Heat Equation: Models, Properties, and Solution Methods

All flashcards

What is the heat equation?

A parabolic partial differential equation modeling temperature evolution, commonly written $u_t=\alpha\Delta u$.

What does the Laplacian measure in the heat equation?

It combines second spatial derivatives and indicates how a value compares with its nearby spatial values.

What is thermal diffusivity?

A material property controlling the rate at which heat differences spread; for a uniform material, $\alpha=k/(\rho c_p)$.

What does Fourier's law say about heat flow?

Heat flux is proportional to the negative temperature gradient: $\mathbf q=-k\nabla u$, so heat flows down temperature gradients.

How does conservation of energy contribute to the heat equation?

It relates the rate of change of stored heat in a region to the net heat flowing into that region.

Quiz questions

1. How is an internal heat source represented in the dimensionless nonhomogeneous heat equation?Answer: By adding a source term: $u_t=\Delta u+f(x,t)$.
2. What equation describes a steady-state temperature field with no internal source?Answer: Laplace's equation, $\Delta u=0$.
3. For a rod with zero temperature at both endpoints, which spatial modes satisfy the boundary conditions?Answer: Sine modes $\sin(n\pi x/L)$ for positive integers $n$.
4. How does the whole-line heat equation evolve initial data $g(x)$ using its heat kernel $\Phi$?Answer: By convolution: $u(x,t)=\int_{-\infty}^{\infty}\Phi(x-y,t)g(y)\,dy$.
5. What does the classical maximum principle imply for an interior temperature maximum under suitable conditions?Answer: It cannot exceed the largest relevant initial or boundary value.