Practice General Steps in the Eigenfunction Expansion Method - 18.3 | 18. Eigenfunction Expansion Method | Mathematics - iii (Differential Calculus) - Vol 2
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18.3 - General Steps in the Eigenfunction Expansion Method

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Learning

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.

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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.

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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.

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