Practice Partial Differential Equations - 12 | 12. One-Dimensional Heat Equation | Mathematics - iii (Differential Calculus) - Vol 2
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12 - Partial Differential Equations

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

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

💡 Hint: Think about what changes with time and position.

Question 2

Easy

State the equation of the One-Dimensional Heat Equation.

💡 Hint: Recall the relationship governing heat diffusion.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

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.

Question 2

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.

Solve 2 more questions and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

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.

Question 2

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.

Challenge and get performance evaluation