+
+
+
-
-
-
- An Intro to DifferentialEquations.jl
+Chris Rackauckas
+ +
-
-
-
-
+
+
-
- 
+
+
+
+
+
+
+
+
-
-
- An Intro to DifferentialEquations.jl
-Chris Rackauckas
- -Basic Introduction Via Ordinary Differential Equations
+Basic Introduction Via Ordinary Differential Equations
This notebook will get you started with DifferentialEquations.jl by introducing you to the functionality for solving ordinary differential equations (ODEs). The corresponding documentation page is the ODE tutorial. While some of the syntax may be different for other types of equations, the same general principles hold in each case. Our goal is to give a gentle and thorough introduction that highlights these principles in a way that will help you generalize what you have learned.
Background
If you are new to the study of differential equations, it can be helpful to do a quick background read on the definition of ordinary differential equations. We define an ordinary differential equation as an equation which describes the way that a variable $u$ changes, that is
@@ -730,17 +727,17 @@First Model: Exponential Growth
retcode: Success Interpolation: Automatic order switching interpolation t: 5-element Array{Float64,1}: - 0.0 + 0.0 0.10042494449239292 0.35218603951893646 - 0.6934436028208104 - 1.0 + 0.6934436028208104 + 1.0 u: 5-element Array{Float64,1}: - 1.0 + 1.0 1.1034222047865465 1.4121908848175448 - 1.9730384275623003 - 2.664456142481452 + 1.9730384275622996 + 2.664456142481451 @@ -755,7 +752,7 @@Analyzing the Solution
-
+
From the picture we see that the solution is an exponential curve, which matches our intuition. As a plot recipe, we can annotate the result using any of the Plots.jl attributes. For example:
@@ -766,7 +763,7 @@Analyzing the Solution
-
+
Using the mutating plot! command we can add other pieces to our plot. For this ODE we know that the true solution is $u(t) = u_0 exp(at)$, so let's add some of the true solution to our plot:
Analyzing the Solution
-
+
In the previous command I demonstrated sol.t, which grabs the array of time points that the solution was saved at:
Analyzing the Solution
5-element Array{Float64,1}:
- 0.0
+ 0.0
0.10042494449239292
0.35218603951893646
- 0.6934436028208104
+ 0.6934436028208104
1.0
@@ -806,11 +803,11 @@ Analyzing the Solution
5-element Array{Float64,1}:
- 1.0
+ 1.0
1.1034222047865465
1.4121908848175448
- 1.9730384275623003
- 2.664456142481452
+ 1.9730384275622996
+ 2.664456142481451
@@ -824,11 +821,11 @@ Analyzing the Solution
5-element Array{Float64,1}:
- 1.0
+ 1.0
1.2038471492789395
1.7643769243364813
- 2.6664820303831105
- 3.664456142481452
+ 2.66648203038311
+ 3.664456142481451
@@ -844,17 +841,17 @@ Analyzing the Solution
retcode: Success Interpolation: Automatic order switching interpolation t: 5-element Array{Float64,1}: - 0.0 + 0.0 0.10042494449239292 0.35218603951893646 - 0.6934436028208104 - 1.0 + 0.6934436028208104 + 1.0 u: 5-element Array{Float64,1}: - 1.0 + 1.0 1.1034222047865465 1.4121908848175448 - 1.9730384275623003 - 2.664456142481452 + 1.9730384275622996 + 2.664456142481451 @@ -867,7 +864,7 @@Analyzing the Solution
-1.5542610480553116 +1.554261048055312@@ -885,24 +882,24 @@
Controlling the Solver
retcode: Success Interpolation: Automatic order switching interpolation t: 9-element Array{Float64,1}: - 0.0 + 0.0 0.04127492324135852 0.14679917846877366 0.28631546412766684 - 0.4381941361169628 - 0.6118924302028597 - 0.7985659100883337 - 0.9993516479536952 - 1.0 + 0.4381941361169628 + 0.6118924302028597 + 0.7985659100883337 + 0.9993516479536952 + 1.0 u: 9-element Array{Float64,1}: - 1.0 + 1.0 1.0412786454705882 1.1547261252949712 1.3239095703537043 1.5363819257509728 1.8214895157178692 2.1871396448296223 - 2.662763824115295 + 2.662763824115295 2.664456241933517 @@ -916,7 +913,7 @@Controlling the Solver
-
+
Notice that by decreasing the tolerance, the number of steps the solver had to take was 9 instead of the previous 5. There is a trade off between accuracy and speed, and it is up to you to determine what is the right balance for your problem.
