Green's_function_animation.gif
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Summary
Description Green's function animation.gif |
English:
An intuitive animation that shows how Green's functions that solve a differential equation subject to a point-like source can be superposed to solve it subject to an arbitrary source.
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Date | |
Source | Own work |
Author | Hersle |
Julia code
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()
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