: E E E E ou geee oe q q q q q q E geee q q q q q q q q q q q q U g Suug Foue lw of couco o he F lw of heoc fo ol ffeel ce elee U g If el eeg e oe U equo ecoe g If ue h hel coucv coeffce eee of e hoogeeou ll eco equo ecoe g whee clle hel ffuv E oe EA ODUIO FORMUAIO I GEERA OORDIAE SYSEM

2 Fo geel cooe e h S c e eee fuco of ce cooe e S If he elo ewee ohgol cooe e 3 u u u ce cooe e gve 3 u u u 3 u u u Y 3 u u u Z he he ffeel legh c e oe : 3 3 u u u u 3 3 u u u u Y 3 3 u u u u Z Suug hee c o he h equo 3 3 u u u S whee u u u e fe ohgol cooe e ecoe: u u q 3 o 3 u q I h ce geel f lw of heoc fo h cooe e ecoe: g u u u u u u Ele ce: Relo ewee ce cooe e clcl cooe e gve : co u u u 3 he cle fco: co co co 3 o o he fe equo ecoe: q q q g Relo ewee ce cooe e hecl cooe e gve : co

3 co co u u 3 u he cle fco: co co co 3 co co co o o he fe equo ecoe: q q q g BOUDARY ODIIOS Fo ffeel ufce elee q l l l l l l q Eeg lce he ufce: e ul=he lo cov q q q q u 4 4 u h q A o o h geele ce eeue uo o he ufce c lo e ecfe uch : =f o S o = o S I o o he ufce e of ou ehe of he cocg ufce c lo ecf ou e couco hough ol =he couco hough ol =he fe co he g

4 h SEPARAIO OF VARIABE MEODS FOR OMOGEEOUS AD O-OMOGEEOUS EA ODUIO e u coe ce cooe e he couco equo Aue eo of vle he fo: h ce ove equo ecoe: he ee fuco ecoe: he ohe e c lo e ee he l w uch : Z Y Suuo of h equo o ove ffeel equo el: Z Y Y Y Z Z Whee Geel oluo of he equo ecoe: e co co co PROBEM OMOGEEOUS SOUIO; ARASIA OORDIAES

5 Bou coo = > = > h F = <=<= Seo of vle: Bou coo = > = > F = <=<= Soluo e: co e 3 e co Alco of ou coo: co = > co = > co equo h oe h oe oluo. e co F = <=<= co F

6 h equo h coeffce. I oe o olve hee coeffce we wll ue oee of ohogol fuco. F co co co F co Dee o vlue of F h ego c e olve ug uecl o lcl eho. co e F co co e PROBEM MAAB SOUIO fuco [] = lh =; fo =: e=e; =+ecoee-eelh; e fuco [] = e =@ F.coe; %hı egl ogı ullılı =qu; =ee++/ee+; =/; e fuco [e] = e PI=4; =-PI+.; = +PI/-.; e=eco@ fuc; e fuco =ecoflu % eco oo fg eho =; e=; e=.; e=.e; whlee>e&&e< =l+u/.; e=e+; f ~= e=u-l/u+l;e fl= fl; f= f; e= flf; f e ==. e=; elef e <. u=; ele l=; e e fe>= ff'mu ue of eo eceee eul gh o e vl""maksimum UMBER OF IERAIO WARIG'; e e fuco [F] = F F=.; e fuco [f] = fuce f=ee-; e

7 fuco [] = e=e; =ee++/ee+; e PROBEM JAVA SOUIO o v.ul.; o v.w.; o v.le.ale; o v.w.eve.; o v.wg.; ulc cl _uv ulc oule ; ; ulc oule foule eoule oule eu emh.e-; ulc _uv =.;=; ulc oule foule oule houle oule // evve eu -f+.h+8.f+h-8.f-h+f-.h/.h; ulc oule eoule oule ++; oule =-Mh.PI+.; oule = +Mh.PI/-.; oule cc=.e-; oule EPS=.e-; oule h=.; MAI=; ; oule folffhfl; oule e; fl=f; fh=f; Se.ou.l" ="++"fl="+fl+"="++"fh="+fh; f fl >. && fh >. fl <. && fh <. Se.ou.l" = "++"="++"="++"o oo ee he gve ego"; f fl ==. eu ; f fh ==. eu ; f fl <. =;=; ele =;=; =.5+; //o o eğe ol=mh.-; =ol; f=f; f=fh; fo =;<=MAI;++ f -f-f-f-f >. Mh..f > Mh.olf //e öle ol=; =.5-; =+; f == && f>eps =+.e-5;eu ; //çöü!!!! ele //ewo-rho ol=; =f/f; e=; -= ; f e == && f<eps =+.e-5;eu ; //çöü!!!!! f Mh. < cc&& f<eps =+.e-5;eu ; //çöü!!!!! f=f; f=fh; f f <. =; ele =;

8 Se.ou.l"Mu eo ue eceee"; =+.e-5; eu ; //og eve eche hee ulc oule oule oule e eu Mh.coe; ulc oule oule eoule oule oule =ee++/ee+; eu ; ulc oule Foule //= ıclı ğılıı oule =.; eu ; ulc oule foule oule eoule oule eu ef; ulc oule egloule eoule oule //Gu Koo egl //e h ı oule e=.e-8; =; Aue=; oole ouc; oule =; oule =; oule egl=; oule A[][]; Aw = ; oule olh = ; oule[] c; oule[] g; oule[] ; = ; g = ; = ; = ; h = ; oule v = ; oule = ; oule = ; oule g = ; oule = ; oule = ; oule = ; Aw = 4; A = ew oule[-+][aw-+]; = 6; g = 5; c = ew oule[]; = ew oule[]; g = ew oule[]; //Gu ılı g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; g[5] = ; g[6] = ; g[7] = ; g[8] = ; g[9] = ; g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; c[] = ; c[] = ; c[] = ;

9 c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[3] =.; //Koo ılı [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [3] = ; fo=-; >=/; -- c[] = -c[--]; fo=-; >=/; -- [] = [--]; fo=g-; >=; -- g[--] = g[]; g[+] = g[]; fo=; <=/; ++ g[] = ; =.5-; =.5+; g = ; = ; fo=; <=-; ++ v = fc[]+e; = +v[];

10 f %== g = g+vg[]; = -.5; g = g-.5; A[][] = Mh.g-; A[][] = ; A[][] = ; A[][3] = ; olh = A[][]; f olh<e ouc = ue; egl = ; Aue = ; eu egl; Aue = ; foh=; h<=-; h++ Aue = h+; ga h Aw; olh = olh-a[h-][]; = A[h-][]; = A[h-][3]; A[h-][] = ; A[h-][3] =.5+; A[h][] =.5+; A[h][3] = ; fo=h-; <=h; ++ =.5A[][3]-A[][]; =.5A[][3]+A[][]; g = ; = ; fo=; <=-; ++ v = fc[]+e; = +v[]; f %== g = g+vg[]; = A[][3]-A[][].5; g = ga[][3]-a[][].5; A[][] = Mh.g-; A[][] = ; olh = olh+a[][]; ca h- Aw; ca h Aw; f olh<e e; ouc = olh<e; egl = ; fo=; <=Aue-; ++ egl = egl+a[][]; eu egl; ulc vo goule A[][] Awh = ; = ; oule = ; c = ; f == eu; fo=; <=Awh-; ++ = A[-][]; A[-][] = A[][]; A[][] = ; //e of fo = ; whle +<- c = +; f +<- f A[+][]>A[+][] c = +;

11 f A[][]<A[c][] fo=; <=Awh-; ++ = A[][]; A[][] = A[c][]; A[c][] = ; = c; ele e; //e of whle ulc vo coule A[][] Awh = ; = ; oule = ; = ; f == eu; = ; whle!= = -/; f A[][]>A[][] fo=; <=Awh-; ++ = A[][]; A[][] = A[][]; A[][] = ; = ; ele e; ulc oule oule eoule oule eu./eegle; ulc oule eoule oule oule lfoule oule oule =; oule l=; fo =;<;++ l=e; oule c=l; Se.ou.l"="+c+"l="+l+"="+; +=clmh.e-lllf; eu ; ulc c vo Sg g[] //f f=ew f; oule lf; =Doule.eDouleJOoPe.howIuDlog" : "; =Doule.eDouleJOoPe.howIuDlog" =h/ / : "; lf=doule.edoulejoope.howiudlog" lf : "; =Doule.eDouleJOoPe.howIuDlog" = : "; =Doule.eDouleJOoPe.howIuDlog" = : "; _uv h=ew _uv; JOoPe.howMegeDlogull" e = "+h.elf "uv ğlı ıı fe : "JOoPe.PAI_MESSAGE; Se.e; PROBEM OMOGEEOUS SOUIO; ARASIA OORDIAES

