MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng s based on the observaton that for small changes of tme h, the change n the temperature s nearly proportonal to h and the dfference between room temperature and the present temperature of the cup Let u be the temperature of the room, u(t be the temperature dependng on tme t, then u (t = C ( u(t u Dscretzng the tme wth t k = k and u k = u(t k and usng u (t k u k+ u k, we obtan the dscrete form u k+ u k = C(u u k u k+ = ( Cu k + C u So the general (dscrete lnear model s ( u k+ = au k + b, whch s called a frst order fnte dfference for the sequence u k To obtan u k, we use telescopng technque and the geometrc summaton + r + r + + r k = rk+ r : ( u k+ = a(au k + b + b = a u k + ab + b = a (au k + b + ab + b = a 3 u k + a b + ab + b = a k+ u 0 + b ak+ a = a k+( u 0 b + b a a Theorem (Steady State Theorem If a, then the soluton of ( s gven by ( If a < then the soluton of ( wll converge to the steady state soluton u = au + b, e, u = b/( a Example Consder a cup of coffee ntally at 00 degree (F and s n a room wth temperature 70 degree After mnute, t cools to 90 degree Predct the temperature of the cup every mnutes Example 3 Consder the setup of prevous example Suppose at every mnute, the temperature of the cup s ncreased for an addtonal degree Predct the temperature every mnutes Error Analyss: Computaton on a computer s done wth floatng pont numbers, thus, there s a roundoff error R k+ at each step Let U k = float(u k, we can wrte U k+ = au k + b + R k+ So U k+ u k+ = a(u k u k + R k+ = a[a(u k u k + R k ] + R k+ = a k+ (U 0 u 0 + a k R + + R k+
Let r = a and let R be the unform bound on R k, then U k+ u k+ r k+ U 0 u 0 + R rk+ r Theorem 4 If a < and the roundoff errors are unformly bounded by R, then the accumulaton error s unformly bounded Heat Dffuson n a Wre Consder a thn electrcal wre that s thermally nsulated on ts surface So the most sgnfcant dffuson s n one drecton x Let u(x, t be the temperature at tme t at the pont x n the wre We derve the heat equaton for u from two physcal laws: The amount of heat energy requred to rase the temperature of a body by du degrees s sm du, where m s the mass of the body and s s a postve physcal constant determned by ts materal, whch s called the specfc heat of the body Fourer heat law: Heat flows from hot to cold The rate at whch heat energy crosses a surface s proportonal to the surface area and to the temperature gradent at the surface The proportonalty constant s called the thermal conductvty and s denoted by κ hot cold So consder a pece of wre of cross-sectonal area A and mass densty ρ We assume the temperature decreases from left to rght The temperature gradents at the rght x x + dx end and the left end are u x(x + x, t and u x(x, t, respectvely So the rate at whch heat energy crosses the rght end and left end are κau x(x + x, t and κau x(x, t because u x(x, t < 0 Snce heat flows from hot to cold regons, heat energy enters the pece through the left end and exts through the rght end So n an nfntesmal tme, the net amount of heat enterng the pece s κau x(x +, t κau x(x, t In addton, suppose the wre has a current gong through t so that there s a source of heat from the electrcal resstance of the wre So we assume that the heat s proportonal to the volume and the length of tme the current travels through the wre Thus, the heat receved s Af x where f s some constant In the same tme nterval, the temperature of ths pece changes by u t(x, t The mass of the pece s ρa x, so the heat energy requred to ncrease the temperature by u t(x, t s sρa xu t(x, t By the energy conservaton law, we have sρa xu t(x, t = Af x + κau x(x + x, t κau x(x, t So sρu t(x, t = f + κ [ u x x (x + x, t u x(x, t ] Lettng x 0, we have the heat equaton (3 u t(x, t = f sρ + κ sρ u xx(x, t
