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.
// The function u(x,t) that we actually plot.
// Cheat at time t=0 by using u0.
function u(x, t) {
if (t == 0) {
return u0_compiled.evaluate({ x: x });
} else {
return u_fourier(x, t);
}
}
// Compute the solution in terms of of the inputs.
u_fourier = uu(
a, // Temperature at left
b, // Temperature at right,
D, // Diffusitivity
x => f.evaluate({ x: x }), // Source
x => u0_compiled.evaluate({ x: x }) // Initial temperature distribution
)
// Use mathjs to parse u0 into computable code.
u0_compiled = math.compile(u0_text)
// Also use mathjs to parse f into computable code.
f = math.compile(f_text)
// This function takes the inputs and returns u(x,t) in terms of the appropriate Fourier series
function uu(a, b, D, f, u0) {
let N = 30;
let pi = Math.PI;
let exp = Math.exp;
let sin = Math.sin;
let cos = Math.cos;
let u;
if (!insulate_left && !insulate_right) {
let f_coeffs = d3
.range(1, N + 1)
.map(k => int(x => f(x) * sin(k * pi * x), 0, 1, 1e-10, 10, 5));
let u0_coeffs = d3
.range(1, N + 1)
.map(k => int(x => u0(x) * sin(k * pi * x), 0, 1, 1e-10, 10, 5));
u = (x, t) => {
return (
a +
(b - a) * x +
2 *
d3.sum(
d3.range(1, N + 1).map(function(k) {
let dkp2 = D * k ** 2 * pi ** 2;
let e = exp(-dkp2 * t);
return (
((f_coeffs[k - 1] * (1 - e)) / dkp2 +
e * (((-1) ** k * b - a) / (k * pi) + u0_coeffs[k - 1])) *
sin(k * pi * x)
);
})
)
);
};
} else if (!insulate_left && insulate_right) {
let f_coeffs = d3
.range(1, N + 1)
.map(k => int(x => f(x) * sin((k - 1 / 2) * pi * x), 0, 1, 1e-10, 10, 5));
let u0_coeffs = d3
.range(1, N + 1)
.map(k =>
int(x => (u0(x) - a) * sin((k - 1 / 2) * pi * x), 0, 1, 1e-10, 10, 5)
);
u = (x, t) => {
return (
a +
2 *
d3.sum(
d3.range(1, N + 1).map(function(k) {
let d2km1p2 = (D * (2 * k - 1) ** 2 * pi ** 2) / 4;
let e = exp(-d2km1p2 * t);
return (
((f_coeffs[k - 1] * 4 * (1 - e)) /
(D * (pi - 2 * k * pi) ** 2) +
e * u0_coeffs[k - 1]) *
sin((k - 1 / 2) * pi * x)
);
})
)
);
};
} else if (insulate_left && !insulate_right) {
let f_coeffs = d3
.range(1, N + 1)
.map(k => int(x => f(x) * cos((k - 1 / 2) * pi * x), 0, 1, 1e-10, 10, 5));
let u0_coeffs = d3
.range(1, N + 1)
.map(k =>
int(x => (u0(x) - b) * cos((k - 1 / 2) * pi * x), 0, 1, 1e-10, 10, 5)
);
u = (x, t) => {
return (
b +
2 *
d3.sum(
d3.range(1, N + 1).map(function(k) {
let d2km1p2 = (D * (2 * k - 1) ** 2 * pi ** 2) / 4;
let e = exp(-d2km1p2 * t);
return (
((f_coeffs[k - 1] * 4 * (1 - e)) /
(D * (pi - 2 * k * pi) ** 2) +
e * u0_coeffs[k - 1]) *
cos((k - 1 / 2) * pi * x)
);
})
)
);
};
} else if (insulate_left && insulate_right) {
let f_coeffs = d3
.range(1, N + 1)
.map(k => int(x => f(x) * cos(k * pi * x), 0, 1, 1e-10, 10, 5));
let u0_coeffs = d3
.range(1, N + 1)
.map(k => int(x => u0(x) * cos(k * pi * x), 0, 1, 1e-10, 10, 5));
let f_int = int(f, 0, 1, 1e-10, 10, 5);
let u0_int = int(u0, 0, 1, 1e-10, 10, 5);
u = (x, t) => {
return (
u0_int +
t * f_int +
2 *
d3.sum(
d3.range(1, N + 1).map(function(k) {
let dkp2 = D * k ** 2 * pi ** 2;
let e = exp(-dkp2 * t);
return (
((f_coeffs[k - 1] * (1 - e)) / dkp2 + e * u0_coeffs[k - 1]) *
cos(k * pi * x)
);
})
)
);
};
}
return u;
}
example_init = ({
"constant, inconsistent endpoints": "1",
lopsided: "5x*(1-x)^2",
"lopsided, with constant source": "5x*(1-x)^2",
"lopsided, inconsistent right": "5x*(1-x)^2",
"step, fixed ends": "x<1/2 ? -1 : 1",
"all zero with asymmetric heat source": "0",
"step, insulated ends": "x<1/2 ? -1 : 1"
})
example_source = ({
"constant, inconsistent endpoints": "0",
lopsided: "0",
"lopsided, with constant source": "1",
"lopsided, inconsistent right": "0",
"step, fixed ends": "0",
"all zero with asymmetric heat source": "10x",
"step, insulated ends": "0"
})
example_left = ({
"constant, inconsistent endpoints": "0",
lopsided: "0",
"lopsided, with constant source": "0",
"lopsided, inconsistent right": "0",
"step, fixed ends": "-1",
"all zero with asymmetric heat source": "0",
"step, insulated ends": "-1"
})
example_right = ({
"constant, inconsistent endpoints": "0",
lopsided: "0",
"lopsided, with constant source": "0",
"lopsided, inconsistent right": "1",
"step, fixed ends": "1",
"all zero with asymmetric heat source": "0",
"step, insulated ends": "1"
})
example_D = ({
"constant, inconsistent endpoints": 0.5,
lopsided: 0.5,
"lopsided, with constant source": 0.5,
"lopsided, inconsistent right": 0.2,
"step, fixed ends": 0.2,
"all zero with asymmetric heat source": 0.5,
"step, insulated ends": 0.2
})
example_insulate_left = ({
"constant, inconsistent endpoints": false,
lopsided: false,
"lopsided, with constant source": false,
"lopsided, inconsistent right": false,
"step, fixed ends": false,
"all zero with asymmetric heat source": false,
"step, insulated ends": "true"
})
example_insulate_right = ({
"constant, inconsistent endpoints": false,
lopsided: false,
"lopsided, with constant source": false,
"lopsided, inconsistent right": false,
"step, fixed ends": false,
"all zero with asymmetric heat source": false,
"step, insulated ends": "true"
})
d3 = require('d3-selection@2', 'd3-array@2', 'd3-format@2')
import { adaptiveSimpson as int } from '@rreusser/integration'
import { slider, text, checkbox, select } from "@jashkenas/inputs"
import { Scrubber } from "@mbostock/scrubber"
functionPlot = require("function-plot@1/dist/function-plot")
math = require('mathjs@7')
Purpose-built for displays of data
Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.