12 Bou coo = > h h = > F = <=<= Seeo of vle: Bou coo = > = > F = <=<= Soluo: co e 3 e co Alco of ou coo: co = > co co co co = > co co

13 co co h equo h ulle oo. F = <=<= co e F co h equo h coeffce. I oe o olve hee coeffce we wll ue oee of ohogol fuco. co co F.5 F co Dee o vlue of F h ego c e olve ug uecl o lcl eho. F co co e PROBEM MAAB SOUIO fuco [] = lh =; fo =: e=e =e =+ecoe+ee-eelh; e fuco [] = e =@ F.ecoe+e; =qu; =.5ee++/ee++; =/; e fuco [e] = e PI=4; =-PI+.; = +PI/-.; e=eco@ fuc; e fuco [F] = F F=.; e fuco [f] = fuce f=e-e+/ee-; PROBEM JAVA SOUIO o v.ul.; o v.w.; o v.le.ale; o v.w.eve.;

14 o v.wg.; ulc cl _uv ulc oule ; ; ulc oule foule eoule oule oule oule e=e; oule B=; eu e-/emh.e-+; ulc _uv =.;=; ulc oule foule oule houle oule oule // üev heı eu -f+.h+8.f+h-8.f-h+f-.h/.h; ulc oule eoule oule oule ++; //lee ol ele öü : eöle-ewo Rho eou //gele; //ö ıılıı oule =-Mh.PI+.; oule = +Mh.PI/-.; oule cc=.e-; //eol ı f u // üçü oluğu çöüe ulşığııı ul eeceğ oule EPS=.e-; oule h=.; //üev e üülüğü MAI=; //Mu eo ıı ; oule folffhfl; oule e; fl=f; fh=f; Se.ou.l" ="++"fl="+fl+"="++"fh="+fh; //Bölgee ö eğe vı ool e f fl >. && fh >. fl <. && fh <. Se.ou.l" = "++"="++"="++"vele ııl çe ö eğe o"; f fl ==. eu ; f fh ==. eu ; f fl <. =;=; ele =;=; =.5+; //o o eğe ol=mh.-; =ol; f=f; //foou oı eğe f=fh; //foou üev oı eğe fo =;<=MAI;++ f -f-f-f-f >. Mh..f > Mh.olf //e öle ol=; =.5-; =+; f == && f>eps =+.e-5;eu ; //çöü!!!! ele //ewo-rho ol=; =f/f; e=; -= ; f e == && f<eps =+.e-5;eu ; //çöü!!!!! f Mh. < cc&& f<eps =+.e-5;eu ; //çöü!!!!! f=f; f=fh; f f <. =; ele =; Se.ou.l"Mu eo ıı şılı"; =+.e-5; eu ; //og u geleel ulc oule oule oule e oule eu emh.coe+mh.e;

15 ulc oule oule eoule oule oule oule =.5ee++/ee++; eu ; ulc oule Foule //= ıclı ğılıı oule =.; eu ; ulc oule foule oule eoule oule oule eu ef; ulc oule egloule eoule oule oule //Gu Koo egl //e h ı oule e=.e-8; =; Aue=; oole ouc; oule =; oule =; oule egl=; oule A[][]; Aw = ; oule olh = ; oule[] c; oule[] g; oule[] ; = ; g = ; = ; = ; h = ; oule v = ; oule = ; oule = ; oule g = ; oule = ; oule = ; oule = ; Aw = 4; A = ew oule[-+][aw-+]; = 6; g = 5; c = ew oule[]; = ew oule[]; g = ew oule[]; //Gu ılı g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; g[5] = ; g[6] = ; g[7] = ; g[8] = ; g[9] = ; g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ;

16 c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[3] =.; //Koo ılı [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [3] = ; fo=-; >=/; -- c[] = -c[--]; fo=-; >=/; -- [] = [--]; fo=g-; >=; -- g[--] = g[]; g[+] = g[]; fo=; <=/; ++ g[] = ; =.5-; =.5+; g = ; = ; fo=; <=-; ++ v = fc[]+e; = +v[]; f %== g = g+vg[]; = -.5; g = g-.5; A[][] = Mh.g-; A[][] = ;

17 A[][] = ; A[][3] = ; olh = A[][]; f olh<e ouc = ue; egl = ; Aue = ; eu egl; Aue = ; foh=; h<=-; h++ Aue = h+; ga h Aw; olh = olh-a[h-][]; = A[h-][]; = A[h-][3]; A[h-][] = ; A[h-][3] =.5+; A[h][] =.5+; A[h][3] = ; fo=h-; <=h; ++ =.5A[][3]-A[][]; =.5A[][3]+A[][]; g = ; = ; fo=; <=-; ++ v = fc[]+e; = +v[]; f %== g = g+vg[]; = A[][3]-A[][].5; g = ga[][3]-a[][].5; A[][] = Mh.g-; A[][] = ; olh = olh+a[][]; ca h- Aw; ca h Aw; f olh<e e; ouc = olh<e; egl = ; fo=; <=Aue-; ++ egl = egl+a[][]; eu egl; ulc vo goule A[][] Awh = ; = ; oule = ; c = ; f == eu; fo=; <=Awh-; ++ = A[-][]; A[-][] = A[][]; A[][] = ; //e of fo = ; whle +<- c = +; f +<- f A[+][]>A[+][] c = +; f A[][]<A[c][] fo=; <=Awh-; ++ = A[][]; A[][] = A[c][]; A[c][] = ;

18 = c; ele e; //e of whle ulc vo coule A[][] Awh = ; = ; oule = ; = ; f == eu; = ; whle!= = -/; f A[][]>A[][] fo=; <=Awh-; ++ = A[][]; A[][] = A[][]; A[][] = ; = ; ele e; ulc oule oule eoule oule oule eu./eegle; ulc oule eoule oule oule oule lfoule oule oule =; oule l=; fo =;<;++ l=e; oule c=l; Se.ou.l"="+c+"l="+l+"="+; +=cmh.e-lllfl; eu ; ulc c vo Sg g[] //f f=ew f; oule lf; =Doule.eDouleJOoPe.howIuDlog" : "; =Doule.eDouleJOoPe.howIuDlog" =h/ / : "; =Doule.eDouleJOoPe.howIuDlog" =h/ / : "; lf=doule.edoulejoope.howiudlog" lf : "; =Doule.eDouleJOoPe.howIuDlog" = : "; =Doule.eDouleJOoPe.howIuDlog" = : "; _uv h=ew _uv; JOoPe.howMegeDlogull" e = "+h.elf "uv ğlı ıı fe : "JOoPe.PAI_MESSAGE; Se.e; PROBEM 3 OMOGEEOUS SOUIO; YIDRIA OORDIAES

19 Bou coo: = > h = > F = <= <= Seeo of vle: R R R R R R R R R R Bou coo: R = > h R = > F = <=<= Soluo: Y J R e 3 e Y J J Y h equo clle Beel fuco. I Fgue ehvo of eel fuco e gve lo.

