Perturbation Theory of Nonlinear Boundary-Value Problems

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Abstract
A systematic perturbation theory is presented for the analysis of nonlinear problems. The lowest-order result is just that obtained by linearizing the problem, and the higher-order terms are the solutions of inhomogeneous linear problems. The essential feature of the method is the procedure for avoiding secular terms, which is based on the Lindstedt-Poincaré technique employed in celestial mechanics. The method is applied to the following nonlinear boundary value problems: (1) temperature distribution due to a nonlinear heat source or sink; (2) self-sustained oscillations of a system with infinitely many degrees of freedom; (3) forced vibrations of a ``string'' with a nonlinear restoring force; (4) superconductivity in a body of arbitrary shape with external magnetic field; (5) superconductivity in an infinite film with parallel magnetic field; (6) comparison of solutions of the Hartree, Fock, and Schrödinger equations for the helium atom. The results in each case are different both qualitatively and quantitatively from those of the linear theory.

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