Journal of Applied & Computational Mathematics

This paper gives two definitions: Star with coefficient a real and Star System with coefficient α in five unknowns. Examples of Star-System and Star-set are given, a relationship between two star-Systems is noted, and some general theorems are proven.

The journal has been started in the year 2012 and it released its first volume in the same year. Till now, the journal has released 8 volumes by the end of 2019 and is successfully running its 9th Volume. The Journal is an open access publication that features scientific works of considerable significance and impact in all branches of mathematics that concerns with mathematical methods that are typically used in science, engineering, business, and industry. Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role in emphasizing algorithms, numerical methods and symbolic methods with the concept of ubiquitous computing. The journal provides an Open Access platform to analyse new computational and applied techniques for solving the scientific or engineering problems.

The journal has been started in the year 2012 and it released its first volume in the same year. Till now, the journal has released 8 volumes by the end of 2019 and is successfully running its 9th Volume. The Journal is an open access publication that features scientific works of considerable significance and impact in all branches of mathematics that concerns with mathematical methods that are typically used in science, engineering, business, and industry. Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role in emphasizing algorithms

In this paper, for a class of complex optical waveguide, the high-precision computation of the propagation constants β are studied. The corresponding Sturm-Liouville (S-L) problem is represented as in an open domain (open on one side), where x is a given value. Firstly, a perfectly matched layer is used to terminate the open domain. Secondly, both the equation and the complex coordinate stretching transformations are constructed. Thirdly, the S-L problem is turned to a simplified form such as in a bounded domain. Finally, the coefficient function is approximated by a piecewise polynomial of degree two. Since the simplified equation in each layer can be solved analytically by the Kummer functions, the approximate dispersion equation is established to the TE case. When the coefficient function is continuous, the approximate solutions converge fast to the exact ones, as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision eigenmodes may be obtained by the Müller's method with suitable initial values.