18 - Partial Differential Equations
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Practice Questions
Test your understanding with targeted questions
What is an eigenfunction?
💡 Hint: Think about functions that remain proportional under transformations.
Define a boundary value problem.
💡 Hint: Consider where conditions are set in a physical system.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What method is commonly used to solve linear PDEs?
💡 Hint: Think about the techniques discussed in class.
True or False: Eigenfunctions are always orthogonal.
💡 Hint: Recall how eigenfunctions relate to inner product properties.
2 more questions available
Challenge Problems
Push your limits with advanced challenges
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.
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.
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