Another common option is to use saveat to make the solver save at specific time points. For example, if we want the solution at an even grid of t=0.1k for integers k, we would use the command:
Controlling the Solver
0.9 1.0 u: 11-element Array{Float64,1}: - 1.0 - 1.1029627851292922 + 1.0 + 1.102962785129292 1.2165269512238264 - 1.341783821227542 + 1.341783821227542 1.4799379510586077 - 1.6323162070541606 - 1.8003833264983586 - 1.9857565541588764 - 2.1902158127997704 - 2.4157257420844966 - 2.664456142481452 + 1.632316207054161 + 1.8003833264983584 + 1.9857565541588758 + 2.1902158127997695 + 2.415725742084496 + 2.664456142481451 @@ -974,8 +971,8 @@Controlling the Solver
0.9 u: 3-element Array{Float64,1}: 1.2165269512238264 - 1.9857565541588764 - 2.4157257420844966 + 1.9857565541588758 + 2.415725742084496 @@ -991,17 +988,17 @@Controlling the Solver
retcode: Success Interpolation: 1st order linear t: 5-element Array{Float64,1}: - 0.0 + 0.0 0.10042494449239292 0.35218603951893646 - 0.6934436028208104 - 1.0 + 0.6934436028208104 + 1.0 u: 5-element Array{Float64,1}: - 1.0 + 1.0 1.1034222047865465 1.4121908848175448 - 1.9730384275623003 - 2.664456142481452 + 1.9730384275622996 + 2.664456142481451 @@ -1020,8 +1017,8 @@Controlling the Solver
0.0 1.0 u: 2-element Array{Float64,1}: - 1.0 - 2.664456142481452 + 1.0 + 2.664456142481451 @@ -1039,7 +1036,7 @@Controlling the Solver
t: 1-element Array{Float64,1}: 1.0 u: 1-element Array{Float64,1}: - 2.664456142481452 + 2.664456142481451 @@ -1059,23 +1056,23 @@Choosing Solver Algorithms
retcode: Success Interpolation: specialized 3rd order "free" stiffness-aware interpolation t: 8-element Array{Float64,1}: - 0.0 + 0.0 0.05653299582822294 0.17270731152826024 - 0.3164602871490142 - 0.5057500163821153 - 0.7292241858994543 - 0.9912975001018789 - 1.0 + 0.3164602871490142 + 0.5057500163821153 + 0.7292241858994543 + 0.9912975001018789 + 1.0 u: 8-element Array{Float64,1}: - 1.0 + 1.0 1.0569657840332976 1.1844199383303913 1.3636037723365293 1.6415399686182572 - 2.043449143475479 - 2.6418256160577602 - 2.6644526430553808 + 2.0434491434754793 + 2.641825616057761 + 2.6644526430553817 @@ -1092,27 +1089,27 @@Choosing Solver Algorithms
retcode: Success Interpolation: specialized 4th order "free" interpolation t: 10-element Array{Float64,1}: - 0.0 + 0.0 0.028970819746309166 - 0.10049147151547619 - 0.19458908698515082 - 0.3071725081673423 - 0.43945421453622546 - 0.5883434923759523 - 0.7524873357619015 - 0.9293021330536031 - 1.0 + 0.10049147151547619 + 0.19458908698515082 + 0.3071725081673423 + 0.43945421453622546 + 0.5883434923759523 + 0.7524873357619015 + 0.9293021330536031 + 1.0 u: 10-element Array{Float64,1}: - 1.0 + 1.0 1.0287982807225062 1.1034941463604806 1.2100931078233779 - 1.351248605624241 - 1.538280340326815 + 1.351248605624241 + 1.538280340326815 1.7799346012651116 - 2.0905717422346277 - 2.4861021714470244 - 2.6644562434913373 + 2.090571742234628 + 2.486102171447025 + 2.6644562434913377 @@ -1200,46 +1197,46 @@Systems of ODEs: The Lorenz Equation
retcode: Success Interpolation: Automatic order switching interpolation t: 1294-element Array{Float64,1}: - 0.0 + 0.0 3.5678604836301404e-5 0.0003924646531993154 0.0032624077544510573 - 0.009058075635317072 - 0.01695646895607931 - 0.0276899566248403 - 0.041856345938267966 - 0.06024040228733675 - 0.08368539694547242 - ⋮ - 99.39403070915297 - 99.47001147494375 - 99.54379656909015 - 99.614651558349 - 99.69093823148101 - 99.78733023233721 - 99.86114450046736 - 99.96115759510786 - 100.0 + 0.009058075635317072 + 0.01695646895607931 + 0.0276899566248403 + 0.041856345938267966 + 0.06024040228733675 + 0.08368539694547242 + ⋮ + 99.39403070915297 + 99.47001147494375 + 99.54379656909015 + 99.614651558349 + 99.69093823148101 + 99.78733023233721 + 99.86114450046736 + 99.96115759510786 + 100.0 u: 1294-element Array{Array{Float64,1},1}: - [1.0, 0.0, 0.0] + [1.0, 0.0, 0.0] [0.9996434557625105, 0.0009988049817849058, 1.781434788799208e-8] - [0.9961045497425811, 0.010965399721242457, 2.146955365838907e-6] - [0.9693591634199452, 0.08977060667778931, 0.0001438018342266937] - [0.9242043615038835, 0.24228912482984957, 0.0010461623302512404] - [0.8800455868998046, 0.43873645009348244, 0.0034242593451028745] - [0.8483309877783048, 0.69156288756671, 0.008487623500490047] - [0.8495036595681027, 1.0145425335433382, 0.01821208597613427] - [0.9139069079152129, 1.4425597546855036, 0.03669381053327124] - [1.0888636764765296, 2.052326153029042, 0.07402570506414284] - ⋮ - [12.999157033749652, 14.10699925404482, 31.74244844521858] - [11.646131422021162, 7.2855792145502845, 35.365000488215486] - [7.777555445486692, 2.5166095828739574, 32.030953593541675] - [4.739741627223412, 1.5919220588229062, 27.249779003951755] - [3.2351668945618774, 2.3121727966182695, 22.724936101772805] - [3.310411964698304, 4.28106626744641, 18.435441144016366] - [4.527117863517627, 6.895878639772805, 16.58544600757436] - [8.043672261487556, 12.711555298531689, 18.12537420595938] + [0.9961045497425811, 0.010965399721242457, 2.146955365838907e-6] + [0.9693591634199452, 0.08977060667778931, 0.0001438018342266937] + [0.9242043615038835, 0.24228912482984957, 0.0010461623302512404] + [0.8800455868998046, 0.43873645009348244, 0.0034242593451028745] + [0.8483309877783048, 0.69156288756671, 0.008487623500490047] + [0.8495036595681027, 1.0145425335433382, 0.01821208597613427] + [0.9139069079152129, 1.4425597546855036, 0.03669381053327124] + [1.0888636764765296, 2.052326153029042, 0.07402570506414284] + ⋮ + [12.999157033749652, 14.10699925404482, 31.74244844521858] + [11.646131422021162, 7.2855792145502845, 35.365000488215486] + [7.777555445486692, 2.5166095828739574, 32.030953593541675] + [4.739741627223412, 1.5919220588229062, 27.249779003951755] + [3.2351668945618774, 2.3121727966182695, 22.724936101772805] + [3.310411964698304, 4.28106626744641, 18.435441144016366] + [4.527117863517627, 6.895878639772805, 16.58544600757436] + [8.043672261487556, 12.711555298531689, 18.12537420595938] [9.97537965430362, 15.143884806010783, 21.00643286956427] @@ -1283,7 +1280,7 @@Systems of ODEs: The Lorenz Equation
