麻省理工学院：《polymer engineering》lec 33

ueatint Ce c at" f 6z丁 =:i w、Pc2b 以 D ≤m- lplace tiu(aw OZT b"2 dy TAt)sI-erf y ℃ excess

Transient temperature profiles >T: =(x, t)->l-erf(x/sqrt(2*alpha*t))i T:=(x1)→1-er val > alpha:=1:pot({T(x,1),(x,4),T(x,9),T(x,16),x=0..10, thickness=3); 08 02

Maple solution of transient heat equation pde: =diff(T(x, t), t)=diff(T(x, t),x, x)i pde: =T(x, t)=T(x, t) t with(DEtools): pdesolve(pde, T(x, t)) pdesolve-T(x,(=-:T(x t),T(xt Maple can t get a solution; try simplifying the problem -convert to ODE with Fourier transform with(inttrans): odel: =fourier(pde, x, w)i ode/:=fourier(T(x, 0). x, w)=-w fourier(T(x, t). x w) ode: =subs( fourier(t(x, t),x, w)=u(t), odel)i de:==,U(t) bc: =U(O)=fourier(T o*Dirac(x), x, w); bc:=U(0)=70 dsolve((ode, bc), U(t)) (t)=T0 Invert back to x-plane assume(t>0)iT(x, t)=simplify (invfourier(rhs(%),w, x))i 冮0 e T(x.t-) simplify(8) /4 70e πyt

Finite ittenewe soliton T(七 STInN 以 t 七4 x 七 ○CA 人七 时丁下41-zT+下x+ +OL△ 8x2 C△N 七1 七 入下 x-1+(1-z入 +入 +1 (M)

1。000 中中======中·…==========+== 1035 0.3 0.25 0.15 600+··=--=- 中= 200+----------+--=--=------=-= T=0.05 000+ 0,000 250 500 1,000 DIMENSIONLESS DI STANCE X