20 A ee fo he lo - = Y = - Y = J = J Befoe gog fuhe le u loo oe oee of eel fuco. Beel fuco c e efe ee. Beel fuco J c e efe ug he followg ee:! J Whee g fuco efe e e e Whee = he Eule-Mcheo co. he Beel fuco Y gve : co J J Y Whe = ove ege he oluo J J - e o eee: he e ele : J J Devve of he J o Y eel fuco c e gve : J J J J Iegl of he J o Y eel fuco c e gve : J J J J Afe h ouco o eel fuco ou coo c e le ow: Y J R R

21 Y J R R R = > J J J J h equo h ulle oo. e J F J F h equo h coeffce. I oe o olve hee coeffce we wll ue oee of ohogol fuco. J J J J F J J F Dee o vlue of F h ego c e olve ug uecl o lcl eho. e e J J F J PROBEM 3 MAAB SOUIO fuco [f] = fuc3e f=eeele-eele; e fuco [e] = gele =+.; = +.; f=fuc3; f=fuc3; =; RY=; FAOR=.; whle ff> && <RY =+FAOR-; f=fuc3; =+; e e=eco@ fuc3; e fuco =ecoflu % eco oo fg eho =; e=; e=.; e=.e; whlee>e&&e<

22 =l+u/.; e=e+; f ~= e=u-l/u+l;e fl= fl; f= f; e= flf; f e ==. e=; elef e <. u=; ele l=; e e fe>= ff'mu ue of eo eceee eul gh o e vl""maksimum UMBER OF IERAIO WARIG'; e e fuco [e] = e3 =; fo =: =gele; e=; e e fuco [] = 3e =@.F3.eele; =qu; J=eele; =JJ/+ee/ee; =/; e fuco [F] = F3 F=.; e fuco [] = 3lh =; =; e=e3; fo =: =3e =+eelee-eelh; e e PROBEM 3 JAVA SOUIO o v.ul.; o v.w.; o v.le.ale; o v.w.eve.; o v.wg.; ulc cl _l oule ; ; ulc _l =; =.; ulc c oule foule eoule oule eu emh.je/mh.je-; ulc c oule foule oule houle oule //üev heı eu -f+.h+8.f+h-8.f-h+f-.h/.h;

23 ulc oule Foule eu.; ulc oule oule eoule oule oule J=Mh.Je; oule =JJ/.+ee/ee; eu ; ulc oule foule oule e eu FMh.Je; ulc oule egloule eoule // Gu Koo egl // e h ı oule e=.e-8; =; Aue=; oole ouc; //egl l oule =; oule =; oule egl=; oule A[][]; Aw = ; oule olh = ; oule[] c; oule[] g; oule[] ; = ; g = ; = ; = ; h = ; oule v = ; oule = ; oule = ; oule g = ; oule = ; oule = ; oule = ; Aw = 4; A = ew oule[-+][aw-+]; = 6; g = 5; c = ew oule[]; = ew oule[]; g = ew oule[]; //Gu ılı g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; g[5] = ; g[6] = ; g[7] = ; g[8] = ; g[9] = ; g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ;

24 c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[3] =.; //Koo ılı [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [3] = ; fo=-; >=/; -- c[] = -c[--]; fo=-; >=/; -- [] = [--]; fo=g-; >=; -- g[--] = g[]; g[+] = g[]; fo=; <=/; ++ g[] = ; =.5-; =.5+; g = ; = ;

25 fo=; <=-; ++ v = fc[]+e; = +v[]; f %== g = g+vg[]; = -.5; g = g-.5; A[][] = Mh.g-; A[][] = ; A[][] = ; A[][3] = ; olh = A[][]; f olh<e ouc = ue; egl = ; Aue = ; eu egl; Aue = ; foh=; h<=-; h++ Aue = h+; ga h Aw; olh = olh-a[h-][]; = A[h-][]; = A[h-][3]; A[h-][] = ; A[h-][3] =.5+; A[h][] =.5+; A[h][3] = ; fo=h-; <=h; ++ =.5A[][3]-A[][]; =.5A[][3]+A[][]; g = ; = ; fo=; <=-; ++ v = fc[]+e; = +v[]; f %== g = g+vg[]; = A[][3]-A[][].5; g = ga[][3]-a[][].5; A[][] = Mh.g-; A[][] = ; olh = olh+a[][]; ca h- Aw; ca h Aw; f olh<e e; ouc = olh<e; egl = ; fo=; <=Aue-; ++ egl = egl+a[][]; eu egl; ulc vo goule A[][] Awh = ; = ; oule = ; c = ; f == eu; fo=; <=Awh-; ++ = A[-][];

26 A[-][] = A[][]; A[][] = ; //e of fo = ; whle +<- c = +; f +<- f A[+][]>A[+][] c = +; f A[][]<A[c][] fo=; <=Awh-; ++ = A[][]; A[][] = A[c][]; A[c][] = ; = c; ele e; //e of whle ulc vo coule A[][] Awh = ; = ; oule = ; = ; f == eu; = ; whle!= = -/; f A[][]>A[][] fo=; <=Awh-; ++ = A[][]; A[][] = A[][]; A[][] = ; = ; ele e; ulc oule oule eoule oule oule c=egle; oule =e; oule =./c; Se.ou.l"c="+c+"="++"="+; eu ; ulc vo geleoule oule //ö ol ölge ul vele ııı geşle =; =+.; RY=; oule []=ew oule[]; oule FAOR=.; ; oule ff; f == Se.ou.l"Gş eğele lış vel"; f=f; f=f; fo =;<=RY;++ f ff <. ++;e; ele

27 f=f += FAOR-; // ve e eğele gel ulc oule eoule oule //lee ol ele öü : eöle-ewo Rho eou gele; //ö ıılıı oule cc=.e-; //eol ı f u // üçü oluğu çöüe ulşığııı ul eeceğ oule EPS=.e-; oule h=.; //üev e üülüğü MAI=; //Mu eo ıı ; oule folffhfl; oule e; fl=f; fh=f; //Se.ou.l" ="++"fl="+fl+"="++"fh="+fh; //Bölgee ö eğe vı ool e f fl >. && fh >. fl <. && fh <. Se.ou.l" = "++"="++"="++"vele ııl çe ö eğe o"; f fl ==. eu ; f fh ==. eu ; f fl <. =;=; ele =;=; =.5+; //o o eğe ol=mh.-; =ol; f=f; //foou oı eğe f=fh; //foou üev oı eğe fo =;<=MAI;++ f -f-f-f-f >. Mh..f > Mh.olf //e öle ol=; =.5-; =+; f == && f>eps =+.;eu ; //çöü!!!! ele //ewo-rho ol=; =f/f; e=; -= ; f e == && f<eps =+.;eu ; //çöü!!!!! f Mh. < cc&& f<eps =+.;eu ; //çöü!!!!! f=f; f=fh; f f <. =; ele =; Se.ou.l"Mu eo ıı şılı"; eu.; //og u geleel ulc oule eoule oule oule lhoule oule oule =;oule l=; fo =;<;++ l=e; //Se.ou.l"l = "+l+" = "+; oule ff=fl; oule c; fmh.ff<.e-4 c=l;

28 Se.ou.l"l ="+l+" = "+c+" ="++" ="++"="++"f="+ff+"f="+f.l+"="+l; +=cmh.e-lllhmh.jl; eu ; ulc c vo Sg g[] _l h=ew _l; oule lh; lh=doule.edoulejoope.howiudlog" lh : "; =Doule.eDouleJOoPe.howIuDlog" =h/ / : "; =Doule.eDouleJOoPe.howIuDlog" l çı = : "; =Doule.eDouleJOoPe.howIuDlog" l oo = : "; =Doule.eDouleJOoPe.howIuDlog" = : "; JOoPe.howMegeDlogull" e = "+h.elh "le ğlı ıı fe : "JOoPe.PAI_MESSAGE; PROBEM 4 OMOGEEOUS SEADY SAE SOUIO; ARESIA OORDIAES W W W W W W 3 Seo of vle: Y Y Y

29 Y Y Y W Y co coh h 4 3 Y Bou vlue: co = co coh h = h 3 h W h equo h coeffce. I oe o olve hee coeffce we wll ue oee of ohogol fuco. W 3 h W h 3 PROBEM 5 O-OMOGEEOUS SEADY SAE SOUIO; ARESIA OORDIAES SUPERPOSIIO PRIIPE

30 W W 3 W 4 W W 3 W 4 ew ole c e coue ueoo of hee hoegeou ole 3 W W W W W 3 W 3 3 W 3 W 3 3 W 3 3 Soluo of ech hoegeou ole c e ce ou l o he evou ole

31 h W W h W h 3 3 PROBEM 6 O-OMOGEEOUS SEADY SAE SOUIO; ARESIA OORDIAES EA GEERAIO g W W h equo c e ve o wo equo o efe hoogeeou ole whee W W g Soluo: g g g g g h g g W h Ohogol coo: g g W h

32 g g W h g g h PROBEM 7 O-OMOGEEOUS IME DEPEDE SOUIO; ARESIA OORDIAES Bou coo h = > h = > F = <=<= Bou coo h h = > h h h h = > = > F F = <=<= Bou coo

33 = > = > F F = <=<= Bou coo = = > Soluo: he f of he ole olve ole. PROBEM 8 O-OMOGEEOUS IME DEPEDE SOUIO; ARESIA OORDIAES Bou coo " q = > h = > F = <=<= Bou coo " q = >

34 = > F F = <=<= Bou coo = > = > F F = <=<= Bou coo q " = = > Soluo: h q q " " he f of he ole olve ole. FIIE DIFFEREE SOUIO PROBEM 9 O-OMOGEEOUS IME DEPEDE SOUIO; ARESIA OORDIAES Bou coo h = > h = >

