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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. This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals).
Broer–Kaup equations; Burgers' equation; Euler equations; Fokker–Planck equation; Hamilton–Jacobi equation, Hamilton–Jacobi–Bellman equation; Heat equation; Laplace's equation
Regularity is a topic of the mathematical study of partial differential equations(PDE) such as Laplace's equation, about the integrability and differentiability of weak solutions. Hilbert's nineteenth problem was concerned with this concept. [1] The motivation for this study is as follows. [2]
PDE-constrained optimization; Perfectly matched layer; Perron method; Petrov–Galerkin method; Phase space method; Poincaré–Lelong equation; Poisson's equation; Population balance equation; Porous medium equation; Potential theory; Primitive equations; Proper orthogonal decomposition; Pseudo-differential operator; Pseudoanalytic function
A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation.
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For a first-order PDE, the method of characteristics discovers so called characteristic curves along which the PDE becomes an ODE. [1] [2] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.
The Crank–Nicolson stencil for a 1D problem. In mathematics, especially the areas of numerical analysis concentrating on the numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the point of interest by using a numerical approximation routine.