翻訳と辞書 Words near each other ・ Laplace formula ・ Laplace functional ・ Laplace invariant ・ Laplace Island ・ Laplace Island (Antarctica) ・ Laplace Island (Western Australia) ・ Laplace limit ・ Laplace no Ma ・ Laplace number ・ Laplace operator ・ Laplace operators in differential geometry ・ Laplace plane ・ Laplace pressure ・ Laplace principle (large deviations theory) ・ Laplace transform ・ Laplace transform applied to differential equations ・ Laplace's demon ・ Laplace's equation ・ Laplace's law ・ Laplace's method ・ LaPlace, Louisiana ・ Laplace-P ・ LaplacesDemon ・ Laplace–Beltrami operator ・ Laplace–Carson transform ・ Laplace–Runge–Lenz vector ・ Laplace–Stieltjes transform ・ Laplacian matrix ・ Laplacian of the indicator ・ Laplacian smoothing Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laplace transform applied to differential equations ： ウィキペディア英語版 Laplace transform applied to differential equations The Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform: :$\backslash mathcal\backslash =s\backslash mathcal\backslash -f(0)$ :$\backslash mathcal\backslash =s^2\backslash mathcal\backslash -sf(0)-f\text{'}(0)$ One can prove by induction that :$\backslash mathcal\backslash =s^n\backslash mathcal\backslash -\backslash sum\_^s^f^(0)$ Now we consider the following differential equation: :$\backslash sum\_^a\_if^(t)=\backslash phi(t)$ with given initial conditions :$f^(0)=c\_i$ Using the linearity of the Laplace transform it is equivalent to rewrite the equation as :$\backslash sum\_^a\_i\backslash mathcal\backslash =\backslash mathcal\backslash$ obtaining :$\backslash mathcal\backslash \backslash sum\_^a\_is^i-\backslash sum\_^\backslash sum\_^a\_is^f^(0)=\backslash mathcal\backslash$ Solving the equation for $\backslash mathcal\backslash$ and substituting $f^(0)$ with $c\_i$ one obtains :$\backslash mathcal\backslash =\backslash frac+\backslash sum\_^\backslash sum\_^a\_is^c\_\}a\_is^i\}$ The solution for ''f''(''t'') is obtained by applying the inverse Laplace transform to $\backslash mathcal\backslash .$ Note that if the initial conditions are all zero, i.e. :$f^(0)=c\_i=0\backslash quad\backslash forall\; i\backslash in\backslash$ then the formula simplifies to :$f(t)=\backslash mathcal^\backslash left\backslash \backslash over\backslash sum\_^a\_is^i\}\backslash right\backslash \}$ ==An example== We want to solve :$f\text{'}\text{'}(t)+4f(t)=\backslash sin(2t)$ with initial conditions ''f''(0) = 0 and ''f′''(0)=0. We note that :$\backslash phi(t)=\backslash sin(2t)$ and we get :$\backslash mathcal\backslash =\backslash frac$ The equation is then equivalent to :$s^2\backslash mathcal\backslash -sf(0)-f\text{'}(0)+4\backslash mathcal\backslash =\backslash mathcal\backslash$ We deduce :$\backslash mathcal\backslash =\backslash frac$ Now we apply the Laplace inverse transform to get :$f(t)=\backslash frac\backslash sin(2t)-\backslash frac\backslash cos(2t)$ 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「Laplace transform applied to differential equations」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.