\n", "
Symmetric matrices are important because they appear in the real world \n", " all the time. One such place is in numerically solving a model for diffusion.\n", "
\n", "\n", " The following partial differential equation models diffusion, and \n", " is known as the heat equation.
\n", "$$\\frac{\\partial}{\\partial t} c(x,t)=\\frac{\\partial^2}\n", " {\\partial x^2}c(x,t) $$\n", "This equation models any sort of diffusion: the diffusion of dye into a \n", " liquid, or the diffusion of heat through a solid. Note that we have omitted constants\n", " required to guarantee the balance of units for simplicity here.
\n", " \n", "To use linear algebra to solve this equation, we need to change this \n", " differential equation into a difference equation.\n", " The partial derivative with respect to $t$\n", " can be approximated as a difference equation as follows:
\n", "$$\\frac{\\partial}{\\partial t} c(x,t) \n", " \\approx \\frac{c(x,t+\\Delta t) - c(x,t)}{\\Delta t}$$
\n", "The second derivative can be approximated as the following difference equation:
\n", "$$\\frac{\\partial^2}{\\partial x^2}c(x,t) \n", " \\approx \\cfrac{\\cfrac{c(x+\\Delta x,t) - c(x,t)}{\\Delta x}- \n", " \\cfrac{c(x,t) - c(x-\\Delta x,t)}{\\Delta x}}{\\Delta x} = \\cfrac{c(x+\\Delta x,t) \n", " - 2c(x,t)+ c(x-\\Delta x,t)}{(\\Delta x)^2}$$
\n", "We can rearrange this difference equation to find an expression \n",
" at the grid point x at the time step $t + \\Delta t$ as \n",
" follows:
$$c(x,t+\\Delta t) = c(x,t) + \\frac{\\Delta t}{(\\Delta x)^2}\n", " (c(x+\\Delta x,t)-2c(x,t)+c(x-\\Delta x,t))$$
\n", "Notice that every term on the right hand side only depends \n",
" on the values at time step t. So we can rewrite this as a matrix \n",
" equation, in terms of a vector c(t) whose entries are points \n",
" $x$ separated\n",
" by $\\Delta x$
$$ c(x,t + \\Delta t) \\approx c(x,t) + \\Delta t \\frac{\\partial}{\\partial t}c(x,t) = c(x,t) + \\Delta t \\frac{\\partial^2}{\\partial x^2}c(x,t).$$
\n", "$$ c(t + \\Delta t) = (I - \\frac{\\Delta t}{(\\Delta x)^2}A)*c(t)$$
\n", "In this problem, we provide you with the code to observe the evolution of this \n",
" differential equation, which describes diffusion. We start by discretizing the \n",
" variable x, which we represent as a vector of 101 points lying on the \n",
" interval from 0 to 1. Thus our meshwidth in space,$\\Delta x$,\n",
" is 0.01. We start with an c(x,0) = c is the initial concentration, \n",
" which we take to be a Gaussian distribution, or a bell curve.
\n",
" Your job is to create a negative second difference matrix A. And, to plot the \n",
" concentration vector C(:,t) for (at least) 3 later times. (See further \n",
" instructions within the code body.)\n",
" \n",
" \n",
"
A note about A:\n", " The dimension of A is 101x101. \n", " The matrix A must enforce boundary conditions!!!! Here, this means we will \n", " make the first row and column zero, and the last row and column zero. You can see\n", " what happens in the plot if you don't enforce this condition!\n", " Note that the diagonal of A will be 2, the super and sub diagonals will be -1, \n", " except at the boundary, where all entries will be zero.\n", "
\n", "$A = \\begin{bmatrix} \n", " 0 & 0 & & \\dots & & & 0 & 0 \\\\\n", " 0 & 2 & -1 & 0 & \\dots & & & \\dots & 0 \\\\\n", " 0 &-1 & 2 & -1 & 0 &\\dots & &\\dots & 0\\\\\n", " 0 & & -1 & 2 & -1 & 0 &\\dots&\\dots & 0\\\\\n", " \\vdots & & &\\ddots & \\ddots & \\ddots & & &\\vdots \\\\\n", " 0 & \\dots & \\dots & 0 & -1 & 2 & -1 & & 0\\\\\n", " 0 &\\dots & &\\dots & 0 & -1 & 2 & -1 & 0\\\\\n", " 0 & \\dots & & & \\dots &0 & -1 & 2 & 0\\\\\n", " 0 & 0 & & & \\dots & & & 0 & 0 \n", " \\end{bmatrix}$\n", "
\n", " \n", "