Partial Differential Equation

From PDEwiki

A partial differential equation or PDE is an identity that relates the independent variables (say $x \in \mathbb{R}^d$ ), the dependent variables (say $u: \mathbb{R}^d \longmapsto \mathbb{R}$ ) and the partial derivatives of $u$ . For example, a general PDE of second order may be expressed

$F (x, u(x), \partial_{x_1} u, \dots, \partial_{x_d} u, \dots , \partial^2_{x_i x_j} u , \dots \partial^2_{x_d x_d} u) = 0$ .

A solution of the PDE is a function $u$ which satisfies the identity. The solution may not be unique. To be unique it may require supplemental boundary conditions.

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Category: Concept