CoCalc Shared FilesPHY213 Coursework / CourseworkGM.ipynb
Author: Gabriel Malvestuto
Views : 32
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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0.07840015669721793 2.353009999999852 0.07761615336388582 2.354009999999852 0.07683181669722036 2.3550099999998517 0.07604714669722157 2.3560099999998516 0.07526214336388945 2.3570099999998515 0.074476806697224 2.3580099999998514 0.0736911366972252 2.3590099999998513 0.07290513336389307 2.360009999999851 0.07211879669722761 2.361009999999851 0.07133212669722881 2.362009999999851 0.07054512336389668 2.363009999999851 0.06975778669723122 2.3640099999998507 0.06897011669723242 2.3650099999998506 0.06818211336390029 2.3660099999998505 0.06739377669723481 2.3670099999998504 0.06660510669723602 2.3680099999998503 0.06581610336390388 2.36900999999985 0.0650267666972384 2.37000999999985 0.0642370966972396 2.37100999999985 0.06344709336390747 2.37200999999985 0.062656756697242 2.3730099999998497 0.06186608669724319 2.3740099999998496 0.06107508336391104 2.3750099999998495 0.060283746697245566 2.3760099999998494 0.05949207669724676 2.3770099999998493 0.05870007336391461 2.378009999999849 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2.4040099999998463 0.03718998336394635 2.405009999999846 0.03638864669728084 2.406009999999846 0.035586976697282 2.407009999999846 0.03478497336394983 2.408009999999846 0.033982636697284324 2.4090099999998458 0.033179966697285485 2.4100099999998457 0.03237696336395331 2.4110099999998456 0.0315736266972878 2.4120099999998454 0.030769956697288953 2.4130099999998453 0.029965953363956774 2.4140099999998452 0.02916161669729126 2.415009999999845 0.028356946697292412 2.416009999999845 0.02755194336396023 2.417009999999845 0.026746606697294714 2.418009999999845 0.025940936697295865 2.4190099999998447 0.02513493336396368 2.4200099999998446 0.02432859669729816 2.4210099999998445 0.023521926697299307 2.4220099999998443 0.02271492336396712 2.4230099999998442 0.021907586697301598 2.424009999999844 0.021099916697302743 2.425009999999844 0.020291913363970553 2.426009999999844 0.01948357669730503 2.427009999999844 0.01867490669730617 2.4280099999998437 0.017865903363973975 2.4290099999998436 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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
#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 0.9998645714507971 0.03001000000000002 0.9998549022175368 0.03101000000000002 0.9998448997335113 0.03201000000000002 0.9998345640046487 0.03301000000000002 0.9998238950370689 0.03401000000000002 0.9998128928370857 0.03501000000000002 0.9998015574112071 0.03601000000000002 0.9997898887661363 0.03701000000000002 0.9997778869087722 0.03801000000000002 0.9997655518462106 0.039010000000000024 0.9997528835857438 0.040010000000000025 0.9997398821348614 0.041010000000000026 0.999726547501251 0.042010000000000026 0.9997128796927977 0.04301000000000003 0.9996988787175849 0.04401000000000003 0.9996845445838943 0.04501000000000003 0.9996698773002063 0.04601000000000003 0.9996548768751997 0.04701000000000003 0.9996395433177522 0.04801000000000003 0.9996238766369404 0.04901000000000003 0.99960787684204 0.050010000000000034 0.9995915439425256 0.051010000000000034 0.9995748779480711 0.052010000000000035 0.9995578788685495 0.053010000000000036 0.9995405467140333 0.05401000000000004 0.9995228814947938 0.05501000000000004 0.9995048832213024 0.05601000000000004 0.9994865519042292 0.05701000000000004 0.9994678875544442 0.05801000000000004 0.9994488901830165 0.05901000000000004 0.9994295598012151 0.06001000000000004 0.9994098964205079 0.06101000000000004 0.9993899000525629 0.062010000000000044 0.9993695707092471 0.06301000000000004 0.9993489084026276 0.06401000000000004 0.9993279131449704 0.06501000000000004 0.9993065849487415 0.06601000000000004 0.9992849238266063 0.06701000000000004 0.9992629297914297 0.06801000000000004 0.9992406028562764 0.06901000000000004 0.9992179430344102 0.07001000000000004 0.9991949503392952 0.07101000000000005 0.9991716247845942 0.07201000000000005 0.9991479663841701 0.07301000000000005 0.9991239751520854 0.07401000000000005 0.9990996511026018 0.07501000000000005 0.9990749942501809 0.07601000000000005 0.9990500046094836 0.07701000000000005 