3×1294 Array{Float64,2}:
1.0 0.999643 0.996105 0.969359 … 4.52712 8.04367 9.97538
- 0.0 0.000998805 0.0109654 0.0897706 6.89588 12.7116 15.1439
+ 0.0 0.000998805 0.0109654 0.0897706 6.89588 12.7116 15.1439
0.0 1.78143e-8 2.14696e-6 0.000143802 16.5854 18.1254 21.0064
@@ -1296,7 +1293,7 @@ Systems of ODEs: The Lorenz Equation
-
+
If we instead want to plot values against each other, we can use the vars command. Let's plot variable 1 against variable 2 against variable 3:
Systems of ODEs: The Lorenz Equation
-
+
This is the classic Lorenz attractor plot, where the x axis is u[1], the y axis is u[2], and the z axis is u[3]. Note that the plot recipe by default uses the interpolation, but we can turn this off:
Systems of ODEs: The Lorenz Equation
-
+
Yikes! This shows how calculating the continuous solution has saved a lot of computational effort by computing only a sparse solution and filling in the values! Note that in vars, 0=time, and thus we can plot the time series of a single component like:
Systems of ODEs: The Lorenz Equation
-
+
Internal Types
The last basic user-interface feature to explore is the choice of types. DifferentialEquations.jl respects your input types to determine the internal types that are used. Thus since in the previous cases, when we used Float64 values for the initial condition, this meant that the internal values would be solved using Float64. We made sure that time was specified via Float64 values, meaning that time steps would utilize 64-bit floats as well. But, by simply changing these types we can change what is used internally.
Internal Types
retcode: Success
Interpolation: Automatic order switching interpolation
-t: 11-element Array{Float64,1}:
- 0.0
- 0.04315877410281615
- 0.10749821507777302
- 0.1798183032337637
- 0.27732622431309095
- 0.3943300853390046
- 0.5271275340166153
- 0.6593505063074551
- 0.7998387276496801
- 0.9661257382228796
- 1.0
-u: 11-element Array{Array{Float64,2},1}:
- [0.306624934095197 0.5338066435533153; 0.19874765604983802 0.6868676641729
-416; 0.6862329122068349 0.13858963914056321; 0.2031754356279607 0.805431744
-4734056]
- [0.2607854140452007 0.3613260168142186; 0.3061733660115813 0.6130920806716
-098; 0.6484554422419746 0.09909176566962785; 0.33528190215003106 0.97147927
-48298231]
- [0.13596497914872052 0.02539146709353668; 0.39275624859948677 0.4140338403
-5184776; 0.6235490412546256 0.1171227635342626; 0.5191531472173252 1.191169
-1282105947]
- [-0.08267310627767702 -0.4573896273601639; 0.3952152446091268 0.0883641473
-6267606; 0.6581128735546947 0.27062599647166163; 0.6975117123588294 1.38567
-9442209257]
- [-0.49551765265574466 -1.2570950504304415; 0.27500883360229983 -0.45014272
-42424515; 0.8439611836541472 0.7435773073443797; 0.8693546862420823 1.53201
-7905622345]
- [-1.1251069448214652 -2.3625200085353635; 0.037325541951849256 -1.08123128
-50121293; 1.3260226380299647 1.765626779016673; 0.9330006745006937 1.480132
-4753451832]
- [-1.9045113695239797 -3.6235436930157006; -0.16220499769796015 -1.46363209
-84855904; 2.246749925213862 3.5332898325322093; 0.7601909703615497 1.043513
-9453705078]
- [-2.5636091056660977 -4.5900120441897645; -0.02547223902998061 -1.09470953
-18805466; 3.510744970192141 5.809641306139268; 0.2774234486076798 0.1395762
-174433236]
- [-2.8588358769826323 -4.8675661833207755; 0.7924931322952213 0.55502678447
-71868; 5.042857181653392 8.434977730075753; -0.6037853546431378 -1.36780693
-54307625]
- [-2.2056076203133 -3.4860281122307057; 3.002992564478892 4.630325770000123
-; 6.583433020832067 10.916694329795456; -2.0967627480607884 -3.801465962226
-624]
- [-1.8905678647543251 -2.903621089761297; 3.6371785566754515 5.759661435694
-562; 6.784809971693031 11.212684216367514; -2.446836669801699 -4.3599865292
-66994]
+t: 10-element Array{Float64,1}:
+ 0.0
+ 0.04507941711835774
+ 0.11847045464115312
+ 0.20200309476486217
+ 0.3080378088437111
+ 0.439160679844409
+ 0.5919747863019895
+ 0.7562936017032477
+ 0.9265969111145607
+ 1.0
+u: 10-element Array{Array{Float64,2},1}:
+ [0.6476678183071587 0.41891793301301194; 0.3373999272887971 0.963528121360
+1437; 0.5468425532557133 0.17626705828817868; 0.9493868101674328 0.09487393
+434072056]
+ [0.4256767292585334 0.41231921313397935; 0.34922238566772923 0.96187293443
+56894; 0.4982465643013043 0.10627425240908965; 1.234722752676834 0.12970737
+289976686]
+ [-0.09290252108890362 0.38465587787593475; 0.19096610037471526 0.921693193
+9107806; 0.5510997009878782 0.00020087347413948187; 1.6534348612956478 0.17
+966764893036533]
+ [-0.9065853042010712 0.3296827654989794; -0.20684622708033182 0.8262651140
+896179; 0.8665055551385252 -0.10280223723656501; 2.0294639518599102 0.22508
+53888535099]
+ [-2.2273319234509366 0.2291244752377739; -0.9053834435996413 0.64545781711
+1385; 1.7526878217563326 -0.19631004525493106; 2.2883756480783064 0.2632704
+732032627]
+ [-4.128848100344415 0.06959791120144104; -1.7172505457746334 0.36581564566
+38496; 3.7098223854217487 -0.24009807935656147; 2.150642100420188 0.2779803
+867598425]
