[Fourier and PDEs] Solving the Heat Equation

2. Nondimensionalizing the Heat Equation

Consider the heat equation \[ \pd{T}{t} = \kappa \pd{^2 T}{x^2}, \]

on the interval $0 \leq x \leq L$ with boundary conditions $T(0, t) = T(L, t) = 0$ and a homogeneous initial temperature $T_0$.

a) Non-dimensionalize the heat equation, being careful to show your new variables are indeed non-dimensional, to give \[ \pd{\tilde{T}}{\tilde{t}} = \pd{^2 \tilde{T}}{\tilde{x}^2} \]

on the interval $0 \leq \tilde{x} \leq 1$ with boundary conditions $\tilde{T} = 0$ at either ends and a homogeneous initial temperature of $1$.

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b) Find the non-dimensional temperature $\tilde{T}(\tilde{x}, t)$.

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c) At approximately what time does the non-dimensional temperature fall below $0.5$ everywhere? What is the corresponding dimensional time? How would the itme change for an identical system which is of length $2L$?

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3. Heat equation with non-uniform initial condition

Consider the heat equation $T_t = \kappa T_{xx}$ on the interval $0 \leq x \leq L$ with boundary conditions $T(0, t) = T(L, t) = 0$ and initial temperature $T(x,0) = x(L - x)$.

a) Find $T(x,t)$ at later times $t > 0$.

b) (i) Find the heat flux at $x = 0$ for $t > 0$ and (ii) what is the heat flux at $x = L/2$.

Spoiler
1) Note that you can guess that the right-hand side is $-\lambda^2$ because if it was positive instead, then solutions would grow exponentially in time.