= Partial differential equation
{title2=$F(x,t,u,u_t,u_x,u_{xx},\ldots)=0$}
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= PDE
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A <partial differential equation> relates an unknown <function> of several independent variables to its <partial derivatives>. For $u(x,t)$, the notation $u_t=\partial u/\partial t$ means changing time while holding position fixed; $u_{xx}=\partial^2u/\partial x^2$ measures how its spatial slope changes. Unlike an <ordinary differential equation>, a <partial differential equation> can describe an entire evolving profile rather than a single trajectory.
The <heat equation> $u_t=D u_{xx}$ smooths spatial variations, the <transport equation> $u_t+c u_x=0$ moves them, and <Laplace's equation> $\Delta u=0$ describes a spatial equilibrium. Here $D>0$ is a <diffusivity>, $c$ is a <velocity>, and $\Delta$ is the <Laplacian>. The <periodic heat and transport comparison> makes the difference visible using identical <initial conditions>.
A <partial differential equation> usually supplies a local rule rather than a complete answer. For example, every $u(x,t)=A+Bx$ satisfies the <heat equation> with zero <PDE source term>; data select which profile is relevant. A complete problem specifies its <domain of a function>, coefficients, <initial conditions>, <boundary conditions>, and meaning of <solution of a partial differential equation>. A <well-posed problem> has existence, uniqueness, and <continuous dependence on initial data> in stated spaces; the equation alone guarantees none of these.https://www.math.toronto.edu/ivrii/PDE-textbook/Chapter1/S1.2.html[Victor Ivrii, Initial and Boundary Value Problems]{ref}
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