They usually have many solutions; one often considers additional boundary conditions which restrict the solution set. Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think that the parameters are function data (informally put, this means that the set of solutions is much larger). That is true fairly generally, unless the equations are heavily over-determined.
Partial differential equations are ubiquitous in science, as they describe phenonena such as fluid flow, gravitational fields, and electromagnetic fields. They are important in fields such as aircraft simulation, computer graphics, and weather prediction. The central equations of general relativity and quantum mechanics are also partial differential equations.
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2 Methods to solve PDEs |
A generalization of Laplace's equation is Poisson's equation:-
The Schrödinger equation is a PDE at the heart of quantum mechanics.
All the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Important non-linear equations include the Navier-Stokes equations describing the flow of fluids and Einstein's field equations of general relativity.
There are no generally applicable methods to solve non-linear PDEs; indeed, many PDEs cannot be solved analytically at all. Nevertheless, some techniques can be used for several types of equations. In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. An alternative are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using high performance supercomputers.