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Jan
28
comment Centre of mass of bounded region conformation of numerical answer
The object is of constant density and symmetric with respect to the $y$-axis, which intuitively justify the $y=0$ component. What would be the cause of it going to the right? Regarding the $z$ calculation, I did it both in euclidean and cylindrical coordinates, and ended up with the same result. Of course, I could be wrong. Why don't you post your calculations so we can compare?
Jan
27
comment Solve the screened Poisson equation $(\Delta-\lambda^2) u(\vec r)=-c$
David, what are the boundary conditions and the domain?
Jan
26
comment Discontinuous solution to first order ODE with delta function coefficients
The object is very strange for undergrads, and I am not the one that calls them "symbolic derivatives". The name was used by Bernard Friedman in his beautiful book Principles and techniques of applied mathematics. It is sad to see that someone thinks his approach deserves a downvote. For me, it is a mystery why some work their way backwards from the complex and rigorous theories to the originally simple mathematical formulations, as if they were born knowing.
Jan
25
comment Discontinuous solution to first order ODE with delta function coefficients
I wouldn't say is a mistake. I'd say that is not formally correct, but not that it is a mistake. To me, it is like saying that thinking the $\delta$-function as $\infty$ in $x=0$ and $0$ otherwise is a mistake: it is formally incorrect, but it is conceptually correct, and it helps a great deal to understand what this strange object is.
Jan
25
comment Centre of mass of bounded region conformation of numerical answer
See edit for answer.
Jan
25
comment Centre of mass of bounded region conformation of numerical answer
See my edited answer for the details. Note that I did have a small error, as I forgot to multiply the integral by two.
Jan
25
comment Centre of mass of bounded region conformation of numerical answer
Let me put it in another way: what is the volume under the surface $z = \sqrt{x^2 + y^2}$, where $x^2 + y^2 = 2 x$?
Jan
24
comment Centre of mass of bounded region conformation of numerical answer
Why should they be different? The only thing that changes is the weight of the integral. You should be careful when a different order of integration has to be done, but I don't think is the case here.
Jan
23
comment Centre of mass of bounded region conformation of numerical answer
Well, $0 < z < r$ is straight forward. As can be seen from the second graph, $0 < r < 2$, and the limitation then means that $\theta$ has to go from $0$ to $\pi \over 2$. You have to be careful with the signs, though.
Jan
22
comment Proving uniqueness of a steady state
Why don't you study the expression $(1+\alpha)\bar{p}^\alpha -\alpha y \bar{p}^{\alpha-1}$?
Jan
20
comment Partial derivatives of a function with conditions dependent on parameters
Why don't you use a smooth characteristic function?
Jan
20
comment Literature on the differential operator $-\frac{d^2}{dx^2}-\frac{2\nu+1}{x}\frac{d}{dx} + x^2$
The operator $\frac{1}{r^\beta}\frac{\partial}{\partial r}\left(r^\beta \frac{\partial}{\partial r}\right)$ has something to do with anomalous diffusion. I can't remember now, but I'll see what I can find.
Jan
20
comment Literature on the differential operator $-\frac{d^2}{dx^2}-\frac{2\nu+1}{x}\frac{d}{dx} + x^2$
You said "any possible literature on the differential operator[...]". What if I had referred you to a Bessel functions book?
Jan
19
comment the solution of a system of nonlinear differential equations
What does the crossoperator stands for?
Jan
11
comment Particular solution to nonhomogeneneous 2nd order ODE
@Ian I agree with all your concerns. In my experience, students tend to omit BCs because they don't understand their role and importance. That's why I'm so emphatic about them, specially on this site, where lack of context is almost a given. If this problem had arisen form a PDE on a cylinder, two out of three students wouldn't know what to do with the missing boundary condition, and the implications on the eigenvalues, eigenfunctions, etc. All the fun starts when BCs are imposed :P
Jan
11
comment Particular solution to nonhomogeneneous 2nd order ODE
@usumdelphini Sorry about that. It has been corrected.
Jan
11
comment What is this ODE called and how to solve it
It can be easily rewritten as an Euler's equation.
Jan
11
comment Particular solution to nonhomogeneneous 2nd order ODE
@Ian True. I've edited accordingly.
Jan
11
comment Particular solution to nonhomogeneneous 2nd order ODE
@Ian The reason we look for homogeneous and particular solutions separately is linearity (the affine space wording is a fancy way of stating this). Didactically, it is a very bad idea to work without bcs: students have a hard time understanding why different bcs lead to different Green functions, what to do with singularities, separation of variables, solvability conditions, etc. This problems grow as they think pdes can be treated the same way, when they have nonmixed bcs and when they work with any kind of nonlinearity.
Jan
11
comment Particular solution to nonhomogeneneous 2nd order ODE
OK, but I have to warn you that in my experience as student and teacher, it's a very undidactic way to learn (and work with) odes and pdes. A differential equation is not properly defined without boundary conditions and a domain of integration.