Practice Canonical (Standard) Forms of Second-Order PDEs - 16.10 | 16. Partial Differential Equations – Basic Concepts | Mathematics (Civil Engineering -1)
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16.10 - Canonical (Standard) Forms of Second-Order PDEs

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

What is the general form of a second-order PDE?

💡 Hint: Look for the highest order derivatives present.

Question 2

Easy

Define discriminant in the context of second-order PDEs.

💡 Hint: Recall how it relates to the coefficients A, B, and C.

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 canonical form of an elliptic PDE?

  • A: \\( \\frac{\\partial^2 u}{\\partial \\xi^2} + \\frac{\\partial^2 u}{\\partial \\eta^2} = 0 \\)
  • B: \\( \\frac{\\partial^2 u}{\\partial \\xi^2} = \\frac{\\partial u}{\\partial \\eta} \\)
  • C: \\( \\frac{\\partial^2 u}{\\partial \\xi \\partial \\eta} = 0 \\)

💡 Hint: Recall the different forms discussed in class.

Question 2

If D = B^2 - 4AC < 0, which type of PDE do you have?

  • True
  • False

💡 Hint: Think about how discriminants categorize the equations.

Solve and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Given the PDE \( 2 \frac{\partial^2 u}{\partial x^2} - 8 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \), classify it and find the canonical form.

💡 Hint: Pay close attention to the coefficients A, B, and C.

Question 2

Derive a transformation for the equation \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \) into its canonical form.

💡 Hint: Focus on recognizing how changes in variables can simplify complex equations.

Challenge and get performance evaluation