Perturbation

perturbation, in mathematics, method for solving a problem by comparing it with a similar one for which the solution is known. Usually the solution found in this way is only approximate.

Perturbation is used to find the roots of an algebraic equation that differs slightly from one for which the roots are known. Other examples occur in differential equations. In a physical situation, an unknown quantity is required to satisfy a given differential equation and certain auxiliary conditions that define the values of the unknown quantity at specified times or positions. If the equation or auxiliary conditions are varied slightly, the solution to the problem will also vary slightly.

The process of iteration is one way in which a solution of a perturbed equation can be obtained. Let D represent an operation, such as differentiation, performed on a function, and let D + εP represent a new operation differing slightly from the first, in which ε represents a small constant. Then, if f is a solution of the common type of problem Df = cf, in which c is a constant, the perturbed problem is that of determining a function g such that (D + εP)g = cg. This last equation can also be written as (D - c)g = -εPg. Then the function g₁ that satisfies the equation (D - c)g₁ = -εPf is called a first approximation to g. The function g₂ that satisfies the equation (D - c)g₂ = -εPg₁ is called a second approximation to g, and so on, with the nth approximation g_n satisfying (D - c)g_n = -εPg_n-1. If the sequence g₁, g₂, g₃, . . ., g_n, . . . converges to a specific function, that function will be the required solution of the problem. The largest value of ε for which the sequence converges is called the radius of convergence of the solution.

Another perturbation method is to assume that there is a solution to the perturbed equation of the form f + εg₁ + ε²g₂ + . . . etc., in which the g₁, g₂, . . . etc., are unknown, and then to substitute this series into the equation, resulting in a collection of equations to solve corresponding to each power of ε.