Merge pull request #615 from ArnoStrouwen/example

some more examples

Author: ChrisRackauckas

docs/Project.toml

@@ -12,7 +12,10 @@ DiffEqBase = "2b5f629d-d688-5b77-993f-72d75c75574e"
 DiffEqCallbacks = "459566f4-90b8-5000-8ac3-15dfb0a30def"
 DiffEqDocs = "d91efeb5-c178-44a6-a2ee-51685df7c78e"
 DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
+Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
+FiniteDiff = "6a86dc24-6348-571c-b903-95158fe2bd41"
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 IncompleteLU = "40713840-3770-5561-ab4c-a76e7d0d7895"
 JLD2 = "033835bb-8acc-5ee8-8aae-3f567f8a3819"
 JumpProblemLibrary = "faf0f6d7-8cee-47cb-b27c-1eb80cef534e"

docs/src/assets/normalized.png

@@ -453,7 +453,8 @@ given by Dual numbers.
 To show this in action, let's say we want to find the Jacobian of solution
 of the Lotka-Volterra equation at `t=10` with respect to the parameters.
 
-```julia
+```@example faq1
+using DifferentialEquations
 function func(du,u,p,t)
   du[1] = p[1] * u[1] - p[2] * u[1]*u[2]
   du[2] = -3 * u[2] + u[1]*u[2]
@@ -471,23 +472,16 @@ be Dual numbers whenever `p` is an array of `Dual` numbers, and we do the same
 for the timespan just to show what you'd do if there was parameters-dependent events.
 Then we can take the Jacobian via ForwardDiff.jl:
 
-```julia
+```@example faq1
 using ForwardDiff
 ForwardDiff.jacobian(f,[1.5,1.0])
-
-2×2 Array{Float64,2}:
-  2.16056   0.188569
- -6.25677  -0.697978
 ```
 
-and compare it to Calculus.jl:
-
-```julia
-Calculus.jacobian(f,[1.5,1.0],:central)
+and compare it to FiniteDiff.jl:
 
-2×2 Array{Float64,2}:
-  2.16056   0.188569
- -6.25677  -0.697978
+```@example faq1
+using FiniteDiff
+FiniteDiff.finite_difference_jacobian(f,[1.5,1.0])
 ```
 
 #### I get Dual number errors when I solve my ODE with Rosenbrock or SDIRK methods

docs/src/assets/vars_plotting_example.png

@@ -145,7 +145,7 @@ i.e. no points outside the interval will be plotted.
 
 ### Example
 
-```julia
+```@example plots
 using DifferentialEquations, Plots
 function lorenz(du,u,p,t)
  du[1] = p[1]*(u[2]-u[1])
@@ -166,26 +166,20 @@ xyz = plot(sol, plotdensity=10000, idxs=(1,2,3))
 plot(plot(xyzt,xyz),plot(xy, xz, yz, layout=(1,3),w=1), layout=(2,1))
 ```
 
-![lorenz_plot](../assets/vars_plotting_example.png)
-
 An example using the functions:
 
-```julia
+```@example plots
 f(x,y,z) = (sqrt(x^2+y^2+z^2),x)
 plot(sol,idxs=(f,1,2,3))
 ```
 
-![norm_plot](../assets/normalized.png)
-
 or the norm over time:
 
-```julia
+```@example plots
 f(t,x,y,z) = (t,sqrt(x^2+y^2+z^2))
 plot(sol,idxs=(f,0,1,2,3))
 ```
 
-![normtime plot](https://user-images.githubusercontent.com/1814174/94351101-4667df80-0023-11eb-987d-aca652e32521.png)
-
 ## Animations
 
 Using the iterator interface over the solutions, animations can also be generated

docs/src/basics/faq.md

@@ -67,8 +67,11 @@ which outputs a tuple.
 
 ### Examples
 
-```julia
+```@example problem
+using DifferentialEquations
 prob = ODEProblem((u,p,t)->u,(p,t0)->p[1],(p)->(0.0,p[2]),(2.0,1.0))
+```
+```@example problem
 using Distributions
 prob = ODEProblem((u,p,t)->u,(p,t)->Normal(p,1),(0.0,1.0),1.0)
 ```
@@ -96,7 +99,7 @@ However, you may want to modify the problem after it is created.  For
 example, to simulate it for longer timespan.  It can be done by the
 `remake` function:
 
-```julia
+```@example problem
 prob1 = ODEProblem((u,p,t) -> u/2, 1.0, (0.0,1.0))
 prob2 = remake(prob1; tspan=(0.0,2.0))
 ```

docs/src/basics/plot.md

@@ -1,6 +1,8 @@
 # Solver Benchmarks
 
-Benchmarks for the solvers can be found at [benchmarks.sciml.ai](https://benchmarks.sciml.ai/stable/). The source code can be found
-at [SciMLBenchmarks.jl](https://github.com/SciML/SciMLBenchmarks.jl).
+Benchmarks for the solvers can be found at
+[benchmarks.sciml.ai](https://docs.sciml.ai/SciMLBenchmarksOutput/stable/).
+The source code can be found at
+[SciMLBenchmarks.jl](https://github.com/SciML/SciMLBenchmarks.jl).
 Many different problems are tested. However, if you would like additional problems to be benchmarked, 
 please open an issue or PR at the SciMLBenchmarks.jl repository with the code that defines the DEProblem.

docs/src/basics/problem.md

@@ -67,7 +67,8 @@ Options:
   orthogonalization procedure (IOP) [^1]. Note that if the linear operator/jacobian is hermitian,
   then the Lanczos algorithm will always be used and the IOP setting is ignored.
 
-```julia
+```@example linear_ode
+using DifferentialEquations
 _A = [2 -1;-3 -5]/5
 A = DiffEqArrayOperator(_A)
 prob = ODEProblem(A, [1.0,-1.0], (1.0, 6.0))
@@ -97,7 +98,7 @@ These methods require ``A`` is only dependent on the independent variable, i.e.
 
 Example:
 
-```julia
+```@example linear_ode
 function update_func(A,u,p,t)
     A[1,1] = cos(t)
     A[2,1] = sin(t)
@@ -128,7 +129,7 @@ These methods can be used when ``A`` is dependent on the state variables, i.e. `
 
 Example:
 
-```julia
+```@example linear_ode
 function update_func(A,u,p,t)
     A[1,1] = 0
     A[2,1] = sin(u[1])
@@ -147,7 +148,7 @@ operators can be solved using specialized adaptive algorithms, like `MagnusAdapt
 
 Example:
 
-```julia
+```@example linear_ode
 function update_func(A,u,p,t)
     A[1,1] = 0
     A[2,1] = 1