Regular perturbation

A regular perturbation admits an asymptotic expansion about the limiting problem without changing its relevant scale or losing solutions or conditions. A regular perturbation of a simple algebraic root follows from the implicit function theorem. A full problem can still be a singular perturbation even when some individual solution branches are regular.

For $(L_0+ϵL_1)u=f(ϵ)$ and $Bu=g(ϵ)$, expand all three quantities in powers of $ϵ$. The order-$n$ problem has $L_0{u}_n=f_n-L_1{u}_{n-1}$ and $B{u}_n=g_n$, with $u_{-1}=0$. In particular, Taylor series boundary coefficients are generally nonzero even when the differential operator depends linearly on the parameter. If the homogeneous realization of $L_0$ is invertible and boundary data admit smooth lifts, the same inverse solves each successive inhomogeneous boundary value problem. Induction establishes unique coefficients at every order. This formal conclusion does not alone assert convergence of the full power series; convergence needs additional bounds on the inverse and the perturbation.