Perturbation theory

Solution to an equation

An equation is a stipulation that two things are equal.

Sometimes equations have free variables. In such a case, we can say that a solution to the equation is an assignment of specific values to its free variables, such that the equation holds true.

There are three types of solutions that an equation might have.

1. Exact solutions

To have an exact solution is a bit of a vague concept. Characteristics of "exact" solutions

• The solution is exact, not approximate. It solves the equation on the nose, rather than solving it up to a small error term.
• We have a simple expression for the solution, in terms of other concepts.
• The above expression is finite.

Loosely speaking, when an equation admits an exact solution, it is called "integrable." [TODO really?]

Exact solutions are the ideal, but they are difficult to find.

The idea of an exact solution quite slippery. For example, if I have an equation E (say, an equation that I can prove admits a solution), I can just define a new concept C = "the solution to equation E," and suddenly---having done nothing at all---we can say that E admits an exact solution C. So the lesson to be learned here is that whether or not an equation admits an exact solution is not an intrinsic property of the equation itself, but rather it depends on the context of your knowledge; it depends on what sorts of concepts you have.

2. Numerical solutions

A numerical solution to an equation could be several things.

• Get some sort of power series like thing, which goes up to the Nth power of some small constant. This doesn't literally solve the equation, but it almost does.
• Discretize spacetime, discretize the equation, then solve it "exactly" in this discrete model