We want to solve the differential equation du/dt = f(t, u), where f(t, u) = α * u, in the interval t ∈ [0, 1]. We know that the solution to this equation is u(t) = u_0 exp(α * t).

In [2]:

using DifferentialEquations, Measurements # Load the packages
α = (1.01 ± 0.01) # Define the α parameter as a number with uncertainty
f(t, u) = α * u # Define the f function
u0 = 1/2 ± 0 # The initial value of the Cauchy problem
tspan = (0.0 ± 0, 1.0 ± 0) # The extrema of the domain in which we want to solve the equation
prob = ODEProblem(f, u0, tspan) # Define the ODEProblem
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8); # Solve the problem

The following is the analytic solution:

In [3]:

u = u0 .* exp.(α .* sol.t);

Let's compare the solution found by DifferentialEquations.jl and the analytic solution:

In [4]:

for i in eachindex(sol)
    println("At time ", Measurements.value(sol.t[i]), ":\n", sol[i], "\n", u[i], "\n")
end

At time 0.0:
0.5 ± 0.0
0.5 ± 0.0

At time 0.012407826196308189:
0.5063053789114713 ± 3.769289486273972e-5
0.5063053789114712 ± 3.7692894862736955e-5

At time 0.042500886481028156:
0.52193026853015 ± 6.72494218852177e-5
0.521930268530135 ± 6.724942188505842e-5

At time 0.0817803932926208:
0.5430526603660298 ± 8.15081766960003e-5
0.5430526603659422 ± 8.1508176695602e-5

At time 0.12887323477424767:
0.5695064255096831 ± 0.00010898724594719012
0.5695064255093786 ± 0.00010898724594568129

At time 0.18409719135533997:
0.6021738925732998 ± 0.00013105341682102388
0.602173892572426 ± 0.00013105341681784647

At time 0.24627365251891972:
0.6412019663929366 ± 0.00015886106381869243
0.64120196639087 ± 0.00015886106381198824

At time 0.31479180297608794:
0.6871467088572084 ± 0.00018569231191653082
0.6871467088529182 ± 0.00018569231190505639

At time 0.38859490981815525:
0.7403247618946622 ± 0.0002155136694726928
0.740324761886693 ± 0.00021551366945420482

At time 0.46685971668181403:
0.8012206780339247 ± 0.0002462164149989751
0.8012206780203366 ± 0.0002462164149718729

At time 0.5487136259449821:
0.8702746579779278 ± 0.00027921963537337163
0.8702746579563262 ± 0.0002792196353356099

At time 0.6334320104824218:
0.9480188907838683 ± 0.00031402163556843257
0.9480188907514014 ± 0.0003140216355183908

At time 0.72035921761696:
1.0350146997549017 ± 0.00035123419392624525
1.035014699708293 ± 0.00035123419386214993

At time 0.8089540967699169:
1.1318986650134468 ± 0.0003908984317072107
1.1318986649489977 ± 0.0003908984316274656

At time 0.8987609607927629:
1.2393677739920392 ± 0.0004334066410309232
1.2393677739056368 ± 0.00043340664093383026

At time 0.9894116178526793:
1.3581976365396038 ± 0.0004789992744179338
1.3581976364267052 ± 0.00047899927430177354

At time 1.0:
1.3728005076225709 ± 0.013728005076303318
1.3728005075084582 ± 0.013728005075084582