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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is an eigenfunction?
💡 Hint: Think about functions that remain proportional under transformations.
Question 2
Easy
Define a boundary value problem.
💡 Hint: Consider where conditions are set in a physical system.
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 method is commonly used to solve linear PDEs?
- Eigenfunction Expansion
- Matrix Method
- Gradient Descent
💡 Hint: Think about the techniques discussed in class.
Question 2
True or False: Eigenfunctions are always orthogonal.
- True
- False
💡 Hint: Recall how eigenfunctions relate to inner product properties.
Solve 2 more questions and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Consider a second-order PDE. Formulate a Sturm-Liouville problem that yields a set of eigenfunctions appropriate for boundary conditions $u(0)=0$ and $u(L)=0$.
💡 Hint: Focus on how the boundary conditions influence the form of the solution.
Question 2
Using the heat equation example, derive how the coefficients in the expansion relate to the initial condition using the inner product.
💡 Hint: Recall how orthogonality simplifies finding coefficients in the expansion.
Challenge and get performance evaluation