+ [-6.257657785592031 -0.1397156646055227; -1.863819609800419 0.028627362227
+357456; 7.171666971637375 -0.17830359561673753; 1.1676069082173322 0.247516
+76873745765]
+ [-7.547964822508019 -0.3506781612472547; 0.022210478300666026 -0.255165558
+48465294; 11.823459685438586 0.019068400607233082; -1.0493451877105624 0.15
+660850772605645]
+ [-6.4608155283414375 -0.4955214737958298; 5.320972335742665 -0.36600230940
+56092; 16.312436567840216 0.32767097272117; -4.609304249155667 0.0007086456
+408165187]
+ [-4.854234662231447 -0.5178857881046955; 8.819130783690689 -0.336592437141
+38454; 17.587712014096375 0.4741650930425332; -6.45918723825745 -0.08411990
+025979836]
@@ -1408,10 +1401,10 @@ Internal Types
4×2 Array{Float64,2}:
- 0.135965 0.0253915
- 0.392756 0.414034
- 0.623549 0.117123
- 0.519153 1.19117
+ -0.0929025 0.384656
+ 0.190966 0.921693
+ 0.5511 0.000200873
+ 1.65343 0.179668
@@ -1425,10 +1418,10 @@ Internal Types
4×2 Array{BigFloat,2}:
- 0.306625 0.533807
- 0.198748 0.686868
- 0.686233 0.13859
- 0.203175 0.805432
+ 0.647668 0.418918
+ 0.3374 0.963528
+ 0.546843 0.176267
+ 0.949387 0.0948739
@@ -1444,68 +1437,55 @@ Internal Types
retcode: Success
Interpolation: Automatic order switching interpolation
-t: 6-element Array{Float64,1}:
- 0.0
- 0.11309299507432245
- 0.34314692151136
- 0.638787912178119
- 0.9249800093396419
- 1.0
-u: 6-element Array{Array{BigFloat,2},1}:
- [0.306624934095196977068553678691387176513671875 0.53380664355331530757098
-3350160531699657440185546875; 0.1987476560498380173669374926248565316200256
-34765625 0.6868676641729416498804994262172840535640716552734375; 0.68623291
-22068349107621543225832283496856689453125 0.1385896391405632144255832827184
-3492984771728515625; 0.203175435627960698781180326477624475955963134765625
-0.805431744473405597517512433114461600780487060546875]
-
-
-
- [0.12195360456574192164308358623413396755370929614133176343692674857595920
-89283282 -0.008123130290881944410204034456520484439167574288627050549707058
-882818710342921869; 0.39634232463333060475342878723397023611671692453560022
-20324236305738155185910297 0.3922711469618407524571385921694548208680191059
-529293489153371312310947138743918; 0.62360874142229614660269413816440946870
-87370580824993887652107972174806719590678 0.1236409384743709259797972837855
-357447990439632298887694654224199400105772431161; 0.53416347348269343867994
-57018811739572433059005299242711070019148084150545073337 1.2083846743849025
-18564024515351644107890176707970397695234248206414887624922368]
- [-0.8361838634484524348538168488462141801511078402805525410153586829197726
-692808571 -1.86702009280275907517976463220718291159544885656355164272253725
-3422383845174623; 0.1453811433020334574482407049325057368670470738112201107
-235484165871856652496102 -0.82134478429799790787504938541531905469749406674
-00386286734161179476112442392394; 1.077771068365623680961642145379083157281
-57412606957580092742711657289164482844 1.2553877009621635276525343870881896
-86801854167614311182221002263583654514473809; 0.927145045985413897479125565
-3032764995985712313551139615976505988068257946842391 1.53723087864554990641
-4797277498265984897737315208334480701715945474510318806641]
- [-2.4781841695365935139189432377168542891767829957296609122941866219442182
-78896451 -4.473442094117027675809815446775958785211969683715454078985814799
-39549300369079; -0.08006364569705258042016347463273375867687218500322261325
-409698105571951190333812 -1.21777113075278823531712886538386589247841244623
-8940453398155952886774203517709; 3.2966315204870468195194565552806077969134
-38470665102008643034908539507467653621 5.4321316110811692416608908089319752
-02884242133404203141998361295001144659681931; 0.374349345424851256103484671
-6131550426370925361654115708204641548117654790737735 0.31282775172215700341
-93926449343762287138753819339916998491633864398598808831969]
- [-2.4975818474654782027132192616937496987044671987510889796984588261379768
-8739931 -4.0447100330112030751965498675045126574427038896461677180992339776
-29276787366896; 2.317330136264621470670536733728552342915045303595855998302
-506298055630934243305 3.393959417472666081039083941359688764565665061035505
-721605949015682598273199206; 6.27290836063364715834215598699423777433297061
-3699990873953739050279600565477863 10.4376619318291287301568866309489208220
-0057863455139268312561945612660615303942; -1.689055448264079339133670789291
-3297438964986327492426444109202910454414785522 -3.1459536741694805543316146
-72864919967432018745488031128508039324887463085414649]
- [-1.8905635653064640482754048965318043472960738675217969596201048475534655
-62833882 -2.903613161693912596462335234152228047992884765645552376599867304
-615746874344366; 3.63719468066918806341320932471563124940388939900708102008
-3798980087192646198465 5.75968789300718216160605225298983954645231221977570
-3753148103416886321220703276; 6.7848191643649295916304519941107790356057684
-47698696199506860813363339546739066 11.212699088749110671833971366870494546
-78391245938108692830749133496976192597072; -2.44684488609446281136992003045
-2810812989399961122674021029527293419007432081828 -4.3600004123377005404380
-03490097475473389116618818531156433722540001840908844477]
+t: 5-element Array{Float64,1}:
+ 0.0
+ 0.17497857103926084
+ 0.4439275929023296
+ 0.7642242008773308
+ 1.0
+u: 5-element Array{Array{BigFloat,2},1}:
+ [0.647667818307158693613700961577706038951873779296875 0.41891793301301194
+49648873342084698379039764404296875; 0.337399927288797085722649171657394617
+7959442138671875 0.9635281213601436522964149844483472406864166259765625; 0.