35 F = <=<= Fo h B B egg he equo Fo Fo Fo Fo Fo Fo elc oluo lc oluo A ohe e oluo Bou coo: A A A h A A h A A A h A A h B egg he equo Fo B Fo B Fo B Fo B Jv oluo: o v.o.; cl FDB ulc c oule [] hooule [][]oule [] //3 =.legh; oule f[]=ew oule[]; oule e[]=ew oule[]; oule g[]=ew oule[]; oule []=ew oule[]; fo =;<;++ f[]=[][]; fo =;<-;++ g[]=[][+]; fo =;<-;++ e[+]=[+][]; fo =;<;++ e[]=e[]/f[-]; f[]=f[]-e[]g[-]; fo =;<;++ []=[]-e[][-]; [-]=[-]/f[-]; fo =-;>=;-- []=[]-g[][+]/f[]; eu ; ulc c oule [][] qoule oule oule [][] //he fe =[].legh; =.legh; oule q[][]=ew oule[][-]; fo =;<;++ fo =;<;++ q[][-]=-[][]-[][-]/;

36 eu q; ulc c oule [][] _cooule f oule oule oule oule []oule lhoule Boule Boule foule f // uv lılığı // u // e // = >= ou eeue // = >= ou eeue =.legh; // e ıı //Se.ou.l="+ oule A[][]=ew oule[][]; oule B[]=ew oule[]; oule =/-; // e uuluğu =/+.; oule Fo=lh/; oule [][]=ew oule[+][]; fo =;<;++ [][]=[]; fo =;<;++ //> wll eeue fo =;<;++ f== B[]=.-.Fof-FoBf[][]+.fFo[][+]+.BFof; ele f==- B[]=.-.Fof-FoBf[][]+.fFo[][-]+.BFof; ele B[]=fFo[][-]+.-.Fof[][]+fFo[][+]; f== A[][]=+.FoB.-f+.Fo.-f; A[][+]=-..-fFo; ele f==- A[][]=+.FoB.-f+.Fo.-f; A[][-]=-..-fFo; ele A[][]=.Fo.-f+; A[][-]=-.-fFo; A[][+]=-.-fFo; [+]=hoab; eu ; ulc c vo Sg g[] oule []=ew oule[8]; oule =.8; // = ıclı ofl =.legh; fo =;<;++ []=3-3.Mh.Mh.PI/; oule f=.; //= > ı ç uv ıclığı ıı şı oule f=5.; //= e > ı ç uv ıclığı ıı şı oule h=5.; //= > ı ç uv ıclığı ıı şı oule h=5.; //= e > ı ç uv ıclığı ıı şı oule =6.; // ol oule =.; // geçe ıı oule lo=.3; // he lo elee ı oule lh=8.4e-5; oule =.; oule =; oule f=.5; oule =.; oule =/.legh; oule h=5.; oule B=h/; oule B=h/; oule [][]=_coflhbbff; oule [][]=M.; oule q[][]=q; elo =ew elolo; //elo =ew eloqlo;.;

37 Ml oluo fuco =gu % Gu elo =legh; c=; ol=; fo =: %vog =; uu=; fo =+: u=; f u > uu uu=u;=;e e f ~= fo =: u=; =; =u; e u=; =; =u; e %gu elee fo =+: c=/; =; fo =+: =-c; e = -c; e e %c uug =/; fo =-:-: ol=; fo =+: ol=ol+;e =-ol/; e e fuco [] = _coflhbbff % wll hce % u e % e e % = >= ou eeue % = >= ou eeue =legh; % ue of e =/-; =/; Fo=lh/; fo =: fo =: f == =; ele =.; e e e A=eo; B=eo; fo =: fo =: f== A=+.FoB.-f+.Fo.-f; A+=-..-fFo; elef== A=+.FoB.-f+.Fo.-f;

38 A-=-..-fFo; ele A=.Fo.-f+; A-=-.-fFo; A+=-.-fFo; e f == B=.-.Fof+fFo++Fof; elef == B=.-.Fof-FoBf+.fFo-+.BFof; ele B=fFo-+.-.Fof+fFo+; e e W=guAB; fo =: +=W; e e e PROBEM O-OMOGEEOUS IME IDEPEDE SOUIO; ARESIA OORDIAES WO DIMESIO h = W h = W h3 3 =

39 4 4 h =W If 4 Bou coo: =.... h = = h h =M = =M =.... h =M = = h h = h = =W h = = =W h h =M= = =W h h Jv oluo: o v.o.; o v.; o v.v3.dlilj3d; o v.v.dliljd; o v..reoeeceo; cl _le_fd ulc c oule [] gu_eeloule [][]oule []oule []oule l //ele e çöüü ef gu_eel eou eou //ı föü l le le velş foül =.legh; =;//u eo ıı oule e=.e-; oule u=; oule e=e-; oule u=; oule ol=;

40 e=; eel=; //Se.ou."e = "+e++" "; fo =;<;++ u=[][]; fo =;<;++ [][]/=u; []/=u; fo =;<;++ u=[]; fo =;<;++ f!=u-=[][][]; []=u; //Se.ou.[]+ " "; //Se.ou.l; e++; o eel=; //Se.ou."e = "+e++" "; fo =;<;++ ol=[]; u=[]; fo =;<;++ f!= u-=[][][]; []=lu+.-lol; feel== && []!=. e=mh.[]-ol/[].; fe<e eel=; //Se.ou.[]+ " "; //Se.ou.l; e++; whleeel== && e<; fe>= Se.ou.l"Mu eo ıı şılı ouç geçel oll e = "+e; eu ; ulc c oule[] voluguoule [][]oule [] //ı volu gu elee öe =.legh; oule []=ew oule[]; oule c=; oule ol=; oule uu; oule u=; //gu elee ; //Se.ou.l"Ol M :\"+M.oSg; fo=;<-;++ //vol =; uu=mh.[][]; fo=+;<;++ u=mh.[][]; fu > uu uu=u;=; f!= fo=;<;++ u=[][]; [][]=[][]; [][]=u; u=[]; []=[]; []=u; //Se.ou.l"volu M : ="++"\"+M.oSg; //gu elee çöe fo=+;<;++ c=[][]/[][]; [][]=; fo=+;<;++ [][]-=c[][];

41 [] =[] -c[]; //gee oğu ee o [-]=[-]/[-][-]; fo=-;>=;-- ol=; fo=+;<;++ ol+=[][][]; []=[]-ol/[][]; eu ; ulc c oule [][] _leco oule oule Woule oule oule oule oule oule houle houle houle houle c // Aue ==co; // eeue of 4 e e ow =/+.; =W/+.; Se.ou.l"="++"="+; =; =; =; oule A[][]=ew oule[][]; oule []=ew oule[]; oule []=ew oule[]; oule ; // ou coo ==; fo=;<;++ fo=;<;++ =; =-; =; =W-; =+; []=+/++/W/.; =++; =-+; // f==&&== A[][]+=-h/c+h/c+.;A[][]+=.;A[][+]+=.;[]+=-h/c-h/c; // 4 ele f==&&==- A[][]+=-h/c+h/c+.;A[][]+=.;A[][-]+=.;[]+=-h/ch/c; // ele f==-&&== A[][]+=-h/c+h/c+.;A[][]+=.;A[][+]+=.;[]+=-h/ch/c; // 3 ele f==-&&==- A[][]+=-h/c+h/c+.;A[][]+=.;A[][-]+=.;[]+=-h/ch/c; // 7 ele f== A[][]+=-.h/c+4.;A[][]+=.;A[][]+=.;A[][+]+=.;[]+=-.h/c; // 5 ele f== A[][]+=-.h/c+4.;A[][]+=.;A[][-]+=.;A[][+]+=.;[]+=-.h/c; // 8 ele f==- A[][]+=-.h/c+4.;A[][]+=.;A[][]+=.;A[][-]+=.;[]+=-.h/c; // 6 ele f==- A[][]+=-.h/c+4.;A[][-]+=.;A[][+]+=.;A[][]+=.;[]+=-.h/c; // ç ölge ele A[][]+=-4.;A[][]+=.;A[][]+=.;A[][-]+=.;A[][+]+=.; //e.a; oule [][]=ew oule[][]; //oule []=couge_ge.couge_gea; //oule []=gu_eela.; oule []=volugua; fo=;<;++ fo=;<;++

