File:Green's function animation.gif

Summary[edit]

Julia code[edit]

using LinearAlgebra
using Plots
using Printf

function solve(f; x1=0, x2=1)
	N = length(f)
	x = Array(range(x1, x2, length=N))
	h = x[2] - x[1]
	diag = fill(+2/h^2, N-2)
	semidiag = fill(-1/h^2, N-3)
	L = Tridiagonal(semidiag, diag, semidiag)
	u = L \ f[2:end-1]
	u = cat([0], u, [0], dims=1)
	return x, u
end

function animate()
	# Store all Green's function solutions
	N = 101
	U = zeros(N, N)
	p = plot()
	for i in 1:N
		f = [i == j ? 1 : 0 for j in 1:N]
		x, u = solve(f)
		U[i,:] = u
		plot!(p, u)
	end

	# Solve a real problem
	f1 = exp.(-(x.-0.5).^2 / (2*0.01))
	x, u1 = solve(f1)
	u = zeros(N)

	barw = x[2]-x[1] # plot bars with no gap between them

	anim = @animate for i in 1:N
	    y = @sprintf("%.2f", (i-1) / (N-1)) # as string
		f2 = [i == j ? 1 : 0 for j in 1:N]
		x, u2 = solve(f2)
		u += u2 * f1[i]

		colors = [i == j ? :black : :red for j in 1:N]

		# for some reason, only (1599, 1600) gives a height that is divisible by 2 during mp4 generation
		plot(layout=(2, 2), size=(1599, 1600), xlims=(0,1), xticks=([0, 0.5, 1], ["\$0\$", "\$x\$", "\$1\$"]), yticks=nothing, bar_width=barw, titlefontsize=40, tickfontsize=40, framestyle=:box, grid=false, legend=nothing, margin=10Plots.mm, top_margin=0Plots.mm)

		# Plot point-source and Green's function solution
		bar!(subplot=1, x[i:i], f2[i:i],         color=:green,      linecolor=:green,     bar_width=barw, ylims=(0, 1.10), title="\$\\delta(x-$y)\$")
		bar!(subplot=2, x,      u2,              color=:darkgreen,  linecolor=:darkgreen, bar_width=barw, ylims=(0, 0.02), title="\$G(x,$y)\$")

		# Plot full source and full solution
		bar!(subplot=3, x[1:i],     f1[1:i],     color=:blue,      linecolor=:blue,       bar_width=barw, ylims=(0, 1.10), title="\$ \\hat{L}\\,(x) u(x) = f(x < $y) \$")
		bar!(subplot=3, x[i+1:end], f1[i+1:end], color=:lightgrey, linecolor=:lightgrey,  bar_width=barw, ylims=(0, 1.10))
		bar!(subplot=4, x,          u1,          color=:lightgrey, linecolor=:lightgrey,  bar_width=barw, ylims=(0, 0.06))
		bar!(subplot=4, x,          u,           color=:darkblue,  linecolor=:darkblue,   bar_width=barw, ylims=(0, 0.06), title="\$ u(x) = {\\int}_{0}^{$y} \\! f(x') \\, G(x,x') \\, \\mathrm{d} x' \$")
	end

	mp4(anim, "green.mp4", fps=5)
	run(`ffmpeg -i green.mp4 -vf "fps=10,scale=640:640:flags=lanczos,split[s0][s1];[s0]palettegen[p];[s1][p]paletteuse" -loop 0 green.gif`)
end

animate()

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:

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You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue