Recent content by Panphobia

Homework Statement I have the heat equation $$u_t=u_{xx}$$ $$u(0,t)=0$$ $$u(1,t) = \cos(\omega t)$$ $$u(x,0)=f(x)$$ Find the stable state solution. The Attempt at a Solution I used a transformation to complex to solve this problem, and then I can just take the real part to the complex...

So when you transform the second with a change of variables you get something like $$v_t=v_{xx} \\ v(0,t)=0 \\ v(\pi, t)=0 \\ v(x,0)=\int \int f(x) dx - Ax - B$$. Solving for this problem $v$ isn't too difficult, but when you transform back, by doing $$u(x,t) = v(x,t)-\int \int f(x) dx + Ax +...

Homework Statement $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$ $$0<x<\pi \\ t>0$$ $$f(x)=\begin{cases} 50 & 0<x<\frac{\pi}{2} \\ 0 & \frac{\pi}{2}\leq x< \pi \end{cases}$$ $$g(x)=\begin{cases} 0 & 0<x<\frac{\pi}{2} \\ 50 & \frac{\pi}{2}\leq...

So the linear operator ##T## in this case just takes in the vector of coefficients and it outputs the corresponding solution? Just wondering how the solution is a vector, it's probably a stupid question.

Ohhhh ok that makes a lot of sense, thank you very much!

Is it necessary for those four vectors to span ##R^4##? Also is that a corollary from some theorem? That if you can write the problem description as a linear combination of other problems, then the solution can also be written as a linear combination of those other problems?

Well yeah the columns are also linearly independent. That means that all of the coefficients for the problem descriptions are linearly independent. But how does that correlate to combining their solutions?

Well to understand it you can think of it like that. For example in your case $$ A \begin{bmatrix} x \\ y \\ z \end{bmatrix}= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} x \\ y \\...

Think about how the matrix multiplication works, then you realize that the ##A## matrix is just a collection of coefficients infront of ##x,y,z## So in this case $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$ To understand this yourself, try to...

Homework Statement If you have the heat equation $$u_{t}-u_{xx}=a \\ u(0,t)=b\\u(1,t)=c\\u(x,0)=d$$ Show that the solution to the above equation can be made up of a linear combination of solutions to $$u_{t}-u_{xx}=a_i \\ u(0,t)=b_i\\u(1,t)=c_i\\u(x,0)=d_i$$ $$i=1,2,3,4$$ if the following...

So I will have to use the quadratic formula to solve for ##\rho##?

Homework Statement Find the triple integral for the volume between a hemisphere centred at ##z=1## and cone with angle ##\alpha##.The Attempt at a Solution What I tried to do first was to get the radius of the hemisphere in terms of the angle ##\alpha##. In this case the radius is ##\tan...

Yeah it was supplied as a hint. According to the hint it's supposed be proven using cases. But I can't see it.

To show that ## r_{k+2} \leq \frac{r_k}{2} ## how would I do that? Contradiction? Cases?

It should be changed to ## min(m, n) \leq 2^k ##