Recent content by Leonardo Machado

Unlike Einstein Toolkit, which is a pretty alive community, LORENE users seems to be hidden somewhere. I cannot found much information about this software anywhere in format of tutorials. I'm having problems to compile codes with make use of LORENE libraries because of the reduced amount of...

Yes. Unfortunately this is part of this problem I am tr I was thinking about expanding the solution, just like you said, but with $$ y(x)=u(x)+r= \sum_n a_n T_n(x) + r. $$ This way of writting turns the condition $$ \frac{dy(infinity)}{dx}=1, $$ into $$ \frac{du(infinity)}{dx}=0, $$...

The problem is that the derivatives evaluated at infinity would still be zero aways, becase sech(infinity)=0. I mean, $$ \sum_n a_n \frac{dT_n(x)}{dx}=\sum_n a_n 2 sech(x) \frac{dT^*_n(z)}{dz} $$ evaluated at infinity still cannot be a finite number.

Hello, I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as $$ y(x)=\sum_{n=0}^{N-1} a_n T_n(x), $$ Being the basis an Chebyshev polynomial with the mapping x in [0,inf]. Then we put this into a general...

I have solved it today. As a computational problem i used a Gaussian integration to take inner products from both sides, as $$ \sum_k a^{(0)}_k (T_n(x),x^l \frac{du}{dx})=a^{(x)}_n (T_n(x),T_n(x)). $$ Using Chebyshev collocation points to solve the inner product integral.

Hi everyone. I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as $$ u(x)=\sum_n a_n T_n(x), $$ then you can also expand its derivatives as $$ \frac{d^q u}{dx^q}=\sum_n...

Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$...

Have you tried runge-kutta methods? I've been using it to solve some classical gravitational dynamics which have this level of difficult.

I took a simple problem which let me know the analytical solution because I want to use this solution to compare with the numerical one, for practicing proposes.

Hello everyone. I'm currently trying to solve the damped harmonic oscillator with a pseudospectral method using a Rational Chebyshev basis $$ \frac{d^2x}{dt^2}+3\frac{dx}{dt}+x=0, \\ x(t)=\sum_{n=0}^N TL_n(t), \\ x(0)=3, \\ \frac{dx}{dt}=0. $$ I'm using for reference the book "Chebyshev and...

I have solved it guys! To operate with inhomogeneous bondary conditions I've used $$ u(r,\theta,t)=v(r,\theta,t)+u_E(r,\theta) $$ being u_E the steady state and "v" the solution of the heat equation.

Hello guys. I am studying the heat equation in polar coordinates $$ u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}) $$ via separation of variables. $$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$ which gives the ODEs $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...

Hi everyone. I'm currently trying to master the use of the formula for nuclear masses from MYERS AND SWIATECKI (1969), https://www.sciencedirect.com/science/article/pii/0003491669902024. $$ E=[-a_1+J\delta^2+0.5(K\epsilon^2-2L\epsilon \delta^2 +M\delta^4)]A+c_2 Z^2 A^{1/3}...

Hi everyone! I'm currently strudying some astrophysical equation of states, some stuff about Fermi's gas and I'm kinda confused about the relation between the energy density and the mass density, $$ \frac{\epsilon}{c^2}=\rho. $$ I don't get why they do not use whole $$...

Hello everyone. I'm currently working on NS mass relations and trying to plot a curve with predicted masses-radii and observations on NS. There are some free data at this website: http://xtreme.as.arizona.edu/NeutronStars/index.php/neutron-star-radii/ . I downloaded the .tar file and tried to...