I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization.
I'm happy to be part of the first Observable Ambassador's cohort.
Published
Edited
Apr 14, 2021
12 stars
// The initial temperature distribution
// This can be edited to experiment with its effect on the results,
// though its values should remain between zero and one, as I haven't
// done anything to scale it.
g = r => 1
// The number of terms in the approximating sum
// The number of terms required for a good approximation depends strongly
// on the initial temperature distribution. The primary choice that's
// illustrated here with g(r) = 1 requires a lot of terms, since it's not
// consistent with the boundary condition
N = 100
// The eigenfunctions
function R(n, r) {
return Math.sin(n * Math.PI * r) / r;
}
// This computation checks that the normalized eigenfunctions are,
// indeed, orthogonal with respect to an inner product. The diagonal
// indicates that the squared-norms are all 1/2
d3
.range(1, 9)
.map(i =>
d3
.range(1, 9)
.map(j => integrate(x => x ** 2 * R(i, x) * R(j, x), 1e-12, 1).toFixed(5))
)
// The coeficients in the approximation
c = d3
.range(1, N)
.map(n => integrate(r => 2 * g(r) * R(n, r) * r ** 2, 1e-12, 1))
// The nth approximation to g using a linear combinations of the eigenfunctions:
function g_approx(n, r) {
return d3.sum(d3.range(0, n).map(n => c[n] * R(n + 1, r)));
}
// The solution
function u(t, r) {
return d3.sum(
d3
.range(c.length)
.map(n => c[n] * Math.exp(-t * Math.PI ** 2 * (n + 1) ** 2) * R(n + 1, r))
);
}
function rgb_to_r_g_b(rgb_string) {
return String(
rgb_string
.split('(')[1]
.split(')')[0]
.split(',')
.map(x => String(parseFloat(x) / 255))
).replace(/,/g, ' ');
}
function integrate(ff, a, b) {
return adaptiveSimpson(ff, a, b, {
tolerance: 1e-11,
maxDepth: 8,
minDepth: 7
});
}
import { adaptiveSimpson } from '@rreusser/integration@3056'
import { funplot } from '@mbostock/funplot'
import { Scrubber } from '@mbostock/scrubber'
import { Range, Radio, Toggle } from '@observablehq/inputs'
d3 = require('d3-selection@2', 'd3-array@2', 'd3-scale-chromatic@2', 'd3-format@2')
x3dom = require('x3dom').catch(() => window['x3dom'])
html`<style>
canvas {
outline: none
}
</style>`
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