PCGA: Polynomial collocation genetic algorithm for singular Poisson-Boltzmann equation arising in thermal explosions

Heat generation as a result of the exothermic reaction reaches the environment mainly due to the conduction through the walls of the vessel. The balance between the heat generated and the heat conducted away, resulting in the explosion is described by the Frank-Kamenetzkii (FK) parameter ρ. The critical value of FK for which the explosion occurs depends upon the shape of the vessel, which requires the solution of governing singular nonlinear Poisson-Boltzmann equation. Owing to the exponential nonlinearity and singularity the analytical exact solution for the non-integer k values does not exist. This work focuses on implementing the polynomial collocation by exploiting the global optimization features of the genetic algorithm to solve the Poisson-Boltzmann equation for integer and non-integer shape factors (k). The governing equation was converted into coupled nonlinear algebraic equations and an objective function was formulated. The method was examined for six different configurations of the control parameters of GA to find the best set of parameters. The solution for temperature distribution is obtained for cylindrical (k = 1), parallelepiped (k = 0.438, 0.694), and an arbitrary shape (k = 0.5) respectively. The solution obtained from Polynomial Collocation Genetic Algorithm (PCGA) remained in good agreement with the corresponding analytical results for k = 1, with the minimum absolute error of 10⁻¹⁰. The critical values of the FK are obtained as 1.5,1.4,and1.7 for shape factor k=0.438,0.5,and0.694 respectively with the convergence of the order of 10⁻⁶to10⁻⁵. The obtained solution is fairly stable over appropriate independent runs with the variation in the fitness value ranging from 10⁻⁰⁵to10⁻⁰³. Further simulations were performed to validate the results through statistical error indices. The diminutive errors of the order of 10⁻⁶ confirm reliable optimum solution, accuracy, and stability.