Iraqi Journal for Computer Science and Mathematics

In this paper, we introduced a new type of operators, which we called ~-operator by cluster system, and studied its properties without conditions and with certain conditions. Through our study of this operator, we defined a new topology, which we called ~-topological space, since it does not, in general, constitute a regular topological space. We also presented ~-interior points, ~-closure, ~-exterior points and reviewed some theories and examples about this concept.

This study aims to present the modified Chebyshev wavelet collocation method (CWCM) to investigate and obtain the numerical approximation of $CO_2$ emissions from the energy sector utilizing the fractional mathematical model. The need for energy rises as the population grows. Burning fossil fuels produces a significant portion of the world's energy, which raises the atmospheric concentration of $CO_2$ and causes global warming. The combination of mathematical modeling studies and numerical simulations allows us to understand the $CO_2$ emissions from the energy sector. Our objective is to build an operational matrix of integration (OMI) based on Chebyshev wavelets and use it to the numerical solution of the $CO_2$ emanation from the energy sector model. Furthermore, these OMI are used to simplify fractional differential equations into an algebraic equation system. Lastly, graphic representations are used to visually present the results and provide empirical support for our theoretical conclusions. Because the CWCM method computes only a few terms and avoids rounding data, it produces correct answers for highly nonlinear situations. We compared the outcomes of the RK4 technique, the ND solver, and the devised method. The numerical results show that CWCM solves the emanation of $CO_2$ from the energy sector with remarkable accuracy and efficiency in both mathematical and environment view. Software called Mathematica has been used for numerical implementation and calculations.