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Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

Partial differential equations (PDEs) contain partial derivatives and usually contain two or more variables; the single-variable cases with normal derivatives are ordinary differential equations.

In general partial differential equation can be written in the form $$f(x, y, ,\dots , u, u_x, u_y, \dots , u_{xx}, u_{xy}, \dots )=0$$involving several independent variables $$x, y, \dots ,$$ an unknown function $$u$$ of these variables, and the partial derivatives $$u_x, u_y, \dots, u_{xx}, u_{xy}, \dots$$, of the function.

Subscripts on dependent variables denote differentiations, e.g., $$u_x\equiv \frac{\partial u}{\partial x},\quad u_{xy}\equiv \frac{\partial^2 u}{\partial x \partial y },$$and so on. The $$f$$ denotes a function defined on a subset of a finite dimensional space. Functional operators on functions like $$f(u):=\frac{du}{dx}$$ are not allowed in this definition. For instance, Laplace's equation in three dimensions $$u_{xx} + u_{yy} + u_{zz} = 0$$ is defined by $$f(x,y,z,\nabla u,\nabla^2 u) = 0$$, where $$f(a,b,c,v,M) := M_{11} + M_{22} + M_{33}.$$

Questions with this tag may be about, among other things:

1. Analysis of existence and uniqueness of classical/strong/weak/viscous/etc. solutions in boundary value problems/Cauchy problems/Riemann problems.
2. Functional analysis related to PDEs, e.g., theories of Sobolev spaces, Bochner spaces, analysis of linear/nonlinear differential operators, and pseudodifferential operators, etc.
3. The stability, or long-term behavior of the solution.
4. Different methods of solving PDEs, separation of variables, Fourier transform, solitons, method of characteristics.
5. The solution technique of the Euler-Lagrange equations from calculus of variations.
6. Equation-relevant theory in other fields, e.g. Hyperbolic conservation laws in fluid/gas dynamics, Maxwell's equations in electromagnetism, Hamilton-Jacobi equation in control theory, etc.

Please consider using more specific tags if your question addresses some of the aspects in that field, e.g., , , , , , , , .

References:

"Partial Differential Equations" by L. C. Evans

" Linear Partial. Differential Equations for Scientists and Engineers" by Tyn Myint-U & Lokenath Debnath

"Differential Equations" by Shepley L. Ross

See also the Wikipedia and MathWorld entries.