Trying to model Heat flow trough different materials with NDsolve

Question

What I'm trying to achieve is model of the heat flow, in this case for the simplest 1D case,its relatively easy to do for the steady state case, but when I try to do it with NDsolve so I get the distribution of heat over time,I fail to come up with a good way to connect the two differential equaions, I recently modeled a gravity field where the same differential equation is applied many times to different bodies, that I got to work, but I dont understand how to link "differeet" ODE/PDE's

This is my code, so far, and I intend to display that solution with Manipulate and Plot:

:ps I added the [] in u[1] and u[2] last minute to see if it changes anything before they were just called u1[] and u2[] respectively, it might be nonsensical but my original question remains

thanks in advance!

heateqs = {D[u[1][x, t], t] == D[u[1][x, t], {x, 2}], 
   D[u[2][x, t], t] == 0.5 D[u[2][x, t], {x, 2}]};

bcs = {u[1][0, t] == 300, u[1][5, t] == u[2][5, t], 
   Derivative[1, 0][u[2]][10, t] == 0};

ics = {u[1][x, 0] == u[2][x, 0] == 0};

sol = NDSolve[{heateqs, bcs, ics}, u, {x, 0, 10}, {t, 0, 10}]

NDSolve::bcedge: Boundary condition u[1][5,t]==u[2][5,t] is not specified on a single edge of the boundary of the computational domain. >>

Answer 1

I think you don't need to have two equation to describe behavior of domain follows same PED. you may need to add UnitStep to the thermal diffusivity factor.

check if this work for you (domain of length=1):

sol = u[x, t] /. NDSolve[{
(1 - 0.5 UnitStep[x - 0.5]) D[u[x, t], t] == D[u[x, t], x, x],
u[x, 0] == 0,
u[0, t] == 300,
    (D[u[x, t], x] /. x -> 1) == 0
    },
   u, {t, 0, 10}, {x, 0, 1}]

With[{sol = sol},
 Manipulate[Plot[sol, {x, 0, 1}, PlotRange -> {0, 350}], {t, 0, 10}]]

you may notice that there is boundary and initial condition inconsistency. you have u=0 when t=0 & any x and at the same time you have u=300 at x=0 and any t.

Answer 2

You don't need two different functions because the position-dependent material parameter can be incorporated into a function that I'll call d[x] and that enters in a single heat conduction equation as follows:

d[x_] := (1 + 4 UnitStep[5 - x])/5.

heateq = d[x] D[u[x, t], t] == D[u[x, t], {x, 2}];

Plot[d[x], {x, 0, 10}, PlotRange -> {0, 1}, Exclusions -> None]

Here, I exaggerated the jump from large to small thermal conductivity so that the interface becomes more easily visible in the plot of the solution below.

To solve the differential equation, we need to add one more condition to the two main constraints that you require (starting height at x=0 is 300, and spatial derivative vanishes at x=10). I added the functional form at t=0 in the form of a narrow Gaussian:

sol = First[NDSolve[{heateq,
     u[x, 0] == 300 Exp[-x^2 10],
     u[0, t] == 300,
     Derivative[1, 0][u][10, t] == 0.}, 
    u[x, t], {x, 0, 10}, {t, 0, 10.}, 
    Method -> {"MethodOfLines", 
      "SpatialDiscretization" -> {"TensorProductGrid", 
        "DifferenceOrder" -> "Pseudospectral"}}]
   ];

frames = Table[
   Plot[Evaluate[u[x, t] /. sol], {x, 0, 10}, 
    PlotRange -> {0, 300}], {t, .001, 10, .2}];

ListAnimate[Show[#, Graphics[Line[{{5, 0}, {5, 300}}]]] & /@ frames]

The vertical line indicates the interface where the jump in d[x] occurs. In the Method option of NDSolve, I chose the method of lines, following the suggestion in this answer.