Matched Asymptotic Expansion Solutions of Nonlinear Partial Differential Equations with a Small Parameter

The initial value problem in a Banach space \[ ε {{du}}{{dt}} = B(t,ε )u + F(t,u,ε ),\]u(0,ε ) = ζ (ε ) is studied. Here, B(t,ε ) is a linear, and F(t,u,ε ) a nonlinear, operator in u, and ε is a small parameter. The solution is found for the case where the null space of B is one-dimensional by a method of matched asymptotic expansions. The form of the solution is shown to depend on the nature of the projection on the null space of B of the coefficient of ε in the expansion of F(t,u,ε ) about ε = 0.