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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Define eigenfunction in the context of PDEs.
💡 Hint: Think about functions that remain unchanged apart from scaling.
Question 2
Easy
What step follows after separating variables in a PDE?
💡 Hint: Identify what your variables relate to.
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 is the first step in the Eigenfunction Expansion Method?
- Combine solutions
- Identify and solve the spatial part
- Express the initial condition
💡 Hint: Think about how we approach separating different parts.
Question 2
True or False: Eigenvalues can be complex.
- True
- False
💡 Hint: Recall the properties of eigenvalues in Sturm-Liouville problems.
Solve 2 more questions and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given the boundary conditions for a rod held at two ends and subjected to a heat flux, derive the eigenfunctions and eigenvalues. Next, use them to find the general solution of the heat equation.
💡 Hint: Focus first on applying the correct boundary conditions to obtain eigenvalues.
Question 2
Consider a wave equation defined on a string of length L with fixed ends. Describe how you would derive the first three eigenfunctions and show how they form a complete basis for the problem.
💡 Hint: Use the knowledge of harmonic frequencies.
Challenge and get performance evaluation