Overview
Linear Algebra can be used to numerically solve ordinary differential equations. Consider the following sample equation
{%
d^2x(t)/ dt^2 = a(t,x) \times dx(t)/dt + b(t,x) \times x(t)
%}
with a set of boundary conditions
{%
x(0) = x_0 \\
dx(0)/dt = x_1 \\
d^2 x(0)/dt^2 = x_2
%}
The key to modeling this equation would be to re-express the equation as a set of linear algebra equations.
First, we form the following vector.
{%
\begin{bmatrix}
\ddot{x}(t+1) \\
\dot{x}(t+1) \\
x(t+1) \\
\end{bmatrix}
%}
Then we express the differential equation as
{%
\begin{bmatrix}
\ddot{x}(t+1) \\
\dot{x}(t+1) \\
x(t+1) \\
\end{bmatrix}
=
\begin{bmatrix}
0 & a(t,x) & b(t,x) \\
\Delta t & 1 & 0\\
0 & \Delta t & 1 \\
\end{bmatrix}
\begin{bmatrix}
\ddot{x}(t) \\
\dot{x}(t) \\
x(t) \\
\end{bmatrix}
%}
Implementation
The following code utilizes the linear algebra library to simulate the evolution of the second order differential equation
{% \frac{d^x(t)}{dt^2} = -x(t) %}
let la = await import('/code/linear-algebra/v1.0.0/linear-algebra.mjs');
let k = 1;
let dt = 0.01;
let x0=1;
let x1=0;
let x2=0;;
let t = 0;
let a = (t, vector) =>0;
let b = (t, vector)=>-1;
let vector = [[x2],[x1],[x0]];
let mult = [[0,a(t, vector[2][0]),b(t, vector[2][0])],[dt, 1, 0],[0, dt, 1]];
let data = [];
for(let i=0;i<1000;i++){
vector=la.multiply(mult, vector);
data.push({t:t, x:vector[2][0]});
mult=[[0,a(t, vector[2][0]),b(t, vector[2][0])],[dt, 1, 0],[0, dt, 1]];
t+=dt;
}