Separation of variables
Main article: Separable partial differential equation
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. The logic of this technique may be confusing upon ?rst acquaintance, but it rests on the uniqueness of solutions to differential equations: as with ODEs, if one can ?nd any solution that solves the equation and satis?es the boundary conditions, then it is the solution. We assume as an ansatz that the dependence of the solution on space and time can be written as a product of terms that each depend on a single coordinate, and then see if and how this can be made to solve the problem.
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ODE 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). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.
This generalizes to the method of characteristics, and is also used in integral transforms.
Method of characteristics
Main article: Method of characteristics
In special cases, one can find characteristic curves on which the equati n reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.
More generally, one may find characteristic surfaces.
An integral transform may transform the PDE to a simpler one, in particular a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example for use of a Fourier integral.
In physics and mathematics, an ansatz (German word roughly corresponding to the English word attempt or approach) is an educated guess that is verified later by its results.
An ansatz is the establishment of:
the starting equation(s),
describing a mathematical or physical problem or solution. It can take into consideration boundary conditions. After an ansatz has been established (constituting nothing more than an assumption), the equations are solved for the general function of interest (constituting a confirmation of the assumption).