Partial Differential Equation

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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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