12. Partial Differential Equations

Try these first

1. 1.

   What does $u(x,t)$ represent in the heat equation?

   Hint

   Think about what changes with time and position.

2. 2.

   State the equation of the One-Dimensional Heat Equation.

   Hint

   Recall the relationship governing heat diffusion.

3. 3.

   What does the thermal diffusivity $\alpha^2$ signify in the heat equation?

   • Rate of temperature change
   • Measure of material's heat capacity
   • Conductivity property of the material

   Hint

   Think about how heat conducts through different materials.

4. 4.

   The equation $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ models which physical phenomenon?

   • True
   • False

   Hint

   Recall the application of this equation.

5. 5.

   Solve the heat equation for a rod of length 2 meters with boundary conditions $u(0,t) = 0$ and $u(2,t) = 0$, and initial condition $u(x,0) = x(2-x)$. What is the solution?

   Hint

   Recall how Fourier series are computed using initial conditions.

6. 6.

   Explain the physical significance of the parameters in the heat equation derived for a metal rod in the context of thermal conductivity.

   Hint

   Connect the parameters to real-world material properties derived from experiments.