Analytic Solutions by Power Series

Many classical boundary value problems have been solved by using power series. Equations leading to Bessel functions and sines and cosines are examples. This technique is useful if the problem is linear, but it is less useful for nonlinear problems. Examples illustrating both linear and nonlinear problems solved by power series are given below.

Consider the linear problem

The series on the right-hand side is known, i.e. the coefficients {b_i} are known numbers. We try to expand the solution in an infinite series in x.

The derivatives are

Putting these expressions into the differential equation gives

Expanded out, this is

The principle of the power series method is to make this equation valid for each power of x; if that is done, then the equation is satisfied. Thus, we set the coefficients of like powers of x equal to each other.

The solution is thus

The last term is now known, and is a solution to the nonhomogeneous equation. We have two constants as yet undetermined, and they are found by applying the two boundary conditions.

This completes the solution. We must show that the infinite series converges for all x (this will may depend on {b_i}). If we want to evaluate y at a specific, x, though, we usually would use a computer to do so. In that case, we have to decide how many terms to use in the evaluation. It is sometimes possible to bound the sum of all terms higher (in powers of x) than a certain power, and that could be used to estimate the error in our evaluation. Usually, however, we simply evaluate the expression for a different number of terms and compare the results. If the results when using 100 terms do not differ appreciably from those when using 200 terms, we accept the results. While we call the solution 'exact', it is clear that at the point of evaluation, there is the possibility of numerical error that must be assessed. By writing the solution down, others can verify that it solves the differential equation; when we evaluate the solution on the computer, though, we must validate our program (to show that it uses the solution correctly) and pick numerical parameters that might affect the accuracy (the number of terms used in the evaluation).

Take Home Message: The power series method is useful sometimes, but we still have to decide the accuracy of the solution when we evaluate it with a finite number of terms. It is, however, the method that provides Bessel functions, sine functions, etc.

See also: Power series method applied to a nonlinear problem.