Recent content by Shackleford

Thanks for the reply. I'll get around to doing the integration by parts myself and will post again if something isn't clear to me.

There's another part I'm not clear on. \begin{gather*} \begin{split} u_t(x,t) - \Delta u(x,t) & = \int_{0}^{t} \int_{\mathbb{R}^n}^{} \Phi(y,s) [(\frac{\partial}{\partial t}-\Delta_x)f(x-y,t-s)] \; dyds \\ & + \int_{\mathbb{R}^n}^{} \Phi(y,t) f(x-y,0) \; dy \\ & = \int_{\varepsilon}^{t}...

I thought that it might have something to do with the fact that the equation on the left is a function of x and t while the integral on the right is with respect to y and s. The substitution in the integral is obvious from the chain rule. For the spatial variable, the function is just being...

This is from Evans page 50. I'm sure it's something simple, but I don't follow the change from $$ \frac{\partial}{\partial t} \quad \text{to} \quad -\frac{\partial}{\partial s}$$ and from $$ \Delta_x \quad \text{to} \quad \Delta_y$$. \begin{gather*} \begin{split} u_t(x,t) - \Delta u(x,t) & =...

I think I figured it out. \begin{gather} \begin{split} u & = |y-x| = \left( \sum_{i=1}^{n} (y_i-x_i)^2 \right)^\frac{1}{2} \\ & = \frac{du}{dy_n} = \frac{1}{2}\left( \sum_{i=1}^{n} (y_i-x_i)^2 \right)^\frac{-1}{2} 2\sum_{i=1}^{n}(y_i-x_i) = \frac{y_n-x_n}{|y-x|^n}. \\ \frac{\partial...

This is from Evans page 37. I seem to be missing a basic but perhaps subtle point. Definition. Green's function for the half-space ##\mathbb{R}^n_+,## is \begin{gather*} G(x,y) = \Phi(y-x) - \Phi(y-\tilde{x}) \qquad x,y \in \mathbb{R}^n_+, \quad x \neq y. \end{gather*} What's the proper way to...

As far as I understand it, I'm using the correct weights and points. After that, it's a fairly straightforward algorithm. Someone with a bit of expertise might notice a subtle detail that I'm overlooking.

Thanks for the reply. I've included a screenshot of the problem so that there's no ambiguity. The analytical/graphical approach certainly has its advantages.

## \int_{-1}^{1} \int_{-1}^{1} e^{-(x^2 + y^2)} cos(2π (x^2 + y^2)\,dx\,dy ## ## I = \int_{-1}^{1} \int_{-1}^{1}f(x,y) \,dx\,dy \approx \sum_{i=0}^{n}\sum_{j=0}^{n} w_i w_j f(x_i, y_j) ## ## = w_0 w_0 f(x_0, y_0) + w_0 w_1 f(x_0, y_1) + w_1 w_0 f(x_1, y_0) + w_1 w_1 f(x_1, y_1) ## ## w_0 =...

Sorry, I missed this reply. For the theory, the books that I have are: An Introduction to the Mathematical Theory of Finite Elements (Dover Books on Engineering) https://www.amazon.com/dp/1614273049/?tag=pfamazon01-20 I'm taking a Numerical Methods for PDEs course this semester. It looks like...

My background is BS Math with Physics minor (3.5) and MA Math (3.7) from the University of Houston. My current plan is to explore the necessary background for doing research in finite element analysis. Over the next few semesters, I plan to take Numerical Methods for PDEs, Statistical Computing...

That's a good suggestion. I should add Numerical Analysis and maybe even a Signals course to the second question's answer. One could apply MCMC Bayesian inference to the error measurement methodology (or error function).

That's my observation as well. To that end, I have A First Look at Rigorous Probability Theory by Jeff Rosenthal.

This program? Zero. I've only seen a PhD Probability program or two in the UK, which are generally research based without all of the prerequisite courses and myriad examinations.

Good question. I suppose there should be a course on Bayesian statistics/inference.