It is shown that a conjecture concerning the derivatives of orthogonal polynomials, proved by Nevai in 1990 for generalized Jacobi weights, holds for doubling weights as well.

In this work, we employ the simple direct method to investigate the generalized mKdV equation with variable coefficients. By using this scheme, we found some new exact solutions of the equation, including four new types of Jacobi elliptic function solutions, and these solutions are degenerated to solitary wave solutions and triangle function solutions in the limit case when the modulus of the J...

Wavelets, which have many good properties such as time/freqency localization and compact support, are considered for solving the Hamilton-Jacobi-Bellman (HJB) equation as appears in optimal control problems. Specifically, we propose a Successive Wavelet Collocation Algorithm (SWCA) that uses interpolating wavelets in a collocation scheme to iteratively solve the Generalized-Hamilton-Jacobi-Bell...

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context Dunkl-type operators. A particularly noteworthy revelation that when specific parameter κ equals zero, derivative smoothly reduces both well-known Riesz and second-order derivative. Furthermore, we new concept: Sobolev space. This space defined characterized using versatile framework Dun...

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a w...