Lindelöf theorem, that is ordinary and Partial Differential Equations PDF from the case for partial differential equations. The Cauchy problem for the Laplace equation is called ill-posed or not well-posed, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

Författare: W. M. Everitt.

Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems. In PDEs, it is common to denote partial derivatives using subscripts. Some linear, second-order partial differential equations can be as parabolic, hyperbolic and elliptic. PDE is second-order in that region.

Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. An example is the wave equation. Elliptic: the eigenvalues are all positive or all negative. Parabolic: the eigenvalues are all positive or all negative, save one that is zero. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.