+5468425532557132928701548735261894762516021728515625 0.17626705828817867960
+36749365157447755336761474609375; 0.949386810167432759755001825396902859210
+968017578125 0.094873934340720555979942218982614576816558837890625]
+ [-0.6188960718824867728111531556212146524949095780924259719976415096764764
+328853145 0.350025724727944325527352372764937485241365601570147495385611666
+7637687099794473; -0.057412938226884432316635849574696464500327781088923530
+80169171374688759803887639 0.8623492198341470171269830534140466995712928986
+42001530629858438609707541610597; 0.730830422039167874412886202539929181834
+9160050300403416934309517614885038098399 -0.0719500558190700839855463189018
+58825992818683047468246048115314298322948926955; 1.921921566676164070426322
+590181246269924957280627105927292830364362298733317165 0.211806529814231327
+651933207068220677681742627450599773431112575139035117075195]
+ [-4.1997669287267147169391976434767379850646610405312981085368073326870987
+25183149 0.0632909583119936047452127680905314781106629595769808862495179252
+6641919769169438; -1.739226243945377437318983891388445419559172390813637142
+588074300773393372319632 0.355094171967641942812311369721542488996880766246
+9011825633568253151317250346457; 3.7994415851571118395489096177427030106542
+30855980402848403493234311687334061043 -0.240040385360714967337281449860542
+1432741151234222669853258429630248013616597714; 2.1342997898784534744128401
+97099997792747907428018703255828946271337100277035398 0.2778129646905650064
+883834316803775809636862936168576804197792563118823674230252]
+ [-7.5635512949525797927949237518032093449428326700525292289291773036652796
+49609056 -0.359589080931816041778274733542243024898769896362352799955755614
+1431160819565173; 0.1862205157110664936856878569541139182155636270857145381
+133507350572428610856458 -0.26515911468581138681760562313054271133704704751
+81561248041565659738152855106581; 12.05431806309802860670638993980073223117
+857302037974047209355120471242496823729 0.031553695837354823179825976343019
+31487277744908679032615947778197444938754169989; -1.18786590343927458966909
+3217063069348886133043261187402566500292679209499910705 0.15071010444668137
+92448321002998387316937047640415334786393236917659415826738098]
+ [-4.8542206937576882564244759249217889671318308807556438856671108319996396
+02571092 -0.517887020234393120656393235045404158630451221929817719450711184
+0483049182615223; 8.8191934867588965211288250323890056203823702461277205988
+46190502079399786992563 -0.336592778003760157180119686025267750085679814246
+8833446195818220913662496243523; 17.587754503667084597189849090570633675689
+01898252112314409231412997820949762859 0.4741671562762803743232805602407211
+8466898851204272954923370182014612097307014; -6.459220498627576364853929880
+791182121539549410534385679122916896743221109454275 -0.08412093627352932890
+56555683394033546032208048602145884638143934414391894387034]
@@ -1516,8 +1496,8 @@ Internal Types
--0.836183863448452434853816848846214180151107840280552541015358682919772669 -2808571 +-4.199766928726714716939197643476737985064661040531298108536807332687098725 +183149@@ -1533,74 +1513,58 @@
Internal Types
retcode: Success
Interpolation: Automatic order switching interpolation
-t: 6-element Array{BigFloat,1}:
- 0.0
-
- 0.113092995074322453107164827202916505594705516625996223111945149913284021
-3685782
- 0.343146921511359988812756981258199791752946439061231935036208672177190552
-5139586
- 0.638787912178119045475686716403453696145703840976671770300362717636169010
-3911218
- 0.924980009339641787803025370802957820064305280645527971524647529343989238
-2198667
- 1.0
-
-u: 6-element Array{Array{BigFloat,2},1}:
- [0.306624934095196977068553678691387176513671875 0.53380664355331530757098
-3350160531699657440185546875; 0.1987476560498380173669374926248565316200256
-34765625 0.6868676641729416498804994262172840535640716552734375; 0.68623291
-22068349107621543225832283496856689453125 0.1385896391405632144255832827184
-3492984771728515625; 0.203175435627960698781180326477624475955963134765625
-0.805431744473405597517512433114461600780487060546875]
-
-
-
- [0.12195360456574192057103450125829479835154212449537040631419147193227177
-07072281 -0.008123130290881946954846439953346604247685795531381076954615265
-216926057964851296; 0.39634232463333060500035154968630411236237809056373749
-67291028820745967835867344 0.3922711469618407507967721492263185395342890073
-81766330685625774594538169690652; 0.623608741422296146622188495992379377326
-1012111860142087916384383100848754432546 0.12364093847437092650170876507509
-88252011078298374602277522448929137058846502853; 0.534163473482693439801594
-9598190797146120758990521065990311201727125884303541186 1.20838467438490251
-984570602012818073090786122054696368793424589674326470922365]
- [-0.8361838634484524302515241377868298585150783921197594796200281898496975
-666674019 -1.86702009280275906714481657557557772254317523145257578870427033
-7325883704383484; 0.1453811433020334592194198867700128021358065155340114402
-40125065565642004128116 -0.821344784297997903320173617608476524831684965801
-178391546925474249475267299101; 1.07777106836562367736866494979442563453648
-6095469728594798970600327565775724293 1.25538770096216352007838104928602742
-9091119897403465064103255124923206814475632; 0.9271450459854138970877764411
