Introduction to Symmetry Analysis Chapter 8 - Ordinary Differential Equations Brian Cantwell Department of Aeronautics and Astronautics Stanford University

8.1 Extension of Lie Groups in the Plane

The Extended Transformation is a Group Two transformations of the extended group Compose the two transformations

The last relation is rearranged to read Differentiating F and G gives Comparing the expressions in parentheses we have The composed transformation is in exactly the same form as the original transformation!

Finite transformation of the second derivative The twice extended finite transformation is

Finite transformation of higher derivatives

The p-th order extended group is where

Infinitesimal transformation of the first derivative Recall the infinitesimal transformation of coordinates where Substitute Expand and retain only the lowest order terms

The once-extended infinitesimal transformation in the plane is where the infinitesimal function fully written out is

Infinitesimal transformation of the second derivative Expand and retain only the lowest order terms where

The infinitesimal function transforming third derivatives is

The infinitesimal transformation of higher order derivatives Expand and retain only the lowest order terms. The p times extended infinitesimal transformation is where

8.2 Expansion of an ODE in a Lie Series - the Invariance Condition for ODEs

The characteristic equations associated with extended groups are

Construction of the general first order ODE that admits a given group - the Ricatti Equation Let the first integral of the group be In principle we can solve for either x or y. Assume we solve for y. The equation takes the following form. where

The general solution of the Ricatti equation can always be determined if a particular solution of the equation can be found. A particular solution in this case is To demonstrate take the differential of f. Dividing by yields the Ricatti equation in terms of f. Now let. and work out the equation that governs h[x].

The general second-order ordinary differential equation is invariant under the twice-extended group if and only if

Consider the case of the simplest second-order ODE The invariance condition is Fully written out the invariance condition is For invariance this equation must be satisfied subject to the condition that y is a solution of

The determining equations of the group are These equations can be used to work out the unknown infinitesimals.

Assume that the infinitesimals can be written as a multivariate power series Insert these series into the determining equations

The coefficients must satisfy the following algebraic system Finally the infinitesimals are

The software package used on Yxx = 0

Example 8.2

Example 8.2 – Using the software

The two parameter group of the Blasius equation The invariance condition

Written out the invariance condition is

Now gather coefficients of like products of derivatives of y

The function y[x] is a solution of the Blasius equation The function y[x] is a solution of the Blasius equation. This is a constraint on the invariance condition that can be used to eliminate the third derivative.

Further simplify and

Finally the determining equations are From which the two parameter group of the Blasius equation is determined to be