Analytic Solutions of Nonlinear Problems by Power Series

Consider the problem

which is a heat transfer problem with a variable thermal conductivity. Write the solution as a power series in x.

The derivatives are then

We hope that

If we put the expressions for the derivatives into the differential equation, we get

or

In the power series method we satisfy this equation for each power of x.

We can combine these equations; c₂ is given in terms of c₀ and c₁.

Likewise c₃ is

The values of c₁ and c₂ are set by the boundary conditions, here

The first condition requires that c₀ be zero. The second condition requires

If we put in the expressions for the various c_i we get

Unfortunately, this gives a nonlinear equation for c₁; furthermore, we need all the terms in order to find c₁. If we truncate at the terms shown one gets c₁ = 1.2956. The approximate solution is

This solution is compared with the exact solution in the figure.

Take Home Message: Clearly, the power series method is not very good for nonlinear problems, and many terms are needed and must be found numerically. In contrast, the Method of Weighted Residuals does not require many more terms to give a good solution.

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