jkwiens.com: Questions

Question: Laplace’s Equation confined to a circle

Wednesday, 26 of March , 2008 at 1:04 pm

Question: Solve Laplace’s equation inside a circle of radius a.

$\nabla ^2 u = \frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial u}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0$

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Category: Differential Equations, Questions

Question: Heat Distribution in a Rod

Monday, 17 of March , 2008 at 9:17 pm

Question: A 1D rod of length L has an initial heat distribution of

$u(x,0) = - cos(\frac{8 \pi x}{L})$

If the rod has insulated ends ( $\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(L,t) = 0$ ) and obeys the heat equation

$\frac{\partial u(x,t)}{\partial t} = k \frac{\partial^2 u(x,t)}{\partial x^2}$ ,

What is the heat distribution of the rod as a function of time?

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Category: Differential Equations, Questions

Question: Solve Laplace’s Equation

Sunday, 9 of March , 2008 at 1:10 pm

Question: Find the solutions to Laplace’s Equation:

$\frac{\partial^2 u(x,y)}{\partial x^2} + \frac{\partial^2 u(x,y)}{\partial y^2} = 0$

Note: Since there are no boundary conditions, there won’t be an exact solution.

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Category: Differential Equations, Questions

Question: Falling Infinite Rope

Sunday, 2 of March , 2008 at 9:42 pm

Question: An infinite rope with a linear density of $\lambda$ is placed on a frictionless table. If the end of the rope is placed at the end of the table and starts falling, what is the velocity of the rope as a function of distance?

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Category: Classical Mechanics, Questions

Question: General Form of Sequence

Sunday, 24 of February , 2008 at 12:34 pm

Question: The solution to $x = \sqrt{10y+1}$ where $x,y \in \mathbb N$ yields the following sequence:

8, 12, 36, 44, 84, 96, 152, 168, 240…

Find the general form for this sequence.

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Category: Algebra, Questions