Practice Fourier Series and Initial Condition - 12.4 | 12. One-Dimensional Heat Equation | Mathematics - iii (Differential Calculus) - Vol 2
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12.4 - Fourier Series and Initial Condition

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

What is the formula to find the Fourier coefficient B_n?

💡 Hint: Remember to consider the limits of integration.

Question 2

Easy

What type of boundary conditions does the Fourier sine series often satisfy?

💡 Hint: Think about the values at the endpoints of the function.

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 is the purpose of Fourier coefficients in heat equation solutions?

  • They define temperature at boundaries
  • They determine stability
  • They relate initial conditions to the solution

💡 Hint: Think about what the coefficients are derived from.

Question 2

True or False: The Fourier series can only represent periodic functions.

  • True
  • False

💡 Hint: Think about the applications of Fourier analysis.

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Challenge Problems

Push your limits with challenges.

Question 1

Solve for B_n if f(x) is defined as a square wave function oscillating between 1 and -1. What would be its Fourier coefficients?

💡 Hint: Remember to analyze the symmetry of the square wave when integrating.

Question 2

If the initial temperature is given as f(x) = sin(2πx/L), derive the coefficients B_n and explain their significance in the context of the heat equation.

💡 Hint: Use the properties of orthogonality of sine functions.

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