Analysis of Malmquist-Takenaka-Christov rational approximations with applications to the nonlinear Benjamin equation
Provides a rigorous convergence analysis for MTC approximations applied to nonlinear PDEs with Fourier multipliers, offering an alternative to spectral methods for problems with non-smooth symbols.
The paper proves rapid convergence of Malmquist-Takenaka-Christov (MTC) approximations for functions with mild asymptotic restrictions, and demonstrates a collocation MTC scheme for the nonlinear Benjamin equation that converges rapidly with efficiency comparable to spectral Fourier methods.
In the paper, we study approximation properties of the Malmquist-Takenaka-Christov (MTC) system. We show that the discrete MTC approximations converge rapidly under mild restrictions on functions asymptotic at infinity. This makes them particularly suitable for solving semi- and quasi-linear problems containing Fourier multipliers, whose symbols are not smooth at the origin. As a concrete application, we provide rigorous convergence and stability analyses of a collocation MTC scheme for solving the nonlinear Benjamin equation. We demonstrate that the method converges rapidly and admits an efficient implementation, comparable to the best spectral Fourier and hybrid spectral Fourier/finite-element methods described in the literature.