Power Series Solution of Differential Equations

Nonlinear Equations

The power series method can be applied to certain nonlinear differential equations, though with much less flexibility. Generally, the power series method will give solutions only to initial value problems (opposed to boundary value problems), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined (by superposition) to solve boundary value problems as well. A further restriction is that the series coefficients will be specified by a nonlinear recurrence (the nonlinearities are inherited from the differential equation).

In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series.

As an example, consider the initial value problem

which describes a solution to capillary-driven flow in a groove. Note the two nonlinearities: the first and second terms involve products. Note also that the initial values are given at, which hints that the power series must be set up as:

since in this way

which makes the initial values very easy to evaluate. It is necessary to rewrite the equation slightly in light of the definition of the power series,

so that the third term contains the same form that shows in the power series.

The last consideration is what to do with the products; substituting the power series in would result in products of power series when it's necessary that each term be its own power series. This is where the identity

$\left(\sum_{i = 0}^{\infty} a_i x^i\right) \left(\sum_{i = 0}^{\infty} b_i x^i\right) = \sum_{i = 0}^{\infty} x^i \sum_{j = 0}^i a_{i - j} b_j$

is useful; substituting the power series into the differential equation and applying this identity leads to an equation where every term is a power series. After much rearrangement, the recurrence

$\sum_{j = 0}^i \left((j + 1) (j + 2) c_{i - j} c_{j + 2} + 2 (i - j + 1) (j + 1) c_{i - j + 1} c_{j + 1}\right) + i c_i + (i + 1) c_{i + 1} = 0$

is obtained, specifying exact values of the series coefficients. From the initial values, and, thereafter the above recurrence is used. For example, the next few coefficients:

$c_2 = -\frac{1}{6} \quad ; \quad c_3 = -\frac{1}{108} \quad ; \quad c_4 = \frac{7}{3240} \quad ; \quad c_5 = -\frac{19}{48600} \ \dots$

A limitation of the power series solution shows itself in this example. A numeric solution of the problem shows that the function is smooth and always decreasing to the left of, and zero to the right. At, a slope discontinuity exists, a feature which the power series is incapable of rendering, for this reason the series solution continues decreasing to the right of instead of suddenly becoming zero.

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Power Series Solution of Differential Equations - Nonlinear Equations