18.1 - Basic Concept
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Practice Questions
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What is an eigenfunction?
💡 Hint: Think of it as remaining unchanged in shape but different in size.
State the importance of orthogonality in eigenfunction expansions.
💡 Hint: Consider why separate dimensions in space help.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the primary purpose of the Eigenfunction Expansion Method?
💡 Hint: Think about what we aim to achieve with PDEs.
True or False: Eigenfunctions must be orthogonal to each other.
💡 Hint: Why is independence of functions beneficial?
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Challenge Problems
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Using a linear PDE, demonstrate the application of the Eigenfunction Expansion Method to solve for temperature distribution in a rod.
💡 Hint: Start with the equation: ∂u/∂t = α²∂²u/∂x².
Prove that the convergence of an eigenfunction expansion is consistent under suitable regularity conditions of the initial function.
💡 Hint: Refer to the properties of Fourier series for continuity implications.
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