Practice Partial Differential Equations (18) - Eigenfunction Expansion Method

Practice: Partial Differential Equations

Lesson 18Eigenfunction Expansion Method

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10 questions

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6 sample questions

Question 1.
What is an eigenfunction?
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Think about functions that remain proportional under transformations.
Question 2.
Define a boundary value problem.
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Consider where conditions are set in a physical system.
3Question 3.
What method is commonly used to solve linear PDEs?
- Eigenfunction Expansion
- Matrix Method
- Gradient Descent
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Think about the techniques discussed in class.
4Question 4.
True or False: Eigenfunctions are always orthogonal.
- True
- False
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Recall how eigenfunctions relate to inner product properties.
Question 5.
Consider a second-order PDE. Formulate a Sturm-Liouville problem that yields a set of eigenfunctions appropriate for boundary conditions $u(0)=0$ and $u(L)=0$ .
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Focus on how the boundary conditions influence the form of the solution.
Question 6.
Using the heat equation example, derive how the coefficients in the expansion relate to the initial condition using the inner product.
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Recall how orthogonality simplifies finding coefficients in the expansion.

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