0.9990246821953703 0.07801000000000005 0.9989990270229013 0.07901000000000005 0.9989730391073361 0.08001000000000005 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0.9971635940255734 0.1320100000000001 0.9971199977934502 0.1330100000000001 0.9970760699382585 0.1340100000000001 0.997031810486214 0.1350100000000001 0.9969872194637315 0.1360100000000001 0.9969422968974242 0.1370100000000001 0.9968970428141042 0.1380100000000001 0.9968514572407823 0.1390100000000001 0.9968055402046679 0.1400100000000001 0.9967592917331692 0.1410100000000001 0.9967127118538931 0.1420100000000001 0.9966658005946449 0.1430100000000001 0.9966185579834288 0.1440100000000001 0.9965709840484474 0.1450100000000001 0.9965230788181019 0.1460100000000001 0.9964748423209921 0.1470100000000001 0.9964262745859165 0.1480100000000001 0.9963773756418717 0.14901000000000011 0.9963281455180532 0.15001000000000012 0.9962785842438547 0.15101000000000012 0.9962286918488686 0.15201000000000012 0.9961784683628856 0.15301000000000012 0.9961279138158948 0.15401000000000012 0.9960770282380838 0.15501000000000012 0.9960258116598382 0.15601000000000012 0.9959742641117424 0.15701000000000012 0.995922385624579 0.15801000000000012 0.9958701762293289 0.15901000000000012 0.9958176359571712 0.16001000000000012 0.9957647648394832 0.16101000000000013 0.9957115629078406 0.16201000000000013 0.9956580301940173 0.16301000000000013 0.9956041667299854 0.16401000000000013 0.995549972547915 0.16501000000000013 0.9954954476801745 0.16601000000000013 0.9954405921593305 0.16701000000000013 0.9953854060181474 0.16801000000000013 0.9953298892895881 0.16901000000000013 0.9952740420068132 0.17001000000000013 0.9952178642031816 0.17101000000000013 0.99516135591225 0.17201000000000014 0.995104517167773 0.17301000000000014 0.9950473480037035 0.17401000000000014 0.9949898484541921 0.17501000000000014 0.9949320185535875 0.17601000000000014 0.9948738583364359 0.17701000000000014 0.9948153678374818 0.17801000000000014 0.9947565470916673 0.17901000000000014 0.9946973961341323 0.18001000000000014 0.9946379150002148 0.18101000000000014 0.9945781037254501 0.18201000000000014 0.9945179623455714 0.18301000000000014 0.9944574908965099 0.18401000000000015 0.9943966894143942 0.18501000000000015 0.9943355579355505 0.18601000000000015 0.9942740964965029 0.18701000000000015 0.9942123051339729 0.18801000000000015 0.9941501838848795 0.18901000000000015 0.9940877327863397 0.19001000000000015 0.9940249518756675 0.19101000000000015 0.9939618411903749 0.19201000000000015 0.9938984007681709 0.19301000000000015 0.9938346306469623 0.19401000000000015 0.9937705308648531 0.19501000000000016 0.9937061014601449 0.19601000000000016 0.9936413424713365 0.19701000000000016 0.9935762539371241 0.19801000000000016 0.9935108358964013 0.19901000000000016 0.9934450883882588 0.20001000000000016 0.9933790114519848 0.20101000000000016 0.9933126051270645 0.20201000000000016 0.9932458694531805 0.20301000000000016 0.9931788044702123 0.20401000000000016 0.993111410218237 0.20501000000000016 0.9930436867375284 0.20601000000000017 0.9929756340685575 0.20701000000000017 0.9929072522519925 0.20801000000000017 0.9928385413286984 0.20901000000000017 0.9927695013397375 0.21001000000000017 0.9927001323263688 0.21101000000000017 0.9926304343300484 0.21201000000000017 0.9925604073924292 0.21301000000000017 0.9924900515553612 0.21401000000000017 0.9924193668608909 0.21501000000000017 0.992348353351262 0.21601000000000017 0.9922770110689149 0.21701000000000017 0.9922053400564865 0.21801000000000018 0.9921333403568108 0.21901000000000018 0.9920610120129182 0.22001000000000018 0.9919883550680361 0.22101000000000018 0.9919153695655881 0.22201000000000018 0.9918420555491949 0.22301000000000018 0.9917684130626735 0.22401000000000018 0.9916944421500374 0.22501000000000018 0.9916201428554968 0.22601000000000018 0.9915455152234584 0.22701000000000018 0.991470559298525 0.22801000000000018 0.9913952751254963 0.22901000000000019 0.9913196627493681 0.2300100000000002 0.9912437222153327 0.2310100000000002 0.9911674535687788 0.2320100000000002 0.991090856855291 0.2330100000000002 0.9910139321206505 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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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