Renormalization methods for higher order differential equations

Abstract

We adapt methodology of statistical mechanics and quantum field theory to approximate solutions to an arbitrary order ordinary differential equation boundary value problem by a second-order equation. In particular, we study equations involving the derivative of a double-well potential such as u − u3 or − u + 2u3. Using momentum (Fourier) space variables we average over short length scales and demonstrate that the higher order derivatives can be neglected within the first cumulant approximation, once length is properly renormalized, yielding an approximation to solutions of the higher order equation from the second order. The results are confirmed using numerical computations. Additional numerics confirm that the main role of the higher order derivatives is in rescaling the length.