Practice Application: Heat Equation in a Semi-Infinite Rod - 10.6.1 | 10. Fourier Cosine and Sine Transforms | Mathematics (Civil Engineering -1)
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Application: Heat Equation in a Semi-Infinite Rod

10.6.1 - Application: Heat Equation in a Semi-Infinite Rod

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Learning

Practice Questions

Test your understanding with targeted questions

Question 1 Easy

Define the heat equation for a semi-infinite rod.

💡 Hint: Think about how temperature changes over time.

Question 2 Easy

What constant temperature is held at x=0?

💡 Hint: This is what you will observe at one end of the rod.

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

What is the general form of the heat equation for a semi-infinite rod?

a) \\( \\frac{\\partial u}{\\partial t} = \\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} \\)
b) \\( \\frac{\\partial u}{\\partial x} = \\alpha^2 \\frac{\\partial^2 u}{\\partial t^2} \\)
c) \\( \\frac{\\partial u}{\\partial t^2} = \\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} \\)

💡 Hint: Look for how temperature changes over time in this differential equation.

Question 2

True or False: The Fourier Cosine Transform is only applicable to bounded domains.

True
False

💡 Hint: Consider where you encounter cosine transformations in boundary value problems.

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Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

In a semi-infinite rod made of copper, calculate the time taken for the temperature to change significantly (by 50%) at a distance of 1 meter from the heated end if the thermal diffusivity is 1.1 × 10^{-4} m²/s.

💡 Hint: Refer to the error function for determining significant temperature changes.

Challenge 2 Hard

Discuss how altering the boundary condition to a time-varying temperature at one end of the rod would impact the solution structure. What complications arise?

💡 Hint: Think about how time-variability influences the interpretation of heat flow.

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