Hi, im currently trying to work on the following formulae for a 1-d transient Heat transfer problem. Basically it is a bar with 2 ends where each end has a fixed temperature after t=0 meaning to say that one end is being heated while the other is cooled to a fixed temperature. i want to get a sketch of temp vs time and the solution for dT/dt my boundary condtions are T(0,0)=301K=T(L,0) and when it reaches steady state at an unknown time, T(0,t)=293 T(L,t)=473 so with L = 5 i tried solving the equation but i kept coming up with the same error. heatsol = NDSolve[{3.25D[temp[x, t], t] == -0.0000447D[ temp[x, t], {x, 2}], temp[0, 0] == 301, temp[5, 0] == 301, temp[5, t] == 473}, temp, {x, 0, 5}, {t, 0, 100}] NDSolve::deqn: Equation or list of equations expected instead of True in the first argument {3.24776 (temp^(0,1))[x,t]==-0.000044705 (temp^(2,0))[x,t],temp[0,0]==301,temp[5,0]==301,True,temp[0,t]==273}. >> but if i assigned a random number to the boundary condition where it reaches steady state, temp[5,100] i get the following error saying that my initial condition is redundant heatsol = NDSolve[{3.247755102 D[temp[x, t], t] == -0.000044705030299999997 D[ temp[x, t], {x, 2}], temp[5, 0] == 301, temp[5, 100] == 473}, temp, {x, 0, 5}, {t, 0, 100}] NDSolve::bcedge: Boundary condition temp[5,0]==301 is not specified on a single edge of the boundary of the computational domain. >>

@Shenghui 1. Both ends are heated/cooled via conduction. so i was wondering if i need to co-relate another transient state equation to the boundary conditions(governed by the heat transfer equation through the respective heating/cooling blocks) 2. Is it possible to use the piecewise function to follow a similar gradient to dTemp/dtime as shown in the formula? because i don't think the actual transient condition is a linear path. also besides using the function Manipulate[ Plot[Evaluate[temp[x, t] /. sol], {t, 0, 100}, PlotRange -> Full], {x, 0, 5}] is it possible to plot a 2D graph of (temp,time) showing at x= 0 and 5 on 1 plot, i couldnt seem to get it using the plot function Plot[Evaluate[temp[x, t] /. sol, [0, t]], {t, 0, 100}, PlotRange -> {Automatic, {270, 480}}] thanks for all your help! i've learnt quite abit!

@Jon 1. If they are both heated gradually, you will need to append two conduction process to this model aka the beam is no longer with uniform thermal property 2. The result really depends on the time scale of transient process. You can in most cases use linear approximation for a system at transient state if 1) peak value is not a big deal and 2) raise time (10% -> 90% of steady state) is relatively short and 3) the response is almost monotonic (or oscillation plays no role). I think linearization should make sense here. Your request may involve some effor beyond the purpose of this forum 3. You just need: Plot[{Evaluate[temp[x1, t] /. sol],Evaluate[temp[x2, t] /. sol]}, {t, 0, 100},<options sit here>]

@Shenghui Hmm.. I'm still trying to learn how Mathematica sees this problem. Because solving it analytically, i just need a PDE, IC and BC. But in Mathematica i need to approixmate the the gradient at 0 and L which i thought the PDE would solve. Do you think if i change the boundary conditions to T(0,t)=473 from t>5 to infinity would it work without using the linear approximation in the piecewise function? Also is there a way to input a function that allows me to put a boundary coundition say t>0 without specifiying it through the piecewise function or how do i specify such boundary condition too. T(0,0)=273 T(5,0)=473 but at the same time,from x=0.5 to x= 4.5 it is T(x,0)=293

Heat Transfer Problem