A computational solution of relativistic Thomas-Fermi equation of an atom with exponential collocation genetic algorithm optimization

Abstract

An Exponential Collocation Genetic Algorithm Optimization (ECGAO) is considered for solving the relativistic Thomas-Fermi equation of the statistical model of an atom. The governing model equation has been transformed into an optimization problem by collocation technique using exponential basis. Then, fitness function is formulated in terms of nonlinear coupled residual equations. The fitness values achieved remained of the order of10−4. The improvement in the ionization energy values for the heavier atoms from the ECGAO came with the small percentage error for Z=80,86and92 as 1.418630%, 0.301381% and 0.018417% respectively. The accuracy and stability of the proposed method were verified using Mean Absolute Deviation (MAD), Theil's Inequality Coefficient (TIC), and Mean Fitness Value (MFV). Approximately 74% of the trial runs have successfully achieved these low statistical error indices values (in the order of10−4to10−3) for the relativistic Thomas-Fermi model. After several experimentations, we have determined that the optimal values for crossover fraction, elite count, and migration fraction interval as 0.4, 16, 0.6, and 10, respectively. The proposed methodology is computationally consistent and more accurate for scientific investigations of the relativistic Thomas-Fermi equation.