42 =+; [][]=[]; eu ; ulc c vo Sg g[] how IOEceoReoeEceo VADEceo oule [][]=_leco...; e.; oouplo =ew oouplo...4 fle"""""=f";.lo; Plo3D...4"""""=f"; Ml oluo: fuco [] = _le_whhhhc =/; =W/; =; =; =; A=eo; B=eo; =eo; % ou coo fo =: fo =: =-+; =+; =-+; % f == && ==

43 A=A-h/c+h/c+.; A=A+.; A+=A++.; B=B-h/c-h/c; % 4 elef ==&& == A=A-h/c+h/c+.; A=A+.; A-=A-+.; B=B-h/c-h/c; % elef == && == A=A-h/c+h/c+.; A=A+.; A+=A++.; B=B-h/c-h/c; % 3 elef == && == A=A-h/c+h/c+.; A=A+.; A-=A-+.; B=B-h/c-h/c; % 7 elef== A=A-.h/c+4.; A=A+.; A=A+.; A+=A++.; B=B-.h/c; % 5 elef== A=A-.h/c+4.; A=A+.; A-=A-+.; A+=A++.; B=B-.h/c; % 8 elef== A=A-.h/c+4.; A=A+.; A=A+.; A-=A-+.; B=B-.h/c; % 6 elef== A=A-.h/c+4.; A-=A-+.; A+=A++.; A=A+.; B=B-.h/c; % el ego ele A=A-4.; A=A+.; A=A+.; A-=A-+.; A+=A++.; e e e =eo; B=B:; =lolveab; fo =: fo =: =-+; =; e e e >> []=_le_...; >> uf

44 PROBEM O-OMOGEEOUS IME IDEPEDE SOUIO; ARESIA OORDIAES WO DIMESIO = W = W = 3 =W 4 If 4 Jv og: o v.o.; o v.; o v.v3.dlilj3d; o v.v.dliljd; o v..reoeeceo; cl _le_fd_ ulc c oule [] gu_eeloule [][]oule []oule []oule l //ele e çöüü ef gu_eel eou eou //ı föü l le le velş foül =.legh; =;//u eo ıı oule e=.e-; oule u=; oule e=e-; oule u=;

45 oule ol=; e=; eel=; //Se.ou."e = "+e++" "; fo =;<;++ u=[][]; fo =;<;++ [][]/=u; []/=u; fo =;<;++ u=[]; fo =;<;++ f!=u-=[][][]; []=u; //Se.ou.[]+ " "; //Se.ou.l; e++; o eel=; //Se.ou."e = "+e++" "; fo =;<;++ ol=[]; u=[]; fo =;<;++ f!= u-=[][][]; []=lu+.-lol; feel== && []!=. e=mh.[]-ol/[].; fe<e eel=; //Se.ou.[]+ " "; //Se.ou.l; e++; whleeel== && e<; fe>= Se.ou.l"Mu eo ıı şılı ouç geçel oll e = "+e; eu ; ulc c oule[] voluguoule [][]oule [] //ı volu gu elee öe =.legh; oule []=ew oule[]; oule c=; oule ol=; oule uu; oule u=; //gu elee ; //Se.ou.l"Ol M :\"+M.oSg; fo=;<-;++ //vol =; uu=mh.[][]; fo=+;<;++ u=mh.[][]; fu > uu uu=u;=; f!= fo=;<;++ u=[][]; [][]=[][]; [][]=u; u=[]; []=[]; []=u; //Se.ou.l"volu M : ="++"\"+M.oSg; //gu elee çöe fo=+;<;++ c=[][]/[][]; [][]=; fo=+;<;++

46 [][]-=c[][]; [] =[] -c[]; //c uuo [-]=[-]/[-][-]; fo=-;>=;-- ol=; fo=+;<;++ ol+=[][][]; []=[]-ol/[][]; eu ; ulc c oule [][] _leco oule oule Woule oule oule oule oule =/+.; =W/+.; =-; =-; =; oule A[][]=ew oule[][]; oule []=ew oule[]; oule []=ew oule[]; oule ; // ou coo ==; fo=;<;++ fo=;<;++ =; =-; =; =W-; =+; []=+/++/W/.; =++; =-+; f== []+=-; f== - []+=-; f== []+=-; ele f==- []+=-; A[][]+=-4.; f==&&== A[][]+=.;A[][+]+=.; ele f==&&==- A[][]+=.;A[][-]+=.; ele f==-&&== A[][]+=.;A[][+]+=.; ele f==-&&==- A[][]+=.;A[][-]+=.; ele f== A[][]+=.;A[][]+=.;A[][+]+=.; ele f== A[][+]+=.;A[][-]+=.;A[][]+=.; ele f==- A[][]+=.;A[][]+=.;A[][-]+=.; ele f==- A[][]+=.;A[][-]+=.;A[][+]+=.; ele A[][]+=.;A[][]+=.;A[][-]+=.;A[][+]+=.; oule [][]=ew oule[+][+]; oule []=couge_ge.couge_gea; fo=;<=;++ fo=;<=;++ f== [][]=; ele f== [][]=; ele f== [][]=; ele f== [][]=; ele =-+-; [][]=[]; eu ; ulc c vo Sg g[] how IOEceoReoeEceo VADEceo

47 oule [][]=_leco ; e.; oouplo =ew oouplo...4 fle"""""=f";.lo; Plo3D...4"""""=f"; ulc c vo Sg g[] how IOEceoReoeEceo VADEceo oule [][]=_leco...; e.; oouplo =ew oouplo...4 fle"""""=f";.lo; Plo3D...4"""""=f"; ulc c vo Sg g[] how IOEceoReoeEceo VADEceo oule [][]=_leco...6..; oouplo =ew oouplo...4 fle"""""=f";.lo; Plo3D...4"""""=f";

48 PROBEM O-OMOGEEOUS IME IDEPEDE SOUIO; YIDRIA OORDIAES OE DIMESIO WI EA GEERAIO M g V R R whee g he geeo V volüe of elee. R / R / V =he legh of he cle. Equo c e ge : g Bou coo: A he cee of he cle: g V R R /

49 V Afe he gee g 4 Seco ou coo covecve ou coo M M A Ah M og h equo wh evou fe ffeece foul : M gve g M M g M MM M h M M h M M Jv Soluo: o v.o.; cl _cle_fd_g ulc c oule [] hooule [][]oule [] //3 =.legh; oule f[]=ew oule[]; oule e[]=ew oule[]; oule g[]=ew oule[]; oule []=ew oule[]; fo =;<;++ f[]=[][]; fo =;<-;++ g[]=[][+]; fo =;<-;++ e[+]=[+][]; fo =;<;++ e[]=e[]/f[-]; f[]=f[]-e[]g[-]; fo =;<;++ []=[]-e[][-]; [-]=[-]/f[-]; fo =-;>=;-- []=[]-g[][+]/f[]; eu ; ulc c oule [] _cle oule oule g[]oule oule oule houle // : cle u //g : he geeşo e u volue M=/; =M+; oule A[][]=ew oule[][]; oule B[]=ew oule[]; oule []=ew oule[]; oule ; // ou coo

50 =; oule GM=eM=gM=; fo=;<=m;++ GM=g[]/; em=++./.h/; gm=+./.h/; f== B[]+=-g[]/; ele f==m B[]+=-.gM-GM; ele B[]+=-g[]/; f== A[][]+=-4.;A[][+]+=4.; ele f==m A[][-]=.;A[][]+=-eM; ele A[][-]+=.-/.;A[][]+=-.;A[][+]+=.+/.; e.a"a"; e.b"b"; //oule []=couge_ge.couge_geab; oule []=hoab; eu ; ulc c vo Sg g[] M=5; oule =.e-; oule =/M; oule g[]=ew oule[m+]; oule []=ew oule[m+]; fo =;<M+;++ g[]=.e6;//w/3 []=; oule =5.;//egee oule h=5.;//w/k oule =38.;//W/K oule []=_cle gh; //coe eul wh lcl ce wh q=co. oule []=ew oule[.legh]; fo =;<M+;++ []=g[]/4..-//+g[]/.h+; oule [][]=; e.; Plo =ew Plo;.D;.ePlel"he fe wh he geeo";.eylel" "" egee c";.lo; Ghc ouu: Soluo lcl oluo o coe he eul he equo: M=5

