A new numerical approach to solve Thomas-Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming.
In this study, a novel bio-inspired computing approach is developed to analyze the dynamics of nonlinear singular Thomas-Fermi equation (TFE) arising in potential and charge density models of an atom by exploiting the strength of finite difference scheme (FDS) for discretization and optimization through genetic algorithms (GAs) hybrid with sequential quadratic programming. The FDS procedures are used to transform the TFE differential equations into a system of nonlinear equations. A fitness function is constructed based on the residual error of constituent equations in the mean square sense and is formulated as the minimization problem. Optimization of parameters for the system is carried out with GAs, used as a tool for viable global search integrated with SQP algorithm for rapid refinement of the results. The design scheme is applied to solve TFE for five different scenarios by taking various step sizes and different input intervals. Comparison of the proposed results with the state of the art numerical and analytical solutions reveals that the worth of our scheme in terms of accuracy and convergence. The reliability and effectiveness of the proposed scheme are validated through consistently getting optimal values of statistical performance indices calculated for a sufficiently large number of independent runs to establish its significance.
Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, Pakistan.
Muhammad Asif Zahoor
0000-0001-9953-822X
MA
Raja
Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650 Pakistan.
Aneela
A
Zameer
Department of Basic Sciences, Riphah International University, Islamabad, Pakistan.
Aziz Ullah
AU
Khan
Department of Mathematics, Saint Xavier University, Chicago, IL 60655 USA.
Abdul Majid
AM
Wazwaz
Atomic physics; Bio-inspired heuristics; Genetic algorithm; Nonlinear singular systems; Sequential quadratic programming; Thomas–Fermi model