Reproducing kernel method for the numerical solution of the 1D Swift-Hohenberg equation

The Swift-Hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems. We study the 1D Swift-Hohenberg equation in order to demonstrate the utility of the reproducing kernel method. The solution is represented in the form of a series in the reproducing kernel space, and truncating this series representation we obtain the n-term approximate solution. In the first approach, we aim to explain how to construct a reproducing kernel method without using Gram-Schmidt orthogonalization, as orthogonalization is computationally expensive. This approach will therefore be most practical for obtaining numerical solutions. Gram-Schmidt orthogonalization is later applied in the second approach, despite the increased computational time, as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the Swift-Hohenberg equation. We demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values.