To dscretze the equaton, we observe a wre wth length L Take > 0, defne h = L n, x = h, t k = k, u k = u(x, t k, and use the approxmatons Then (3 mples u k+ u k u k+ u t(x, t k uk+ u k u xx(x, t k uk + + uk uk h = f sρ + κ u k + + uk uk sρ = f sρ h + κ sρh (u k + + u k + ( κ sρh u k =: ( αu k + α(u k + + u k + β Boundary condtons: suppose ntal temperature u 0 for = 0,, n and temperatures at two ends (u k 0, uk n for k N are gven, defne the matrces u k α α 0 0 β + αu k 0 u k u k α α α β, A = 0 α, b k = u k n α β β + αu 0 α α k n Then we have u k+ = Au k + b k If u k 0 and uk n are constant for all k, then b k = b s also a constant Theorem 5 (steady state theorem Consder the frst order fnte dfference equaton u k+ = Au k + b where A s a square matrx If A k converges to the zero matrx and u = Au + b, then for every ntal choce u 0, the sequence u k converges to u Proof We have u k+ u = (Au k + b (Au + b = A(y k u = A (u k u = A k+ (u 0 u 0 Dffuson n a Wre wth Lttle Insulaton We consder heat dffuson n a thn electrcal wre that s not thermally nsulated on ts lateral surface Let u be the surroundng temperature The heat loss through lateral surface s assume to be proportonal to the change n tme, the lateral surface area and the temperature dfference, e, cπr x[u(x, t u], where r s the cross-secton radus So energy conservaton law mples sρa xu t(x, t = Af x + κau x(x + x, t κau x(x, t cπr x[u(x, t u] Notce that A = πr, so u t(x, t = f sρ + κ ( u sρ x x(x + x, t u x(x, t c [u(x, t u] sρr 3
Lettng x 0, we have (4 u t(x, t = ( f sρ + cu + κ sρr sρ u xx(x, t c u(x, t sρr To dscretze the equaton, we observe a wre wth length L Take > 0, defne h = L n, x = h, t k = k, u k = u(x, t k, and use the approxmatons Then (4 mples u k+ u t(x, t k uk+ u k u k ( f = sρ + cu sρr = ( sρ f + cu r u k+, u xx(x, t k uk + + uk uk h + κ u k + + uk uk sρ h c sρr uk + κ (u k sρh + + u k + ( κ c sρh sρr =: ( α du k + α(u k + + u k + β Boundary condtons: suppose ntal temperature u 0 for = 0,, n and temperatures at two ends (u k 0, uk n for k N are gven, defne the matrces u k α d α 0 0 β + αu k 0 u k u k α α d α β, A = 0 α, b k = u k n α β β + αu 0 α α d k n Then we have u k+ = Au k + b k If u k 0 and uk n are constant for all k, then b k = b s also a constant u k 3 Flow and Decay of a Pollutant n a Stream Consder a rver that has been polluted upstream The concentraton wll decay and dsperse downstream Assume the stream s movng from left to rght wth velocty v > 0 Let A be the cross sectonal area of the stream Let u(x, t be the concentraton at poston x at tme t change n amount n tme (amount enterng from upstream n tme (amount leavng to downstream n tme (amount decayng n tme Consder the volume A x at poston x, n an nfntesmal tme So The amount of pollutant enterng the left sde s The amount of pollutant leavng the rght sde s The amount of pollutant n the volume A x at tme t s The amount of pollutant decayed (wth rate r s A v u(x, t, A v u(x + x, t, A x u(x, t, r A x u(x, t A x u(x, t + A x u(x, t = A v u(x, t A v u(x + x, t r A x u(x, t 4
So u(x, t + u(x, t u(x + x, t u(x, t = v ru(x, t x Lettng x, 0, we have the contnuous model (5 u t(x, t = vu x(x, t ru(x, t To dscretze, we consder the stream wth length L Take > 0, defne h = L n, x = h, t k = k, u k = u(x, t k and use the approxmatons Then (5 mples u t(x, t k uk+ u k, u x(x, t k uk uk h u k+ u k = v uk uk ru k h ( v u k + h u k+ = ( v h r u k =: ( α du k + αu k Boundary condtons: suppose we are gven the ntal concentraton at all locatons u 0 for = 0,, n and the upstream concentraton u k 0 at all tme k N, defne the matrces u k α d 0 0 u k u k α α d 0 αu k 0, A = 0 α, b k 0 u k n 0 0 0 α α d Then we have u k+ = Au k + b k If u k 0 s constant for all k, then bk = b s also a constant 4 Heat and Mass Transfer n Two Drectons Consder heat dffuson n a thn D coolng fn where there s dffuson n both x and y drectons, whle any dffuson n z drecton s mnmal and can be gnored The objectve s to determne the temperature n the nteror of the fn gven the ntal temperature and the temperature on the boundary Let u(x, y, t be the temperature at poston (x, y on the fn at tme t Smlar to the prevous secton, we arrve at the D contnuous heat model u t(x, y, t = f sρ + κ ( u sρ xx(x, y, t + u yy(x, y, t, wth boundary condtons u(x, y, 0 are gven for all (x, y, and u(x, y, t are gven for all t on the regon boundary For the D pollutant model, let u(x, y, t be the pollutant concentraton at poston (x, y at tme t, then we have the model u t(x, y, t = v u x(x, y, t v u y(x, y, t ru(x, y, t where v and v are velocty n x and y drecton, respectvely Boundary condtons: u(x, y, 0 are gven for all (x, y, and u(x, y, t for all t are gven on the upwnd boundary 5
References [] J Matthews, K Fnk, Numercal Methods Usng Matlab, 4th ed, Pearson Educaton (004 [] C Meyer Matrx Analyss and Appled Lnear Albegra SIAM (000 [3] RE Whte Computaton Mathematcs: Models, Methods, and Analyss wth Matlab, CRC Press (004 6