51 If he e ole olve M= ce wll ecoe : A - ee ol eel of hel coucv =4 W/K hee eleccll he ge of eleccl cue whch heee eeg wh he o e of g=4 6 W/ 3. e e fo he ufce of he o coveco wh he fe coeffce h=4 W/ K o e eeue =. B vg he Ru o 5 equl evelo olve he fe ffeece equo. M=5 vo ulc c vo Sg g[] M=5; oule =5.e-; oule =/M; oule g[]=ew oule[m+]; oule []=ew oule[m+]; fo =;<M+;++ g[]=4.e6;//w/3 []=; oule =.;//egee oule h=4.;//w/k oule =4.;//W/K oule []=_cle gh; //coe eul wh lcl ce wh q=co. oule []=ew oule[.legh]; fo =;<M+;++ []=g[]/4..-//+g[]/.h+; oule [][]=; e.; Plo =ew Plo;.D;.ePlel"he fe wh he geeo";.eylel" "" egee c";.lo;

52 M= vo PROBEM 3 O-OMOGEEOUS IME IDEPEDE SOUIO; YIDRIA OORDIAES OE DIMESIO WI EA GEERAIO A log clcl fuel ele of Ru = - hel coucv =5 W/K geee eeg co e of g=5 8 W/ 3. he ou ufce = c e ue o e e. Aug oe-eol l he flow clcule he l eeue uo he fuel elee fe ffeece eho. Pole l o he evou ole ece h eeue ou gve he oue. heefoe he Bou coo e:

53 M ow So he ou coo wll ecoe: fo =M- Jv Soluo: o v.o.; cl _cle_fd_g g ulc c oule[] voluguoule [][]oule [] //ı volu gu elee öe =.legh; oule []=ew oule[]; oule c=; oule ol=; oule uu; oule u=; //gu elee ; //Se.ou.l"Ol M :\"+M.oSg; fo=;<-;++ //vol =; uu=mh.[][]; fo=+;<;++ u=mh.[][]; fu > uu uu=u;=; f!= fo=;<;++ u=[][]; [][]=[][]; [][]=u; u=[]; []=[]; []=u; //Se.ou.l"volu M : ="++"\"+M.oSg; //gu elee çöe fo=+;<;++ c=[][]/[][]; [][]=; fo=+;<;++ [][]-=c[][]; [] =[] -c[]; //c uuo [-]=[-]/[-][-]; fo=-;>=;-- ol=; fo=+;<;++ ol+=[][][]; []=[]-ol/[][]; eu ; ulc c oule [] _cle oule oule g[]oule oule oule // : cle u //g : he geeşo e u volue M=/+.; oule A[][]=ew oule[m][m]; oule B[]=ew oule[m]; oule []=ew oule[m+]; oule ; // ou coo M

54 =; fo=;<m;++ f== B[]+=-g[]/; ele f==m- B[]+=-g[]/-+/.; ele B[]+=-g[]/; f== A[][]+=-4.;A[][+]+=4.; ele f==m- A[][-]+=.-/.;A[][]+=-.; ele A[][-]+=.-/.;A[][]+=-.;A[][+]+=.+/.; e.a"a"; e.b"b"; //oule []=couge_ge.couge_geab; oule []=voluguab; eu ; ulc c vo Sg g[] M=; oule =.e-; oule =/M; oule g[]=ew oule[m+]; oule []=ew oule[m+]; fo =;<=M;++ g[]=5.e8;//w/3 []=; oule =.;//egee oule =5.;//W/K Se.ou.l"="++"gg="+g[]/; oule []=_cle g; oule []=ew oule[.legh+]; //ec oluo oule []=ew oule[.legh+]; [M]=; [M]=; fo =;<M;++ []=[]; []=g[]/4..-[]/[]/+; oule [][]=; e.; Plo =ew Plo;.D;.ePlel"he fe wh he geeo";.eylel" "" egee c";.lo;

55 FIIE EEME MEODS B ug le o o-le le elee ufce cuve fg c e cheve. he le geoec he h c e eloe o oe egul ufce he gle oe of he oul elee cuel ue fe elee clculo. he wo-eol le gul elee lo ow le elee eeee 3 whee he olol le co hee coeffce. Sce le gle h hee oe he vlue of α α α 3 e eee fo = = = whch eul he followg: A A A 3 whee A he e of he gle gve e A Suug he vlue of o he equo collg he coeffce of we ge c A

56 c A c A c c c If we evlue oe whee he cooe e he we o = ll = = hu we ee h he he fuco hve vlue of u he ege vee eo ll ohe vece. I ole o how h + + = evewhee he elee clug he oue. e le gve u elvel le w of eolg o he ufce le eolo. So he eeue ofle of he le elee c e clcule : ] [ ] [ he evve of he eeue fel c e gve : ] [ B c c c A g heefoe he fe c e gve : c c c A B q q ] [ Iego of he fe elee c e ce ou : l c c l c l!!!! A c c A c A!!!! ow geel e e geel he couco equo c e vege covee o egl equo g Bou coo = o ufce S q l o ufce S h l o ufce S 3 ow h gl ego eho wll e le o ou ffeeo equo. he eho eque h he followg eeo wll e fe w whee he volue of he elee.

57 g oe h fo fe elee =[] egl c e covee l ego ule vu uv uv o S S B lg hee equo ecoe S g Fo ou coo: S h S h qs S h q S S h S h qs g If h equo we fo: ] [ f K whee S S h K] [ S S h qs g f If equo u o he fo: S S h B D B K ] [ ] [ ] ][ [ ] [ ] [ whee D ] [ S S h qs g f ] [ ] [ ] [ l l l!!! 6!!! l l l l 6!!! l l l l M e c e oee fo wo eol gul elee : ] [ hl hl hl c c c c c c c c c c c c c c c A K 3 l h l h l h ql ql ql ga f PROBEM 4 O-OMOGEEOUS SEADY SAE SOUIO; ARESİA OORDIAES FIIE EEME SOUIO

58 A que le wh eeue of hee e 5 he fouh e gve. Ple e e gve. lcule he eeue ofle Jv og: ulc cl gul_elee ulc oule ; //hce of wo eol elee ulc oule [][]c[]; ulc oule [][];//cooe of o ulc oule cc; //cee o ulc oule A; //Ae ulc oule []B[][]; //She fco evve ulc oule []; //eeue oe ulc _o[]; //o ue ulc _elee; //elee ue ulc oule l[]; ulc oole cov[]; ulc oole heflu[]; ulc oole hege; ulc oule h[]; ulc oule f[]; ulc oule qflu[]; ulc oule qge; ulc gul_elee _elee _o[]oule []oule [] =ew oule[3]; =ew oule[3]; =ew oule[3]; l=ew oule[3];//:l:l:l cov=ew oole[3]; heflu=ew oole[3]; _o=ew [3]; c=;c=; hege=fle; h=ew oule[3]; f=ew oule[3]; qflu=ew oule[3]; _elee=_elee; fo =;<3;++ _o[]=_o[]; []=[];[]=[]; c+=[];c+=[]; cov[]=fle; heflu[]=fle; h[]=.; f[]=.; qflu[]=.; qge=.; c/=3.; c/=3.; =ew oule[3]; =ew oule[3]; c=ew oule[3]; =ew oule[3]; B=ew oule[][3]; []=[][]-[][]; []=[][]-[][]; []=[][]-[][]; []=[]-[]; []=[]-[];

59 []=[]-[]; c[]=[]-[]; c[]=[]-[]; c[]=[]-[]; A=.5[][]-[][]+[][]-[][]+[][]-[][]; fo =;<3;++ B[][]=./.A[]; B[][]=./.Ac[]; l[]=mh.q[][]+c[]c[]; l[]=mh.q[][]+c[]c[]; l[]=mh.q[][]+c[]c[]; ulc oule[][] Koule oule [] =; oule =/4.A[]; oule =/4.A[]; oule =h[]l[]/6.; oule =h[]l[]/6.; oule =h[]l[]/6.; oule K[][]=[][]+c[]c[]++. [][]+c[]c[]+ [][]+c[]c[]+ [][]+c[]c[]+ [][]+c[]c[]++ [][]+c[]c[]+ [][]+c[]c[]+ [][]+c[]c[]+ [][]+c[]c[]++ ; eu K; ulc oule[] foule oule [] =; oule =qgea/3.; oule =-qflu[]l[]/.; oule =-qflu[]l[]/.; oule =-qflu[]l[]/.; oule c= h[] f[]l[]/.; oule c= h[] f[]l[]/.; oule c= h[] f[]l[]/.; oule f[]=+++c+c+++c+c+++c+c; eu f; ulc ei cooule oule oule Woule W =; fco== //= f[]== && []== =; ele f[]== && []== =; ele f[]== && []== =; ele =-; ele fco== //= f[]== && []== =; ele f[]== && []== =; ele f[]== && []== =; ele =-; ele fco== //= f[]==w && []==W =; ele f[]==w && []==W =; ele f[]==w && []==W =; ele =-; ele fco==3 //=W f[]==w && []==W =; ele f[]==w && []==W =; ele f[]==w && []==W =; ele =-; eu ;

