An approximation of one-dimensional nonlinear Kortweg de Vries equation of order nine

This research presents the approximate solution of nonlinear Korteweg-de Vries equation of order nine by a hybrid staggered one-dimensional Haar wavelet collocation method. In literature, the underlying equation is derived by generalizing the bilinear form of the standard nonlinear KdV equation. The highest order derivative is approximated by Haar series, whereas the lower order derivatives are attained by integration formula introduced by Chen and Hsiao in 1997. The findings are shown in the form of tables and a figure, demonstrating the proposed technique’s convergence, robustness, and ease of application in a small number of collocation points.

Reconstruction of martingales and applications to multiple Haar series

Reconstruction theorems for martingales with respect to regular filtration are proved provided that the majorant of the martingale satisfies some specified condition. The ob-tained results are applied to obtain formulas for restoration of coeffcients for multiple Haar series.

On the absolute convergence of Fourier series with respect to general orthonormal systems

The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.