Partial differential equation
To discuss such existence and uniqueness theorems, it is necessary to be precise about the domain of the "unknown function." Otherwise, speaking only in terms such as "a function of two variables," it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.
The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity," in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
When writing PDEs, it is common to denote partial derivatives using subscripts. For example:
Nearest to linear PDEs are semilinear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semilinear PDE in two variables is
In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.
An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.