gibson:teaching:fall-2016:math753:finitediff

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gibson:teaching:fall-2016:math753:finitediff [2016/11/17 06:41] gibson |
gibson:teaching:fall-2016:math753:finitediff [2016/12/12 18:49] (current) gibson |
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=== First derivative df/dx === | === First derivative df/dx === | ||

- | Given a set of evenly space gridpoints $x_1, x_2, \ldots$, where $x_i = x_1 + (i-1) h$, and a function evaluated at the gridpoints $f(x_1), f(x_2), \ldots$, we can approximate the first derivative of $f$ at the gridpoints several ways | + | Given a set of evenly space gridpoints $x_1, x_2, \ldots$, where $x_i = x_1 + (i-1) h$, and a function $y(x)$ evaluated at the gridpoints $y_1 = y(x_1), y_2 = y(x_2), \ldots$, we can approximate the first derivative of $y(x)$ at the gridpoints several ways |

**One-sided finite differencing for the first derivative**, rightwards | **One-sided finite differencing for the first derivative**, rightwards | ||

\begin{equation*} | \begin{equation*} | ||

- | \frac{df}{dx}(x_i) = \frac{x_{i+1} - x_{i}}{h} + O(h) | + | \frac{dy}{dx}(x_i) = \frac{y_{i+1} - y_{i}}{h} + O(h) |

\end{equation*} | \end{equation*} | ||

**One-sided finite differencing for the first derivative**, leftwards | **One-sided finite differencing for the first derivative**, leftwards | ||

\begin{equation*} | \begin{equation*} | ||

- | \frac{df}{dx}(x_i) = \frac{x_{i} - x_{i-1}}{h} + O(h) | + | \frac{dy}{dx}(x_i) = \frac{y_{i} - y_{i-1}}{h} + O(h) |

\end{equation*} | \end{equation*} | ||

**Centered finite differencing for the first derivative** | **Centered finite differencing for the first derivative** | ||

\begin{equation*} | \begin{equation*} | ||

- | \frac{df}{dx}(x_i) = \frac{x_{i+1} - x_{i-1}}{2h} + O(h^2) | + | \frac{dy}{dx}(x_i) = \frac{y_{i+1} - y_{i-1}}{2h} + O(h^2) |

\end{equation*} | \end{equation*} | ||

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=== Second derivative === | === Second derivative === | ||

- | To approximate the second derivative $d^2f/dx^2$, we use the | + | To approximate the second derivative $d^2y/dx^2$, we use the |

**Centered finite differencing for the second derivative** | **Centered finite differencing for the second derivative** | ||

\begin{equation*} | \begin{equation*} | ||

- | \frac{d^2f}{dx^2}(x_i) = \frac{x_{i+1} - 2 x_i + x_{i-1}}{h^2} + O(h^2) | + | \frac{d^2y}{dx^2}(x_i) = \frac{y_{i+1} - 2 y_i + y_{i-1}}{h^2} + O(h^2) |

\end{equation*} | \end{equation*} | ||

gibson/teaching/fall-2016/math753/finitediff.txt ยท Last modified: 2016/12/12 18:49 by gibson