-093660000440062290278553621204170193599798676166655 1.537230878645549906894
-162691393353603227149486073413270940164965502830507778154]
- [-2.4781841695365933879454296869576851158794725492090979510087154475624816
-40308525 -4.473442094117027500962149810242483127063290604571647473071064187
-993636195979255; -0.0800636456970526473426825590296719382059392407749978855
-3243695332637183329478918 -1.2177711307527883897259917368996975453640322015
-70350741827901448233547018592178; 3.296631520487046521606572939309413704811
-029124693202574218616309645150411280569 5.432131611081168714399861849631519
-193841227727803785362948080853293664273521618; 0.37434934542485138684240887
-51458359168829629652339837772895985107618118489826407 0.3128277517221572388
-320863277424571593898950652132176086684904308392850230973752]
- [-2.4975818474654785620583333419028132327401174606322107203958802825370885
-2923473 -4.0447100330112037811789973414846617842896582299344829841646119747
-41954017475663; 2.317330136264620532155919713301594091246140146802376310965
-55742941879474848447 3.3939594174726643759505719833718108393819666973928514
-25070913540376507610733073; 6.272908360633646656799832472605245173020138797
-91124780627884217093459523595315 10.437661931829127942742116637101708750848
-09021359675225845126531385156749683626; -1.68905544826407875647187275626951
-5308655502536388607395692873313231436862205155 -3.1459536741694796133885914
-09304003319814855469322625609339840227466509054260104]
- [-1.8905635653064639047282991343505179655840798093162889698003912501902176
-05059431 -2.903613161693912334222297344693476087420852212725142231232173351
-309153331868035; 3.63719468066918833601693286989159200951330508996604387659
-1769828411954450994223 5.75968789300718264451102885315708689832163972296572
-6432333575598555749412180328; 6.7848191643649296626210778813473088663956547
-37119855069667803522335464117944441 11.212699088749110772509816303551421403
-75641742621155666612423517643312409626039; -2.44684488609446295706038840307
-4582141844252660843255139959116702928450319605342 -4.3600004123377007720866
-8282413065848764416194625002658163522925887031917076989]
+t: 5-element Array{BigFloat,1}:
+ 0.0
+ 0.174978571039260839955971362922932311969587766184311595600003814987688917
+670033
+ 0.443927592902329549262955543201866942448156891600626374133386916591989198
+2452429
+ 0.764224200877330725867312601549041343518066108894817582022589715719257638
+689101
+ 1.0
+u: 5-element Array{Array{BigFloat,2},1}:
+ [0.647667818307158693613700961577706038951873779296875 0.41891793301301194
+49648873342084698379039764404296875; 0.337399927288797085722649171657394617
+7959442138671875 0.9635281213601436522964149844483472406864166259765625; 0.
+5468425532557132928701548735261894762516021728515625 0.17626705828817867960
+36749365157447755336761474609375; 0.949386810167432759755001825396902859210
+968017578125 0.094873934340720555979942218982614576816558837890625]
+ [-0.6188960718824867315600479934744149366024904665644964231142436361894342
+932688088 0.350025724727944328386746030968719973583575229449002128134621080
+6703142928360151; -0.057412938226884411332353995061985664386953700876801540
+8273367293620859214807343 0.86234921983414702215936734373922280062509104587
+65074257239842556045873829623679; 0.730830422039167856677939018752674284467
+6068724526786605920150319560591324997474 -0.0719500558190700791931994663477
+2401178797807082308463123640778450194963692561123; 1.9219215666761640532952
+4923324445596764154340974752640998224422930766289174934 0.21180652981423132
+55670980185662902931712622332530585206029867133617204212242951]
+ [-4.1997669287267142313732045539404047484114194120399489722683023800035925
+08971228 0.0632909583119936480334383159105773197835051978243704967829547607
+1331680393624141; -1.739226243945377289547481971522409139376310510051609221
+275123959693022486052091 0.355094171967642016298116584668849797403972933138
+5504713998517761256381025786839; 3.7994415851571112213500563836346512582811
+54770211912120721589582030845806947245 -0.240040385360714968142107266138929
+7746469210161750599060258190115042253622508182; 2.1342997898784535892996564
+40491445438293150252246450473226832977722872269962572 0.2778129646905650078
+06693269877821917414661056116102160054686955469100703646363]
+ [-7.5635512949525796987736116184057879839282156909780960499963338551134003
+99481257 -0.359589080931815977341912800027407129310897454992861429003331810
+2208311625259592; 0.1862205157110652691300800139213029913215325082910294572
+287295496316040266450176 -0.26515911468581131538767688062084618206650343063
+10704184720096603820931966510419; 12.05431806309802692414194582081648616481
+42785753061834077106199125537720282214 0.0315536958373547311934658544097954
+3916551258182702592654753178204743926662259857; -1.187865903439273568242938
+436493941128695861300927184146190376350392473205034539 0.150710104446681422
+7984372925104960180497308147302288119152868894315476594225174]
+ [-4.8542206937576882564245579234967520920780649707785056491252596842641804
+60786032 -0.517887020234393120656396709159393988348694698314043201847566049
+4330752288113321; 8.8191934867588965211288403482626033541157741314168154153
+21895369048580961583098 -0.336592778003760157180120554804844891963046986902
+8211880744559125485187905114471; 17.587754503667084597190022349283335243876
+07308932141404430253539110346898271463 0.4741671562762803743232862592247530
