Interdisciplinary Applied Mathematics

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1. For I A — Ac |= O(e), the center manifold theory applies.

2. For | A — Ac I= 0(1), the Galerkin methods apply.

According to the center manifold theory (Troger and Steindl, 1991), the field variable can be decomposed into a form:

nc

u(x,t) = uc(x,t) + us(x,t) = ^2 qi(t)xi(x) + U(q(t), x),    (17.25)

i=1

where xi (x) are the active spatial modes, obtained from the solution of the eigenvalue problem related to the linear system

u = L(Ac)u.

Also, qi(t) are the time-dependent amplitudes, and us(x,t) can be represented by    an infinite    sum.    The key    point    is    that    the    influence    sum    of

the higher modes contained in us(x,t) can be expressed in terms of the lower-order modes by the function U(qi(t), x).

In applying the Galerkin methods, the field variable u(x, t) is expressed in the form

m

u(x,t) = X) qj (t)^j(x)

j=i

by a set of m comparison vectors фj (x) called the Galerkin basis, which satisfies the geometric and natural boundary conditions. Two subdivisions in the Galerkin methods exist, namely, the standard (linear) Galerkin method and the nonlinear Galerkin method. In the linear Galerkin method one neglects us(x,t) in equation (17.25). Therefore, the fast dynamics taken into account by the center manifold theory are completely ignored in the reduction process. Nonlinear Galerkin methods take into consideration the influence of higher modes and are also known by the name inertial manifolds in the mathematical literature. Two nonlinear Galerkin methods that have gained importance are (Steindl and Troger, 2001):

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