Tuesday, May 12, 2020
Sunday, May 10, 2020
Picard iterative method to solve a differential equation
\[ \frac{dy}{dx} = f(x,y) , \qquad y(x_0)=y_0 \]
The Picard iterative process constucts a sequence of functions which will get closer and closer to the desired solution.
1. \[ y_0(x) = y_0 \]
2. the recurrent formula
\[ y_{n+1}(x) = y_0 + \int_{x_0}^x f(t,y_n(t))dt \,\qquad for n\ge 1\]
Example:
\[ y' = x(3-2y), \qquad y(0)=1 \]
\[y_0 =1\]
\[y_1 = 1 + \int_0^x t(3-2*1) dt = 1 + x^2/2 \]
\[y_2 = 1+ \int_0^x t(3-2*y_1) dt = 1 + x^2/2 - x^4/4\]
\[ \ldots\]
\[\lim_{n\to \infty} y_n(x) = 3/2 -1/2 e^{-x^2} \]
Reference:
Thursday, May 7, 2020
Shock compression ratio
The derivation of the shock compression factor.
\[ \rho v = const \]
\[ \rho v^2 + P = const \]
\[ \rho v(\frac{1}{2}v^2 + \frac{\gamma}{\gamma-1}\frac{P}{\rho}) = const \]
Let 1(2) to indicates the shock upstream (downstream) quantity.
The compression factor is defined by
\[ r = \frac{\rho_2}{\rho_1} = \frac{v_1}{v_2} .\]
\[ P_2 = \rho_1 v_1^2 + P_1 - \rho_2 v_2^2 \]
\[ \frac{P_2}{\rho_2} = \frac{P_1}{\rho_2} + \frac{\rho_1 v_1^2}{\rho_2} - v_2^2 = \frac{P_1}{\rho_1 r} + \frac{v_1^2}{r} - (\frac{v_1}{r})^2 \]
\[ \frac{1}{2}v_1^2+\frac{\gamma}{\gamma-1}\frac{P_1}{\rho_1} = \frac{1}{2}v_2^2+ \frac{\gamma}{\gamma-1}( \frac{P_1}{\rho_1 r} + \frac{v_1^2}{r} - (\frac{v_1}{r})^2) \]
multiply by $\frac{2}{v_1^2}$,
\[ r^2(1+\frac{1}{\gamma-1}\frac{2}{M^2}) -\frac{2}{\gamma-1}(\frac{1} {M_1^2}+\gamma)r+\frac{2\gamma}{\gamma-1}-1 \,* \]
It is easy to verify that $r_1=1$ is one of the solution. Thus, we can write the equation as
\[ (r-1)(r-r_2) = r^2-(r_2-1)r+r_2 \]
Comparing with equation *, we can get
\[ r_2 = (\frac{2\gamma}{\gamma-1}-1)/(1+\frac{2}{(\gamma-1)M_1^2}) = \frac{(\gamma +1)M_1^2}{(\gamma-1)M_1^2+2} \]
cosmic rays modified shock
Wednesday, April 22, 2020
TERMUX INSTALL MATPLOTLIB error: ft2build.h / fthread.h
error: can not find ft2build.h
ft2build.h in the below directory: /data/data/com.termux/files/usr/include/freetype2
ft2build.h in the below directory: /data/data/com.termux/files/usr/include/freetype2
solution:
Reference:
Monday, April 20, 2020
From pitch angle diffusion to spatial diffusion coefficient
\[ dp/dt = qv/c \times (B_0 + \delta B) \]
\[dp_{\parallel}/dt = q/c (v_{\perp} \times \delta B) \]
\[p_{\perp} = \mu p \]
\[ d\mu/dt = q/(pc)v(1-\mu^2)^{1/2} \delta B \cos(\Omega t -kx +\psi) \]
\[ \delta B(x,t) = \delta B \cos(\Omega t -kx +\psi) \]
Averaging over random phase of the waves $\psi$,
\[ \int_0^{2\pi} \cos^2\psi d\psi = \pi\]
\[\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]
in the frame in which the wave is at rest, we can write $x=\mu v t$,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{\psi} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \cos[(\Omega -kv\mu)(t-t'')] \].
Integrating over time,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{t} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \int dt' \int dt'' \cos[(\Omega -kv\mu)(t-t'')] \\
= \frac{q^2 v(1-\mu^2)(\delta B)^2}{c^2 p^2 \mu} \delta(k-\Omega/(v\mu)) \delta t \].
\[dp_{\parallel}/dt = q/c (v_{\perp} \times \delta B) \]
\[p_{\perp} = \mu p \]
\[ d\mu/dt = q/(pc)v(1-\mu^2)^{1/2} \delta B \cos(\Omega t -kx +\psi) \]
\[ \delta B(x,t) = \delta B \cos(\Omega t -kx +\psi) \]
Averaging over random phase of the waves $\psi$,
\[ \int_0^{2\pi} \cos^2\psi d\psi = \pi\]
\[\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]
in the frame in which the wave is at rest, we can write $x=\mu v t$,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{\psi} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \cos[(\Omega -kv\mu)(t-t'')] \].
Integrating over time,
\[ \langle \delta \mu (t')\delta \mu (t'') \rangle_{t} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \int dt' \int dt'' \cos[(\Omega -kv\mu)(t-t'')] \\
= \frac{q^2 v(1-\mu^2)(\delta B)^2}{c^2 p^2 \mu} \delta(k-\Omega/(v\mu)) \delta t \].
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