+11648517464552072337100017336327754204010767; -6.45922049862757636485396767
+1076753760616070448084440787847954446988697661809018 -0.0841209362735293289
+0565569920493050960722113902499021749143000529257674882024245]
@@ -1624,52 +1588,48 @@ Internal Types
retcode: Success
Interpolation: Automatic order switching interpolation
-t: 11-element Array{Float64,1}:
- 0.0
- 0.04373840556447778
- 0.10504494381022488
- 0.1832067917446593
- 0.27624508173685597
- 0.38220516978577257
- 0.5082351570065224
- 0.6464984228378687
- 0.7904627863642966
- 0.9604154264967177
- 1.0
-u: 11-element Array{StaticArrays.SArray{Tuple{4,2},Float64,2,8},1}:
- [0.9556033595934601 0.4445826506822814; 0.6653733603188241 0.3599066578409
-216; 0.08928453186876584 0.9161879858955384; 0.5023138864356134 0.410089319
-6093318]
- [0.8612814796873004 0.35095022356361616; 0.6884157053916526 0.481918985400
-9987; -0.0435701516313082 0.8681552671858593; 0.7245715179296826 0.60524795
-99696545]
- [0.639416506953087 0.14149186863102034; 0.6045055288378626 0.5539149485415
-189; -0.17604607024753233 0.8509308437858583; 1.0234926858206546 0.85811132
-14950639]
- [0.2045828895696477 -0.2525041978120318; 0.3189536789846515 0.498253464653
-9205; -0.21859428329425323 0.9428260910353048; 1.3676916752649069 1.1293421
-70650124]
- [-0.5247863084275708 -0.8853558155271741; -0.22904535724602598 0.272081830
-017429; -0.022586423462697486 1.2658051861544248; 1.6930321068954122 1.3473
-66838750421]
- [-1.596241487638124 -1.7650129829900383; -1.0096364160682132 -0.0777405483
-9070577; 0.6111815116390642 1.9710492033139315; 1.9013867799889466 1.410428
-616184992]
- [-3.088864730689933 -2.8884867482243854; -1.89939215748847 -0.377417741616
-9476; 2.0255573851318265 3.3106758294172733; 1.8401389914264719 1.159055788
-0944177]
- [-4.741278898243673 -3.9335065059007217; -2.3837684701411317 -0.1628264011
-8413036; 4.420117802509115 5.328489736198788; 1.271812824420278 0.396970463
-19250806]
- [-6.0300291310205925 -4.367029492312431; -1.74274192279828 1.1540427220849
-248; 7.657029927596874 7.728063705591205; 0.00947891820170188 -0.9832882854
-497728]
- [-6.137700521949925 -3.2666820278834194; 1.3340496794841403 4.697017393957
-745; 11.717909327425811 10.089825057925813; -2.4251766777177335 -3.31580501
-33186405]
- [-5.826048266206638 -2.666854071203864; 2.487015014508281 5.86393368970059
-7; 12.557283203988641 10.417972995877923; -3.129665835379061 -3.94069849695
-08274]
+t: 10-element Array{Float64,1}:
+ 0.0
+ 0.044268383288701384
+ 0.112594141115787
+ 0.18849010870990623
+ 0.2848505016507512
+ 0.4019790700207936
+ 0.5407550483819644
+ 0.6894509765240777
+ 0.8445730185805572
+ 1.0
+u: 10-element Array{StaticArrays.SArray{Tuple{4,2},Float64,2,8},1}:
+ [0.24033312390823647 0.8979401730035852; 0.46674384786229717 0.72621673279
+51474; 0.6104238201344643 0.3564577178285062; 0.9933518244309794 0.90221948
+93683779]
+ [0.004891504631234167 0.7027669540137237; 0.4147689965386876 0.71967685855
+92731; 0.6361562875932304 0.25986913047222704; 1.1814558531925952 1.1814686
+208833447]
+ [-0.4570244736237513 0.264050800328172; 0.23273686958692968 0.545386869904
+514; 0.7847439780439889 0.21852350075142657; 1.425727970417176 1.5773611564
+460381]
+ [-1.0948377870594164 -0.4114194170312348; -0.07478740208120283 0.153324394
+74780513; 1.1332151743269427 0.37028301689515775; 1.6127509706668235 1.9444
+745564676607]
+ [-2.0481704597424226 -1.5222571419277677; -0.5285242778822297 -0.549821128
+9609227; 1.8954182650067346 0.9386246285108104; 1.6863087081882662 2.255157
+0527888907]
+ [-3.3028943639184885 -3.151523554454985; -0.9627583143880756 -1.4833533435
+024806; 3.336483768095359 2.2961545121423343; 1.466931077543047 2.315648359
+1747077]
+ [-4.6378194222266504 -5.200697805959124; -0.8911555816171014 -2.2062830447
+743305; 5.7130589428286775 4.9113682246370605; 0.6779574942196687 1.7994681
+0124736]
+ [-5.366478203424986 -6.9869154537664855; 0.4860214587912063 -1.74242477710
+86843; 8.734535354674204 8.748840064233864; -0.8784957112072949 0.382019905
+12031037]
+ [-4.606328876422249 -7.522771131283662; 3.9540001108450564 1.0291461918840
+121; 11.618323762719408 13.224470966242185; -3.258974462675836 -2.142263438
+2850085]
+ [-1.4659632576115578 -5.610718738601163; 9.788349464540074 6.8738327944165
+49; 12.919692049714163 16.869256407991536; -6.186179892711854 -5.6455665591
+70115]
@@ -1682,10 +1642,10 @@ Internal Types
4×2 StaticArrays.SArray{Tuple{4,2},Float64,2,8} with indices SOneTo(4)×SOne
To(2):
- 0.639417 0.141492
- 0.604506 0.553915
- -0.176046 0.850931
- 1.02349 0.858111
+ -0.457024 0.264051
+ 0.232737 0.545387
+ 0.784744 0.218524
+ 1.42573 1.57736
@@ -1702,36 +1662,82 @@ Conclusion
Computer Information:
Julia Version 1.3.0
-Commit 46ce4d7933 (2019-11-26 06:09 UTC)
+Julia Version 1.4.2
+Commit 44fa15b150* (2020-05-23 18:35 UTC)
Platform Info:
- OS: Windows (x86_64-w64-mingw32)
- CPU: Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz
+ OS: Linux (x86_64-pc-linux-gnu)
+ CPU: Intel(R) Xeon(R) CPU @ 2.30GHz
WORD_SIZE: 64
LIBM: libopenlibm
- LLVM: libLLVM-6.0.1 (ORCJIT, skylake)
+ LLVM: libLLVM-8.0.1 (ORCJIT, haswell)
Environment:
- JULIA_EDITOR = "C:\Users\accou\AppData\Local\atom\app-1.42.0\atom.exe" -a
+ JULIA_CUDA_MEMORY_LIMIT = 536870912
+ JULIA_DEPOT_PATH = /builds/JuliaGPU/DiffEqTutorials.jl/.julia
+ JULIA_PROJECT = @.
