Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...

In this work, we present the notion of a weakly symmetric function. We show that subset all elements an arbitrary vector space functions is itself. Moreover, some algebra algebra. Also consider on complex Banach $L_p[0,1]$ Lebesgue measurable complex-valued $[0,1]$ for which $p$th power absolute value integrable. every continuous linear functional $L_p[0,1],$ where $p\in (1,+\infty),$ can be ap...

CRYSCOR is a periodic post-Hartree–Fock program based on local functions in direct space, i.e., Wannier functions and projected atomic orbitals. It uses atom centered Gaussians as basis functions. The Hartree–Fock reference, as well as symmetry information, is provided by the CRYSTAL program. CRYSCOR presently features an efficient and parallel implementation of periodic local second order Møll...

The notion of causal operators is extended to periodic boundary value problems with nonlinear boundary conditions in this paper. By utilizing the monotone iterative technique and the method of lower and upper solutions (resp. weakly coupled lower and upper solutions), we establish the existence of the extremal solutions (resp. weakly coupled extremal quasi-solutions) for nonlinear periodic boun...

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integro-differential equation (FIDE) with a weakly singular kernel‎. ‎First‎, ‎a collocation method based on Haar wavelets (HW)‎, ‎Legendre wavelet (LW)‎, ‎Chebyshev wavelets (CHW)‎, ‎second kind Chebyshev wavelets (SKCHW)‎, ‎Cos and Sin wavelets (CASW) and BPFs are presented f...