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

P. Bakhtiari, S. Abbasbandy and R.A. Van Gorder

Applied Mathematics and Computation, 2018, vol. 339, issue C, 132-143

Abstract: 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.

Keywords: Swift–Hohenberg equation; Reproducing kernel method; Boundary value problem; Convergence analysis (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:339:y:2018:i:c:p:132-143

DOI: 10.1016/j.amc.2018.07.006

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