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- Der Separationsansatz dient der Lösung partieller Differentialgleichungen mit mehreren Variablen. Der Produktansatz ist ein Spezialfall. (de)
- A separable partial differential equation is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (PDE) can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations. The most common form of separation of variables is simple separation of variables in which a solution is obtained by assuming a solution of the form given by a product of functions of each individual coordinate. There is a special form of separation of variables called -separation of variables which is accomplished by writing the solution as a particular fixed function of the coordinates multiplied by a product of functions of each individual coordinate. Laplace's equation on is an example of a partial differential equation which admits solutions through -separation of variables; in the three-dimensional case this uses 6-sphere coordinates. (This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of integrals; see separation of variables.) (en)
- 可分離變數的偏微分方程（PDE）是指一種偏微分方程，在求解時可以用分離變數法分離為一組階數較低的微分方程。這一般是因為偏微分方程滿足某種形式或是對稱。因此可以利用求解一組較簡單的偏微分方程來求解原問題，若可以簡化為一維的問題，甚至可以用變成常微分方程。 分離變數法最常見的形式是其解可以假設為幾個函數的積，而每個函數只有一個自變數。例如給予一個 元函數 的偏微分方程，猜想解答的形式為 。 這是一種特別的分離變數法，稱為-分離變數法，此方式是將解寫成和座標有關的固定函數，以及以各座標為自變數函數的乘積。上的拉普拉斯方程是一個可以用-分離變數法求解的偏微分方程的例子，在三維空間下會用來求解。 偏微分方程的分離變數法和常微分方程的分離變數法不同，後者是指問題可以變成二個積分相等的形式。 (zh)
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- Der Separationsansatz dient der Lösung partieller Differentialgleichungen mit mehreren Variablen. Der Produktansatz ist ein Spezialfall. (de)
- 可分離變數的偏微分方程（PDE）是指一種偏微分方程，在求解時可以用分離變數法分離為一組階數較低的微分方程。這一般是因為偏微分方程滿足某種形式或是對稱。因此可以利用求解一組較簡單的偏微分方程來求解原問題，若可以簡化為一維的問題，甚至可以用變成常微分方程。 分離變數法最常見的形式是其解可以假設為幾個函數的積，而每個函數只有一個自變數。例如給予一個 元函數 的偏微分方程，猜想解答的形式為 。 這是一種特別的分離變數法，稱為-分離變數法，此方式是將解寫成和座標有關的固定函數，以及以各座標為自變數函數的乘積。上的拉普拉斯方程是一個可以用-分離變數法求解的偏微分方程的例子，在三維空間下會用來求解。 偏微分方程的分離變數法和常微分方程的分離變數法不同，後者是指問題可以變成二個積分相等的形式。 (zh)
- A separable partial differential equation is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (PDE) can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations. (en)
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- Separationsansatz (de)
- Separable partial differential equation (en)
- 可分離變數的偏微分方程 (zh)
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