SharedLab4-turnin.sagewsOpen in CoCalc
Author: Brian Clark
Views : 4
# Lab 4 # Name: Brian Clark # I worked on this code with: Christy Seguritan and Ana Leon # Please do all of your work for this week's lab in this worksheet. If # you wish to create other worksheets for scratch work, you can, but # this is the one that will be graded. You do not need to do anything # to turn in your lab. It will be collected by your TA at the beginning # of (or right before) next week’s lab. # Be sure to clearly label which question you are answering as you go and to # use enough comments that you and the grader can understand your code.
#1 # Step 1: Start with defining the differential equation # Step 2: Plug in the initial N value into the differential equation # Step 3: Compute the new N by adding the initial N + stepsize*N' # Step 4: Save the new N and add it a storage # Step 5: Treat the new N as the intial N and repeat from step 2
#2.a N_prime(n) = 0.2*n*(1-(n/100)) NewN(n) = Nprime(n)*0.1 + n NewN(10)
10.1800000000000
#2.b N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 N_init = [10] iterations = srange(0,1000) for i in iterations: n_new = N_init[i]+delta_t*N_prime(N_init[i]) N_init.append(n_new) N_init
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#3 N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 N_init = [101] iterations = srange(0,1000) for i in iterations: n_new = N_init[i]+delta_t*N_prime(N_init[i]) N_init.append(n_new) N_init
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N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 N_init = [310] iterations = srange(0,1000) for i in iterations: n_new = N_init[i]+delta_t*N_prime(N_init[i]) N_init.append(n_new) N_init
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N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 N_init = [1001] iterations = srange(0,1000) for i in iterations: n_new = N_init[i]+delta_t*N_prime(N_init[i]) N_init.append(n_new) N_init
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#4.a N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 listpop= [10] count = srange(0,100,.1) for i in iterations: n_new = listpop[i]+delta_t*N_prime(listpop[i]) listpop.append(n_new) list_plot(zip(count,listpop), axes_labels=["time","population"]) N_prime(n) = 0.2*n*(1-(n/100)) delta_t = .1 listpop= [10] count = srange(0,100,.1) for i in iterations: n_new = listpop[i]+delta_t*N_prime(listpop[i]) listpop.append(n_new) list_plot((count,listpop), axes_labels=["time","population"]) # If you omit "zip" it would not display a graph, but rather a 3-d object. The object being displayed is a cube.
3D rendering not yet implemented
#5 def euler(differentialeq,initialval,stepsize,numsteps): vals = [initialval] tfinal= numsteps*stepsize for i in srange(0,tfinal,stepsize): vals.append(differentialeq(vals[-1])*stepsize+vals[-1]) return vals euler(.2*x*(1-x/100),10,.1,100)
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#6 f(N) = .2*N(1-(N/100)) N = [10] h = .1 for i in srange (0,100,h): Nnew = N[-1] + f(N[-1])*h N.append(Nnew) tlist = srange(0,100,h) zip(tlist,N)
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#7 def euler(differentialeq,initialval,stepsize,maxsize): nvals = [initialval] count = srange(0, maxsize) for i in count: a = differentialeq(nvals[-1]) nvals.append(nvals[-1]+a*stepsize) return zip(count, nvals) differentialeq(n) = .2*n(1-n/100) euler(differentialeq,10,.1,100) list_plot(zip(count,listpop),axes_labels= ["time","state values"])
[(0, 10), (1, 10.0180000000000), (2, 10.0359964000000), (3, 10.0539892007200), (4, 10.0719784028799), (5, 10.0899640071993), (6, 10.1079460143978), (7, 10.1259244251950), (8, 10.1438992403099), (9, 10.1618704604619), (10, 10.1798380863698), (11, 10.1978021187525), (12, 10.2157625583287), (13, 10.2337194058171), (14, 10.2516726619359), (15, 10.2696223274035), (16, 10.2875684029380), (17, 10.3055108892575), (18, 10.3234497870796), (19, 10.3413850971222), (20, 10.3593168201028), (21, 10.3772449567387), (22, 10.3951695077474), (23, 10.4130904738459), (24, 10.4310078557511), (25, 10.4489216541799), (26, 10.4668318698491), (27, 10.4847385034751), (28, 10.5026415557744), (29, 10.5205410274633), (30, 10.5384369192578), (31, 10.5563292318739), (32, 10.5742179660276), (33, 10.5921031224344), (34, 10.6099847018099), (35, 10.6278627048695), (36, 10.6457371323285), (37, 10.6636079849021), (38, 10.6814752633051), (39, 10.6993389682524), (40, 10.7171991004588), (41, 10.7350556606387), (42, 10.7529086495066), (43, 10.7707580677767), (44, 10.7886039161631), (45, 10.8064461953799), (46, 10.8242849061408), (47, 10.8421200491596), (48, 10.8599516251497), (49, 10.8777796348247), (50, 10.8956040788977), (51, 10.9134249580820), (52, 10.9312422730903), (53, 10.9490560246357), (54, 10.9668662134308), (55, 10.9846728401881), (56, 11.0024759056201), (57, 11.0202754104389), (58, 11.0380713553569), (59, 11.0558637410858), (60, 11.0736525683376), (61, 11.0914378378239), (62, 11.1092195502563), (63, 11.1269977063463), (64, 11.1447723068050), (65, 11.1625433523437), (66, 11.1803108436732), (67, 11.1980747815045), (68, 11.2158351665482), (69, 11.2335919995148), (70, 11.2513452811149), (71, 11.2690950120587), (72, 11.2868411930563), (73, 11.3045838248177), (74, 11.3223229080527), (75, 11.3400584434711), (76, 11.3577904317824), (77, 11.3755188736961), (78, 11.3932437699213), (79, 11.4109651211673), (80, 11.4286829281431), (81, 11.4463971915575), (82, 11.4641079121192), (83, 11.4818150905367), (84, 11.4995187275186), (85, 11.5172188237731), (86, 11.5349153800084), (87, 11.5526083969324), (88, 11.5702978752530), (89, 11.5879838156779), (90, 11.6056662189148), (91, 11.6233450856710), (92, 11.6410204166539), (93, 11.6586922125706), (94, 11.6763604741280), (95, 11.6940252020332), (96, 11.7116863969928), (97, 11.7293440597134), (98, 11.7469981909015), (99, 11.7646487912633)]
#8 EulersX= euler(-2.1*x,1,1,10) list_plot(EulersX,axes_labels=["Time","X"],plotjoined=True) # Oscillation occurs within the positive values as it becomes increasingly more positive and negative as the value time goes on.
#9 f(x)=-2.1*x def euler(diffeq,initial,f,time): newn=[diffeq] hold=[] for i in srange(0,initial,time): hold.append(i) newn.append(newn[-1]+(.1*f(newn[-1]))) return list_plot(zip(newn,hold),plotjoined=true) euler(10,10,f,.5) #Using a function thats takes a final time instead of a number of steps as an input