Add heat equation example (#876)

Author: jishnub

docs/make.jl

@@ -15,15 +15,16 @@ function replace_includes(str, included)
     return str
 end
 
-for (example, included) in [
-            ("ODE.jl", ["ODE_BVP.jl", "ODE_increaseprec.jl"]),
-            ("PDE.jl", ["PDE_Poisson.jl", "PDE_Helmholtz.jl"]),
-            ("Sampling.jl", ["Sampling1.jl"]),
-            ("Periodic.jl", ["Periodic1.jl"]),
-            ("Eigenvalue.jl", ["Eigenvalue_standard.jl", "Eigenvalue_symmetric.jl"]),
-            ("NonlinearBVP.jl", ["NonlinearBVP1.jl", "NonlinearBVP2.jl"]),
-            ("system_of_eqn.jl", ["System1.jl"])
-            ]
+function get_included_files(filename)
+    v = [l for l in eachline(joinpath(example_dir, filename)) if contains(l, "include")]
+    strip.(getindex.(split.(getindex.(split.(v, "include("), 2), ")"), 1), '\"')
+end
+
+file_with_includes(filename) = (filename, get_included_files(filename))
+
+for (example, included) in map(file_with_includes,
+                ["ODE.jl", "PDE.jl", "Sampling.jl", "Periodic.jl",
+                "Eigenvalue.jl", "NonlinearBVP.jl", "system_of_eqn.jl"])
     filename = joinpath(example_dir, example)
     Literate.markdown(filename, output_dir, documenter=true,
         preprocess = str -> replace_includes(str, included))
@@ -32,7 +33,7 @@ end
 makedocs(
             format = Documenter.HTML(),
             sitename = "ApproxFun.jl",
-            authors = "Sheehan Olver",
+            authors = "Sheehan Olver, Jishnu Bhattacharya and contributors",
             pages = Any[
                     "Home" => "index.md",
                     "Usage" => Any[

docs/src/library.md

@@ -13,10 +13,6 @@ Circle
 Curve
 ```
 
-```@docs
-ApproxFunFourier.Disk
-```
-
 ```@docs
 Segment
 ```

examples/PDE.jl

@@ -15,3 +15,14 @@ include("PDE_Helmholtz.jl")
 # Plot the solution
 import Plots
 Plots.plot(u; st=:surface, legend=false)
+
+include("PDE_Heat.jl")
+
+# Plot the solution
+import Plots
+xplot = -1:0.02:1
+p = Plots.plot(xplot, u.(xplot, 0), label="t=0", legend=true, linewidth=2)
+for t in [0.05, 0.1, 0.2, 0.5, 0.8]
+	Plots.plot!(xplot, u.(xplot, t), label="t=$t")
+end
+p

examples/PDE_Heat.jl

@@ -0,0 +1,35 @@
+# ## Heat equation
+
+# We solve the heat equation
+# ```math
+# \frac{\partial}{\partial t} u(x,t) = \frac{\partial^2}{\partial x^2} u(x,t)
+# ```
+# on the spatial domain `-1..1` from ``t=0`` to ``t=1``,
+# with zero boundary conditions, and the initial condition
+# ``u(x,0) = u_0(x)``, where we choose ``u_0(x)`` to be a sharp Gaussian.
+
+using ApproxFun
+using LinearAlgebra
+
+dx = -1..1;
+dt = 0..1;
+
+# We construct a 2D domain that is the tensor product of x and t
+d = dx × dt;
+
+# The initial condition
+u0 = Fun(x->exp(-x^2/(2*0.2^2)), dx);
+
+# We define the derivatives on the 2D domain
+Dx = Derivative(d,[1,0]);
+Dt = Derivative(d,[0,1]);
+
+# The operator along with the initial and the boundary conditions
+A = [Dt - Dx^2; I⊗ldirichlet(dt); bvp(dx)⊗I];
+
+# We collect the initial condition along with zeros for the two boundary conditions and the equation
+rhs = [0.0; u0; 0.0; 0.0];
+
+# Solve the equation with an arbitrary tolerance.
+# A lower tolerance will increase accuracy at the expense of execution time
+u = \(A, rhs, tolerance=1e-4);