# How can solve this partial differential equation (PDE) and plot?

How can plot and solve this partial differential equation in mathematica?

$$K \frac{\partial^2 T}{\partial x^2}- h (T-T_m) = \frac{\partial T}{\partial t}$$

$$Tm = 25$$

$$k= 47$$

$$h= 1.5$$

this equation is for a bar 1 meter long the boundary conditions

$$T(0,t) = 0 °C$$

$$T(L,t) = 0 °C$$

And the initial conditions are

$$T(x,0) = 42 °C$$

And how can solve whit the finite difference method?

I try with this

Ecuacion = D[\[Theta][x, s], x, x] - (h + s)/k \[Theta][x, s] == -(1/ s) (h Ta + 41)

SolED = DSolve[{Ecuacion, \[Theta][0, s] == 0, \[Theta][L, s] == 0}, \[Theta][x, s], x]

• Could you please give us your code so that people can use it to answer your question. – Tugrul Temel Oct 31 '18 at 23:23
• Yes I put the code in the question – Conan Oct 31 '18 at 23:28

Try this:

Clear[x, Ecuacion, s];

Ecuacion = D[\[Theta][x, s], {x,2}] == -(1/s) (h Ta + 41) + (h + s)/k \[Theta][x, s];
SolED    = DSolve[{Ecuacion, \[Theta][0, s] == 0, \[Theta][L, s] == 0}, \[Theta][x,s], x]
//FullSimplify

(*
The solution is:
{{\[Theta][x, s] -> -((
E^(-((Sqrt[h + s] x)/Sqrt[
k])) (-1 + E^((Sqrt[h + s] x)/Sqrt[
k])) (-E^(((L Sqrt[h + s])/Sqrt[k])) + E^((Sqrt[h + s] x)/Sqrt[
k])) k (41 + h Ta))/((1 + E^((L Sqrt[h + s])/Sqrt[k])) s (h +
s)))}}
*)

• Yes, but when I do the Laplace transfform dont give me a solution. thanks for the answer. – Conan Nov 5 '18 at 20:28
• @Conan: Please post the code you applied so that I can try to answer. – Tugrul Temel Nov 5 '18 at 20:42
• the code is in my question, what I did was to use Laplace and solve the equation, once the equation was solved I used inverse transform in Mathematica to be able to return to the time domain. Thanks :) – Conan Nov 7 '18 at 21:52
• I try to use this: pde = 47* D[T[x, t], {x, 2}] - D[T[x, t], t] - 47*(T[x, t] - 25) == 0 S2 = NDSolve[{pde, T[0, t] == 0 , T[1.5, t] == 0, T[x, 0] == 42}, T[x, t], {x, t}] – Conan Nov 10 '18 at 17:56
• @Conan: You have a second order differential equation, therefore, you need to give the required first order initial values, otherwise, you system of ODEs will be undetermined. The reason is that the number of equations is less than the number of dependent variables. – Tugrul Temel Nov 10 '18 at 21:42