60 ulc vo eoveco oule h oule f f>= && <= h[]=h;f[]=f;cov[]=ue; ulc vo eeflu oule q f>= && <= qflu[]=q;heflu[]=ue; ulc vo eegeeooule qge qge=qge;hege=ue; ulc vo eoule oule fo =;<3;++ []=./.A[]+[]+c[]; ulc oule[] qoule oule q[]=ew oule[]; fo =;<3;++ q[]+=-b[][][]; q[]+=-b[][][]; eu q; ulc oule[] qoule oule [] //he fe e; oule q[]=ew oule[]; fo =;<3;++ q[]+=-b[][][]; q[]+=-b[][][]; eu q; ulc vo eoule [] e[][]; ulc vo eoule [] fo =;<3;++ []=[]; ulc oule oule oule //eolo oule =; e; fo =;<3;++ +=[][]; eu ; ulc oule oule []oule oule //e g o eeue ge eole vlue e; e; oule =; fo =;<3;++ +=[][]; eu ; ulc oule oule []oule [] eu [][]; ulc c vo Sg g[] _elee=; _o[]=; oule []=.4; oule []=.5; gul_elee fe=ew gul_elee_elee_o; oule []=..; fe.e; oule []=57; fe.e; oule =fe.[][]; oule q[]=fe.q.;

61 ulc cl elee_ue ue; ; ulc elee_ue ue ue=ue; f<= =; ele =;Se.ou.l"wog vlue"; ulc oole equlelee_ue e fe.ue==ue && e.== eu ue; ele eu fle; ulc geue eu ue; ulc gei eu ; ulc [] ouu []=ue; eu ; o v.ul.; o v.w.; o v.wg.; o v.w.eve.; ulc cl eelee A<elee_ue> e; o[]; []; ulc eelee e=ew A<elee_ue>; ulc vo elee_ue e=ew elee_ue; e; ulc vo elee_ue // =e.e; oole =fle; elee_ue ; Ieo =e.eo; whle.he&&! =elee_ue.e; =.equl; f! e.; ulc [][] ouu =e.e; [][]=ew [][]; Ieo =e.eo; =; whle.he elee_ue el=elee_ue.e;[]=el.ouu;++; eu ; ulc c vo Sg g[] eelee e=ew eelee; elee_ue e=ew elee_ue; elee_ue e=ew elee_ue; elee_ue e3=ew elee_ue3; elee_ue e4=ew elee_ue3; e.e; e.e; e.e3; e.e4; [][]=e.ouu; e.;

62 o v.wg.; o og.h.lo.; o c v.lg.mh.; o c og.h..doulea.; o v.; o v.v3.dlilj3d; o v.v.dliljd; o v..reoeeceo; o v.w.; o v.wg.; o v.w.eve.; ulc cl gul_g ulc gul_elee e[]; ulc eelee e[]; ulc MKKKM; ulc oule [][]; ulc oule [][]; ulc oule [][]; ulc oule []; ulc oule []; ulc oule []; ulc oole []; ulc oule ; ulc B[]BM[]B[]B[]; ulc oule W; ulc oule oule [][] oule =e; fo =;<.legh;++ fo =;<[].legh;++ f[][]< =[][]; eu ; ulc oule oule [][] oule =-e; fo =;<.legh;++ fo =;<[].legh;++ f[][]< =[][]; eu ; ulc gul_goule [][]oule [][] =; =; =.legh-; M=[].legh-; =ew oule[+][m+]; =; W=; eelee; ulc vo geeegulgoule oule W M ; =;M=M; =; W=W; =ew oule[+][m+]; =ew oule[+][m+]; =ew oule[+][m+]; eelee; ulc vo eelee ; =/M; =W/; K=M; K=K; K=K+; //g o

63 =ew oule[k]; =ew oule[k]; =ew oule[k]; =ew oole[k]; e=ew eelee[k]; fo=;<k;++ e[]=ew eelee; e=ew gul_elee[k]; =-===; fo=;<=;++ fo=;<=m;++ =++; [][]=;[][]=; []=[][]; []=[][]; []=fle; []=.; fo=;<;++ fo=;<m;++ =++; =+++; = +; oule []=[][][][+][+][]; oule []=[][][][+][+][]; ++; _o[]=+; e[].; e[+].; e[].; e[]=ew gul_elee_o; oule []=[][+][+][+][+][]; oule []=[][+][+][+][+][]; ++; _o[]=+; e[+].; e[].; e[].; e[]=ew gul_elee_o; ulc gul_goule oule W M geeegulgwm; ulc f_eleeoule [] eu f_elee[][]; ulc f_eleeoule oule oule R=Mh.q+; =; oule =.e5; fo =;<K;++ e[].e; oule =Mh.qe[].[]-e[].[]-+e[].[]-e[].[]-; oule =Mh.qe[].[]-e[].[]-+e[].[]-e[].[]-; oule =Mh.qe[].[]-e[].[]-+e[].[]-e[].[]-; oule =Mh.q++; oole =>=e[].[] && <=e[].[]; oole =>=e[].[] && <=e[].[]; oole = && ; f< =;=; eu ; ulc vo eoveco cooule oule oule Woule W oule h oule f //co : == //co : == //co : ==W //co 3: ==W =; fo =;<K;++ =e[].eicoww; e[].eovecohf;

64 ulc vo eeflu cooule oule oule Woule W oule q //co : == //co : == //co : ==W //co 3: ==W =-; fo =;<K;++ =e[].eicoww; e[].eefluq; ulc vo eeeue cooule oule oule Woule W oule //co : == //co : == //co : ==W //co 3: ==W fo =;<K;++ fco== f[]== []=;[]=ue; ele fco== f[]== []=;[]=ue; ele fco== f[]==w []=;[]=ue; ele fco==3 f[]==w []=;[]=ue; ulc vo eegeeo oe[]oule qge =oe.legh; fo =;<;++ e[oe[]].eegeeoqge; ulc vo olveoule oule [] oule K[][]=ew oule[k][k]; oule KK[][]=ew oule[k][k]; oule ff[]=ew oule[k]; oule K[][]=ew oule[3][3]; oule f[]=ew oule[k]; oule f[]=ew oule[3]; fo =;<K;++ K=e[].K; f=e[].f; =e[]._elee; []=e[]._o; fo =;<3;++ fo =;<3;++ K[[]][[]]+=K[][];KK[[]][[]]=K[[]][[]]; f[[]]+=f[]; ff[[]]=f[[]]; //eeg K f veco f ow //eeue ou gve // =; fo =;<K;++ fo =;<K;++ f[] f== f[]=[];k[][]=.; ele K[][]=.; //Se.ou.l"="++"="++"="+[]+"="+[]+"f="+f[]; //e of fo //e of fo //Se.ou.l"K =\"+M.oSgK; //Se.ou.l"f =\"+M.oSgf; oule []=ABKf; =; fo =;<=;++ fo =;<=M;++

65 =++; [][]=[]; //U ecooo eho ulc c oule[][] Uoule c[][] [] [] //eu U ecooo of c e oule [][]; =c.legh; =ew oule[][]; oule vv[]; vv=ew oule[]; oule uuge; ; ; =; oule =.e-4; =; fo=;<=;++ fo=;<=;++ [-][-]=c[-][-]; []=; fo=;<=;++ g=.; fo=;<=;++ fmh.[-][-]>g g=mh.[-][-]; fg== Se.ou.l"gul ";eu ; vv[-]=./g; fo=;<=;++ fo=;<;++ u=[-][-]; fo=;<;++ u-=[-][-][-][-]; [-][-]=u; g=; fo=;<=;++ u=[-][-]; fo=;<;++ u-=[-][-][-][-]; [-][-]=u; u=vv[-]mh.u; fu>=g =; g=u; //e of = f!= fo=;<=;++ u=[-][-]; [-][-]=[-][-]; [-][-]=u; []=-[]; vv[-]=vv[-]; //e of f [-]=;

