Practice Partial Differential Equations (12) - One-Dimensional Heat Equation

Practice: Partial Differential Equations

Lesson 12One-Dimensional Heat Equation

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10 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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Quiz
4 questions · about 3 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What does $u(x,t)$ represent in the heat equation?
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Think about what changes with time and position.
Question 2.
State the equation of the One-Dimensional Heat Equation.
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Recall the relationship governing heat diffusion.
3Question 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
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Think about how heat conducts through different materials.
4Question 4.
The equation $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ models which physical phenomenon?
- True
- False
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Recall the application of this equation.
Question 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?
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Recall how Fourier series are computed using initial conditions.
Question 6.
Explain the physical significance of the parameters in the heat equation derived for a metal rod in the context of thermal conductivity.
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Connect the parameters to real-world material properties derived from experiments.

6 more questions in this section. Enroll free to practise them all.