Thermal field 1D, Dirichlet

Example: a transient Dirichlet problem

Consider a concrete example of a transient Dirichlet problem. Take a rod on the segment [α,β][\alpha, \beta] =[0,1]= [0, 1] with thermal diffusivity a2a^2 =0.05= 0.05. The rod is initially cold, and the end temperatures are not imposed at once but grow over time: the ends heat up linearly from zero to 2020 and 8080, respectively, over the first 5 seconds and are then held. Inside the rod we place a point heat source f(x)f(x) =P⋅δ(x−x0)= P \cdot \delta(x - x_0) of power PP =10= 10 at the point x0x_0 =0.35= 0.35. The problem statement reads

∂T(x,t)∂t\displaystyle \frac{\displaystyle \partial T(x, t)}{\displaystyle \partial t} =a2⋅∂2T(x,t)∂x2\displaystyle = a^2 \cdot \frac{\displaystyle \partial^2 T(x, t)}{\displaystyle \partial x^2} +P⋅δ(x−x0),\displaystyle + P \cdot \delta(x - x_0),T(x,0)\displaystyle T(x, 0) =0,\displaystyle = 0,T(α,t)\displaystyle T(\alpha, t) =φα(t),\displaystyle = \varphi_\alpha(t),T(β,t)\displaystyle T(\beta, t) =φβ(t),\displaystyle = \varphi_\beta(t),
(7.1)

where the end temperatures depend on time

φα(t)\displaystyle \varphi_\alpha(t) ={4t,0≤t≤5,20,t>5,φβ(t)\displaystyle = \begin{cases} 4t, & 0 \le t \le 5,\\ 20, & t > 5, \end{cases} \qquad \varphi_\beta(t) ={16t,0≤t≤5,80,t>5.\displaystyle = \begin{cases} 16t, & 0 \le t \le 5,\\ 80, & t > 5. \end{cases}
Fig. 7.1. Transient Dirichlet problem: heating of the rod over time. The blue curve is the temperature profile T(x,t)T(x, t), the orange marker is the point source.