Recent content by stanley.st

1. Euler equation in Polar coordinates

Hello. I have 2D Euler equation for fluids. I cant derive it in polar coordinates. I defined functions u(x,y,t) = u'(r, theta, t) and v(x,y,t) = v'(r, theta, t). I started by computing derivatives \frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial...
2. Mass conservation

I need to find two functions I_1, I_2 constant on charakterstics and write general solution u(x,y,t)=\varphi(I_1,I_2) I found one function I_1=x_1^2+x_2^2 I don't know to find second one with t. Thx
3. Mass conservation

Hello i want to solve \frac{\partial \rho}{\partial t}=\frac{\partial v_1\rho}{\partial x_1}+\frac{\partial v_2\rho}{\partial x_2} for v_1 = -x_2 and v_2=x_1 i obtain equation \frac{\partial \rho}{\partial t}+x_2\frac{\partial\rho}{\partial x_1}-x_1\frac{\partial \rho}{\partial...
4. First order PDE with two conditions?

Hello, I have a problem in the form \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+e^{x}u=0 with conditions u(x,0)=u_0(x) u(0,t)=\int_{0}^{\infty}f(x)u(x,t)dx Im confused, because in first order PDE i require only 1 condition. How to solve this for two conditions?
5. Navier stokes

Thank you ! Navier-Stokes without pressure? It is strange, because in order to get particles move, we have to include pressure considerations. How would you interpretate that ?
6. Navier stokes

Thank you so much. I have no specific problem to solve. I wanted to find general solution of NS in 1D. What is an example of such information?
7. Navier stokes

Hello, I have Navier stokes in 1D \rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\frac{\partial^2u}{\partial x^2} Condition of imcompressibility gives \frac{\partial u}{\partial x}=0 So I have Navier stokes...
8. Solving specific PDE

Hello, I derived a model in the form \begin{array}{rcl}\frac{\partial U(\vec{x},t)}{\partial t}&=&\gamma^2\Vert\nabla U(\vec{x},t)\Vert,\\\int_{\Omega}U(\vec{x},t)\, d \Omega&=&U_0,\quad\forall t\\U(\vec{x},0)&=&f(\vec{x}).\end{array} I don't know to solve that. THanks for help.
9. Wrong Time Dilation

Thanks for help so much to all. But any of these replies didn't satisfied me, but now I solved my problem. To yuiop: The problem is, what the hell is L\sqrt{1-v^2/c^2} :-) Of course, I know what is it, but we didn't derive it here. To PAllen: This was that obstacle :-) I really...

11. Wrong Time Dilation

Hello ! Can you tell me what's wrong? I suppose two observers. They measure speed of light. One observer is in rocket of length L. Observer in rocket measured speed C'=\frac{L}{t'} and t' is time from observer's view. Second observer is outside. He measured C=\frac{L+vt}{t} where v is...
12. Proving continuity of f(x,y) = g(x)p(y)

We want to prove \forall \varepsilon>0\quad\exists\delta>0\quad\forall (x,y)\in B_{\delta}(x_0,y_0):f(x,y)\in B_{\varepsilon}(f(x_0,y_0)) Let fix epsilon and we want to find such delta. From definition of ball B, we compute...
13. Proof - Substitution, Jacobian, etc.

Hello! I recently tried to prove following theorem: Let \phi:B\to\mathbb{R}^2 be a diffeomorphism (regular, injective mapping). Then \int_{\phi(B)}f(\mathbf{x})\,\mathrm{d}x=\int_{B}f(\phi(\mathbf{t}))\left|{\mathrm{det}}\mathbf{J}_{\phi}\right|\mathrm{d}t With following I can't proof...
14. Open set (differential equation)

Hello ! When I'm reading something about differential equations everywhere it's about open sets. For example when we define special kind of equation x'=f(t,x)\,;\;f:\Omega\subset\mathbb{R}\times\mathbb{R}\to\mathbb{R} Omega is open. Why Omega must be open? Thanks
15. Energy conservation law

Hello, I nowhere find general form of energy conservation law, but in one book i found this (*)\hspace{1cm}E=E_{in}-E_{out}+E_{generated} where E(in) is energy flow into system, E(out) is energy flow out of system and E(generated) is energy generated. It was in sense of heat...