66 f[-][-]== [-][-]=; f!= u=./[-][-]; fo=+;<=;++ [-][-]=u; //ef //e fo = eu ; ulc c oule[] Uoule [][]oule [] [] //olve A=B e of le equo of U ecooe //clcule eho U =; ll=; oule u=; =.legh; oule []; =ew oule[]; fo=;<=;++ [-]=[-]; fo=;<=;++ ll=[-]; u=[ll-]; [ll-]=[-]; f!= fo=;<=-;++ u-=[-][-][-]; ele fu!= =; [-]=u; fo=;>=;-- u=[-]; f< fo=+;<=;++ u-=[-][-][-]; [-]=u/[-][-]; eu ; ulc vo oouplo oouplo =ew oouplo 6ue"Fe elee""""";.lo; ulc vo Plo3Doule oule oule Woule W how ReoeEceo VADEceo Plo3D.lo3D_DWW""""" egee "; ulc vo GPlooule oule oule Woule W Plo3DPel lo = ew Plo3DPel"SOU"; oule =-/M; oule =W-W/; oule[] = cee ; // =.:.:. oule[] = ceeww; lo.gplo"fe elee G" ; JFe fe = ew JFe" lo el"; fe.ese6 6; fe.eoepelo; fe.evleue; ulc c oule[] ABoule [][]oule []

67 //Soluo of e of le equo U eho // oe h he e clculo c e oe ve eho. =.legh; oule c[]=ew oule[]; []=; []=ew []; oule e[][]=ew oule[][]; e=u; c=ue; eu c; //e of AB ulc c vo Sg g[] how ReoeEceo VADEceo oule =.; oule W=.; =6M=6; oule h=.; gul_g g=ew gul_gwm; co=;//== co=;//== co=;//== co3=3;//==w oule =.; oule =.; oule =.; oule 3=5.; g.eeeueco..w; g.eeeueco..w; g.eeeueco..w; g.eeeueco3..w3; //g.eovecoco3..wh3; oule []=55; g.olve; e.g.; g.oouplo; g.plo3d..w; g.gplo..w;

68 PROBEM 5 O-OMOGEEOUS IME IDEPEDE SOUIO; ARESİA OORDIAES FIIE EEME SOUIO A eco ole: I h oe hee e eeue ecfe he h e covecve he fe ee. ulc c vo Sg g[] how ReoeEceo VADEceo oule =.; oule W=.; =6M=6; oule h=.; //w/k gul_g g=ew gul_gwm; co=;//== co=;//== co=;//== co3=3;//==w oule =.; //egee oule =.; //egee oule =.; //egee oule 3=5.; //egee g.eeeueco..w; g.eeeueco..w; g.eeeueco..w; g.eovecoco3..wh3; oule []=55; // W/K g.olve; e.g.; g.oouplo; g.plo3d..w; g.gplo..w;

69 he e ole olve cool ulhc fe elee cge: PROBEM 6 O-OMOGEEOUS IME IDEPEDE SOUIO; ARESIA OORDIAES FIIE EEME SOUIO A h ole: I h oeoee eeue ecfe eco h e covecve he fe ee. I he fouh e he flu gve ulc c vo Sg g[] how ReoeEceo VADEceo oule =.; oule W=.; =6M=6; oule h=.; gul_g g=ew gul_gwm; co=;//== co=;//== co=;//== co3=3;//==w oule =.; //egee

70 oule =.; //egee oule =.; //egee oule q=-5.; //W/ g.eeeueco..w; g.eovecoco..wh; g.eovecoco..wh; g.eefluco3..wq; oule []=55; g.olve; e.g.; g.oouplo; g.plo3d..w; g.gplo..w; ow e eee geel he couco equo c e vege fe elee equo g Bou coo = o ufce S l q > o ufce S

71 h l > o ufce S 3 = o = Gl ego eho wll e le o ou ffeel equo. he eho eque h he followg eeo wll e fe w whee he volue of he elee. g oe h fo fe elee =[] ] [ egl c e covee l ego ule vu uv uv o S S B lg hee equo ecoe S g Fo ou coo: S h S h qs S h q S S h qs g S h If h equo we fo : ] [ ] [ f K whee ] [ S S h K] [ S S h qs g f ] [ ] [ ] [ S S h B D B K ] [ ] [ ] ][ [ ] [ ] [ whee

72 D ] [ S S h qs g f ] [ ] [ ] [ l l l!!! 6!!! l l l l 6!!! l l l l A c c A c A!!!!!!! A A A A!!! A A A A he c equo ll ffeel equo e heefoe houl e olve ug fe ffeece eho. ] [ ] [ ] [ f K K whee lc elc hee. If I full elc chee If full lc chee. 5. clle -colo eho. oe h oee hg q c e chge ech e fuco of eeue u ech gle e clculo he c e coee co. Whe equo ge fo + e e uow eeue ecoe ] [ ] [ ] [ ] [ f K K M e c e oee fo wo eol gul elee : ] [ A ] [ hl hl hl c c c c c c c c c c c c c c c A K 3 l h l h l h ql ql ql ga f Il eeue ofle wll e efe loe o he og he M wll e olve eeue ofle fo ech e e wll e clcule. AAYI SOUIO MEODS: DUAME S EOREM oe he hee eol ohoogeeou he couco ole ego R wh e eee ou coo fuco he geeo he fo g ego R > f h o ou S F fo = h ole c o e olve wh eeo of vle eho ece evou ole ecue of he ohoegeeou e g f. heefoe e of olvg h ole ecl ole wll e eee e of oluo of le ul ole follow: e e oluo of he ove oleo he uo h he ohoogeeou e g f o o ee o e; el he vle eel ee u o e vle. he he oluo of he followg ul ole g ego R >

73 f h o ou S F fo = h oe c e olve ug eho uch eeo of vle. he Duhel heoe ele he oluo he followg gl equo h eul c lo e eee he leve fo efog he ffeel F If he e fuco h F fuco wh coue uch : Equo ecoe: f Afe hee l efo ow A he fe ole c e vege A l of hce ll eo eeue. Fo e > ou ufce = e eo eeue whle he ufce = uece o e vg eeue f efe fo fo F f equo wh ou coo: = = > = f = > = = Becoe = = > = = > = = Pole ve o wo Whee = = = =

74 = = = = = = = > = = > = - = Seeo of vle: Bou coo = > = > co 3 e co e Alco of ou coo: co = = = e = - = fo oogol coo

75 co co co co u v u v v vu uv uv co co co 4 co e e he fuco oe elcg equo e Duhel heoe ow c e le o olve he ole. Reee h fuco o e coee w fo fo F f Fo ou coo = f = > he oluo fo e f fo ecfc fuco fo fo F f e he ego efoe o gve

76 e 3 he oluo fo e f Jv oluo fo h ole gve follow: o v.ul.; cl fc ee f_ ulc oule fucoule oule ff=; f== f<=ff=.; f=.; ele eu ff; ff=.; ulc cl _uhel f ffo ecfc fuco //ego fuco ulc c oule foule ooule oule oule oule lh oule f=/; oule =; luu=-; fo =;<;++ =Mh.PI/; f+=./mh.e-lh-oluu/mh.; luu=-; //oule =/Mh.q4.Mh.PIlhfo/Mh.ow-o.5Mh.e-/4.lh-o; eu f; // lc oluo fuco ulc c oule foule oule oule oule lh oule f=/; oule =; luu=-; fo =;<;++ =Mh.PI/; f+=4./luu/lh.-mh.e-lhmh.; luu=-; //oule =/Mh.q4.Mh.PIlhfo/Mh.ow-o.5Mh.e-/4.lh-o; eu f; //egl //Gu Koo ego foul ulc c oule eglf_ f oule oule oule oule oule oule lh //e h =5; oule e=e-; Aue=; oole ouc; oule egl=; oule A[][]; Aw = ; oule olh = ; oule[] c; oule[] g; oule[] ; = ; g = ; = ;

77 = ; h = ; oule v = ; oule = ; oule = ; oule g = ; oule = ; oule = ; oule = ; Aw = 4; A = ew oule[-+][aw-+]; = 6; g = 5; c = ew oule[]; = ew oule[]; g = ew oule[]; //Gu l g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; g[5] = ; g[6] = ; g[7] = ; g[8] = ; g[9] = ; g[] = ; g[] = ; g[] = ; g[3] = ; g[4] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[] = ; c[] = ; c[] = ; c[3] = ; c[4] = ; c[5] = ; c[6] = ; c[7] = ; c[8] = ; c[9] = ; c[3] =.; //Koo l [] = ; [] = ; [] = ; [3] = ; [4] = ; [5] = ; [6] = ; [7] = ; [8] = ; [9] = ; [] = ; [] = ;