JULIA_NUM_THREADS = 4
Package Information:
-Status `~\.julia\dev\DiffEqTutorials\Project.toml`
+Status `/builds/JuliaGPU/DiffEqTutorials.jl/Project.toml`
+[2169fc97-5a83-5252-b627-83903c6c433c] AlgebraicMultigrid 0.3.0
+[7e558dbc-694d-5a72-987c-6f4ebed21442] ArbNumerics 1.0.5
+[6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf] BenchmarkTools 0.5.0
+[be33ccc6-a3ff-5ff2-a52e-74243cff1e17] CUDAnative 3.1.0
+[159f3aea-2a34-519c-b102-8c37f9878175] Cairo 1.0.3
+[3a865a2d-5b23-5a0f-bc46-62713ec82fae] CuArrays 2.2.1
+[55939f99-70c6-5e9b-8bb0-5071ed7d61fd] DecFP 0.4.10
+[abce61dc-4473-55a0-ba07-351d65e31d42] Decimals 0.4.1
+[ebbdde9d-f333-5424-9be2-dbf1e9acfb5e] DiffEqBayes 2.15.0
+[eb300fae-53e8-50a0-950c-e21f52c2b7e0] DiffEqBiological 4.3.0
+[459566f4-90b8-5000-8ac3-15dfb0a30def] DiffEqCallbacks 2.13.3
+[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.21.0
+[77a26b50-5914-5dd7-bc55-306e6241c503] DiffEqNoiseProcess 4.2.0
+[9fdde737-9c7f-55bf-ade8-46b3f136cc48] DiffEqOperators 4.10.0
+[1130ab10-4a5a-5621-a13d-e4788d82bd4c] DiffEqParamEstim 1.14.1
+[055956cb-9e8b-5191-98cc-73ae4a59e68a] DiffEqPhysics 3.2.0
+[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.14.0
+[31c24e10-a181-5473-b8eb-7969acd0382f] Distributions 0.23.4
+[497a8b3b-efae-58df-a0af-a86822472b78] DoubleFloats 1.1.12
+[587475ba-b771-5e3f-ad9e-33799f191a9c] Flux 0.10.4
+[f6369f11-7733-5829-9624-2563aa707210] ForwardDiff 0.10.10
+[7073ff75-c697-5162-941a-fcdaad2a7d2a] IJulia 1.21.2
+[23fbe1c1-3f47-55db-b15f-69d7ec21a316] Latexify 0.13.5
+[c7f686f2-ff18-58e9-bc7b-31028e88f75d] MCMCChains 3.0.12
+[eff96d63-e80a-5855-80a2-b1b0885c5ab7] Measurements 2.2.1
+[961ee093-0014-501f-94e3-6117800e7a78] ModelingToolkit 3.1.1
+[2774e3e8-f4cf-5e23-947b-6d7e65073b56] NLsolve 4.4.0
+[8faf48c0-8b73-11e9-0e63-2155955bfa4d] NeuralNetDiffEq 1.5.0
+[429524aa-4258-5aef-a3af-852621145aeb] Optim 0.21.0
+[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.41.0
+[65888b18-ceab-5e60-b2b9-181511a3b968] ParameterizedFunctions 5.3.0
+[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.4.0
+[d330b81b-6aea-500a-939a-2ce795aea3ee] PyPlot 2.9.0
+[731186ca-8d62-57ce-b412-fbd966d074cd] RecursiveArrayTools 2.4.4
+[47a9eef4-7e08-11e9-0b38-333d64bd3804] SparseDiffTools 1.8.0
+[684fba80-ace3-11e9-3d08-3bc7ed6f96df] SparsityDetection 0.3.2
+[90137ffa-7385-5640-81b9-e52037218182] StaticArrays 0.12.3
+[f3b207a7-027a-5e70-b257-86293d7955fd] StatsPlots 0.14.6
+[789caeaf-c7a9-5a7d-9973-96adeb23e2a0] StochasticDiffEq 6.23.1
+[c3572dad-4567-51f8-b174-8c6c989267f4] Sundials 4.2.3
+[1986cc42-f94f-5a68-af5c-568840ba703d] Unitful 1.2.1
+[44d3d7a6-8a23-5bf8-98c5-b353f8df5ec9] Weave 0.10.2
+[b77e0a4c-d291-57a0-90e8-8db25a27a240] InteractiveUtils
+[37e2e46d-f89d-539d-b4ee-838fcccc9c8e] LinearAlgebra
+[44cfe95a-1eb2-52ea-b672-e2afdf69b78f] Pkg
-
-
-
-
-
+
+