Partial differential equation

A partial differential equation (PDE) is an equation for an unknown function $u$ of several independent variables involving its partial derivatives. A finite-order equation can be written schematically as

F\bigl(x,(\partial^\alpha u(x))_{|\alpha|\le m}\bigr)=0.

Its order is the highest derivative order on which the relation actually depends. A system consists of several such relations, often for a vector-valued unknown.

Time and space

For an evolution equation, one variable is distinguished as time $t$ , with the remaining variables denoted $x$ . The heat equation $\partial_tu=\nu\Delta u$ is first order in time and second order in space. The way a solution satisfies the equation must be specified; classical solutions satisfy it pointwise.

References

John K. Hunter, Notes on Partial Differential Equations.