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變數分離法

變數分離法是求解常微分方程和偏微分方程的一種方法。

對於一個常微分方程

(dy)/(dx)=g(x)f(y),
(1)

其中 f(y) 在初始值附近非零，解由下式隱式給出

int(dy)/(f(y))=intg(x)dx.
(2)

如果積分可以以閉合形式完成，並且得到的方程可以求解 y (這兩個都是相當大的“如果”)，那麼就得到了問題的完整解。這種技術最適用的最重要的方程是 y^'=ay，指數增長和衰減方程 (Stewart 2001)。

對於函式 Phi(x,y,...) 和變數 x, y, ... 的偏微分方程，可以透過進行如下形式的替換來應用變數分離法

Phi(x,y,...)=X(x)Y(y)...,
(3)

將得到的方程分解為一組獨立的常微分方程，求解這些方程得到 X(x), Y(y), ...，然後將它們代回原始方程。

這種技術之所以有效，是因為如果獨立變數函式的乘積是常數，則每個函式必須單獨為常數。成功需要選擇合適的座標系，並且可能並非在所有情況下都可實現，具體取決於方程。變數分離法最早由洛必達於 1750 年使用。它在求解數學物理中出現的方程時特別有用，例如拉普拉斯方程、亥姆霍茲微分方程和薛定諤方程。

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亥姆霍茲微分方程, 拉普拉斯方程, 偏微分方程, Stäckel 行列式 在 課堂中探索此主題

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參考文獻

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變數分離法

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Renze, John 和 Weisstein, Eric W. “變數分離法。” 來自 —— 資源。 https://mathworld.tw/SeparationofVariables.html

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