nonlinear finite difference example

content/bvps/finite-difference.ipynb

@@ -352,6 +352,26 @@
     "```"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example: nonlinear BVP\n",
+    "\n",
+    "So far we've seen how to handle a linear boundary value problem, but what if we have a **nonlinear** BVP? This is going to be trickier, because our work so far relies on using linear algebra to solve the system of (linear) equations.\n",
+    "\n",
+    "For example, consider the 2nd-order ODE\n",
+    "\\begin{equation}\n",
+    "y^{\\prime\\prime} = 3y + x^2 + 10 y^3 \n",
+    "\\end{equation}\n",
+    "with the boundary conditions $y(0) = y(1) = 0$. This is nonlinear, due to the $y^3$ term on the right-hand side.\n",
+    "\n",
+    "To solve this, let's first convert it into a discrete form, by replacing the second derivative with a finite difference and any $x$/$y$ present with $x_i$ and $y_i$:\n",
+    "\\begin{equation}\n",
+    "\\frac{y_{i-1} - 2y_i + y_{i+1}}{\\Delta x^2} - 3y_i = x_i^2 + 10 y_i^3\n",
+    "\\end{equation}\n"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},