“partial differential equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

partial differential equation

释义

partial differential equation

Let z be a function of variables x and y let zx, zy, zxx, zxy, zyy, zxxx,… denote the partial derivatives and higher-order partial derivatives of z. A partial differential equation (PDE for short) is an equation involving x, y, z and its partial derivatives (compare ordinary differential equation, stochastic differential equation). The order of the PDE is the highest total order of any partial derivative that appears. These notions readily generalize to functions of more than two variables.

As the differential operators ∂/∂x and ∂/∂y respectively send functions of y only and x only to 0, a PDE of order n might be expected to have n arbitrary functions, rather than n arbitrary constants, in its general solution. As with ODEs, this is at best a rule of thumb and does not always hold. For example, the first order PDE zx + zy = 0 has general solution z = f(x–y), and the second order PDE zxy = 0 has general solution z = f(x) + g(y).

See heat equation, Laplace's equation, parabolic differential equation, wave equation.

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