|
In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives. FDMs are thus discretization methods. Today, FDMs are the dominant approach to numerical solutions of partial differential equations. == Derivation from Taylor's polynomial == First, assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor's theorem, we can create a Taylor Series expansion : where ''n''! denotes the factorial of ''n'', and ''R''''n''(''x'') is a remainder term, denoting the difference between the Taylor polynomial of degree ''n'' and the original function. We will derive an approximation for the first derivative of the function "f" by first truncating the Taylor polynomial: : Setting, x0=a we have, : Dividing across by ''h'' gives: : Solving for f'(a): : Assuming that is sufficiently small, the approximation of the first derivative of "f" is: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「finite difference method」の詳細全文を読む スポンサード リンク
|