CoCalc Shared FilesPHY213 Coursework / CourseworkGM.ipynbOpen in CoCalc with one click!
Author: Gabriel Malvestuto
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In [1]:
#Import required modules import numpy as np import matplotlib.pyplot as plt import sympy as sp from sympy import * from scipy.integrate import odeint from scipy import optimize
In [2]:
#Define five second order non-linear differential equations of Lane-Emden Equation #Represents n=0 to n=5 indexes
In [3]:
def dydt_dzdt(yz,t): n = 0 y = yz[0] z = yz[1] dydt = z dzdt = -(2/t)*z - y**n return [dydt, dzdt] #Define initial conditions y0 = 1 z0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 dz = 0.0001 t = np.arange(t0, tend, dz) #Call odeint yz = odeint(dydt_dzdt, [y0, z0], t)
In [4]:
x=0.00001 y=1. u=0.0 h=0.001
In [5]:
while (x<2.5): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**0)) x=x+h print(x,y)
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In [6]:
def dydt_dadt(ya,t): n = 1 y = ya[0] a = ya[1] dydt = a dadt = -(2/t)*a - y**n return [dydt, dadt] #Define initial conditions y0 = 1 a0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 da = 0.0001 t = np.arange(t0, tend, da) #Call odeint ya = odeint(dydt_dadt, [y0, a0], t)
In [7]:
x=0.00001 y=1. u=0.0 h=0.001
In [8]:
while (x<3.5): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**1)) x=x+h print(x,y)
0.00101 1.0 0.00201 0.999999 0.00301 0.9999989801980198 0.00401 0.9999979801005024 0.00501 0.9999966445206278 0.00601 0.9999949750673988 0.00701 0.9999929720659201 0.00801 0.9999906356256988 0.00901 0.9999879657945657 0.010010000000000002 0.9999849625972996 0.011010000000000002 0.9999816260485308 0.012010000000000003 0.9999779561579101 0.013010000000000004 0.9999739529324608 0.014010000000000005 0.9999696163777564 0.015010000000000006 0.9999649464985539 0.016010000000000007 0.9999599432991574 0.017010000000000008 0.9999546067836347 0.01801000000000001 0.9999489369559541 0.01901000000000001 0.9999429338200725 0.02001000000000001 0.9999365973799936 0.02101000000000001 0.9999299276398088 0.022010000000000012 0.9999229246037248 0.023010000000000013 0.9999155882760837 0.024010000000000014 0.9999079186613783 0.025010000000000015 0.999899915764262 0.026010000000000016 0.9998915795895573 0.027010000000000017 0.9998829101422624 0.028010000000000018 0.9998739074275551 0.02901000000000002 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In [9]:
def dydt_dbdt(yb,t): n = 2 y = yb[0] b = yb[1] dydt = b dbdt = -(2/t)*b - y**n return [dydt, dbdt] #Define initial conditions y0 = 1 b0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 db = 0.0001 t = np.arange(t0, tend, db) #Call odeint yb = odeint(dydt_dbdt, [y0, a0], t)
/ext/anaconda-2019.03/lib/python3.7/site-packages/scipy/integrate/odepack.py:247: ODEintWarning: Excess work done on this call (perhaps wrong Dfun type). Run with full_output = 1 to get quantitative information. warnings.warn(warning_msg, ODEintWarning)
In [10]:
x=0.00001 y=1. u=0.0 h=0.001
In [11]:
while (x<4.5): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**2)) x=x+h print(x,y)
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In [12]:
def dydt_dcdt(yc,t): n = 3 y = yc[0] c = yc[1] dydt = c dcdt = -(2/t)*c - y**n return [dydt, dcdt] #Define initial conditions y0 = 1 c0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 dc = 0.0001 t = np.arange(t0, tend, dc) #Call odeint yc = odeint(dydt_dcdt, [y0, a0], t)
In [13]:
x=0.00001 y=1. u=0.0 h=0.001
In [14]:
while (x<7): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**3)) x=x+h print(x,y)
WARNING: some intermediate output was truncated. WARNING: 2 intermediate output messages were discarded. WARNING: some intermediate output was truncated. WARNING: some intermediate output was truncated.
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WARNING: 2 intermediate output messages were discarded. WARNING: some intermediate output was truncated. WARNING: some intermediate output was truncated. WARNING: some intermediate output was truncated.
In [15]:
def dydt_dedt(ye,t): n = 4 y = ye[0] e = ye[1] dydt = e dedt = -(2/t)*e - y**n return [dydt, dedt] #Define initial conditions y0 = 1 e0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 de = 0.0001 t = np.arange(t0, tend, de) #Call odeint ye = odeint(dydt_dedt, [y0, e0], t)
In [16]:
x=0.00001 y=1. u=0.0 h=0.001
In [17]:
while (x<15.5): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**4)) x=x+h print(x,y)
WARNING: some intermediate output was truncated. WARNING: 11 intermediate output messages were discarded.
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WARNING: 11 intermediate output messages were discarded. WARNING: some intermediate output was truncated.
In [18]:
def dydt_dpdt(yp,t): n = 5 y = yp[0] p = yp[1] dydt = p dpdt = -(2/t)*p - y**n return [dydt, dpdt] #Define initial conditions y0 = 1 p0 = 0 #Define t (xi) t0 = 0.0001 tend = 16 dp = 0.0001 t = np.arange(t0, tend, dp) #Call odeint yp = odeint(dydt_dpdt, [y0, p0], t)
In [19]:
x=0.00001 y=1. u=0.0 h=0.001
In [20]:
while (x<15.5): yn=y y=y+(h*u) u=u+(h*((-(2./x)*u)-(yn)**5)) x=x+h print(x,y)
WARNING: some intermediate output was truncated. WARNING: 11 intermediate output messages were discarded.
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