By making use of the concept ofq-calculus, various types of generalized starlike functions of orderαwere introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types ofq-starlike functions of orderα. We also introduce and study a new subclass ofq-starlike functions of orderα. Moreover, we give some properties of thoseq-starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.

Two numerical models to obtain the solution of the KdV equation are proposed. Numerical tools, compact fourth‐order and standard fourth‐order finite difference techniques, are applied to the KdV equation. The fundamental conservative properties of the equation are preserved by the finite difference methods. Linear stability analysis of two methods is presented by the Von Neumann analysis. The new methods give second‐ and fourth‐order accuracy in time and space, respectively. The numerical experiments show that the proposed methods improve the accuracy of the solution significantly.