16.7 - Solving PDEs with Boundary and Initial Conditions
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Practice Questions
Test your understanding with targeted questions
What is an initial condition in a PDE?
💡 Hint: Think about what information you need at time t = 0.
Name the three types of boundary conditions.
💡 Hint: Recall the DNR acronym.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What do Dirichlet conditions specify?
💡 Hint: Recall the type of condition that dictates what is happening right at the edge.
True or False: A well-posed problem is one that is stable.
💡 Hint: Consider what well-posed means in connection with solutions.
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Challenge Problems
Push your limits with advanced challenges
A cylindrical rod of length L is held at fixed temperatures at both ends (Dirichlet conditions). The initial temperature distribution is given as u(x,0) = sin(πx/L). Solve the PDE for different times t > 0.
💡 Hint: Consider using Fourier series to express the initial condition.
A string fixed at both ends vibrates according to the wave equation with initial displacement u(x,0) = f(x) and initial velocity ∂u/∂t (x,0) = g(x). Derive the general solution of the wave equation subject to Neumann boundary conditions.
💡 Hint: Focus on how the wave behavior is altered based on the boundary at the fixed ends.
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