Comparative experiment on the numerical solutions of Hammerstein integral equation arising from chemical phenomenon

In this paper, efficient numerical techniques have been proposed to solve nonlinear Hammerstein integral equation arising from chemical phenomenon. The following integral equation forms the basis for the conductor like screening model for real solvent which appeared in chemical phenomenon. Our aim is to solve the Hammerstein nonlinear integral equation μS(σ)=−RTln[∫PS(σ′)exp(−Eint(σ,σ′)−μS(σ′)RT)dσ′] where RR is the gas constant, TT is the temperature and the term Eint(σ,σ′)Eint(σ,σ′) denotes the interaction energy expression for the segments with screening charge density σσ and σ′σ′ respectively, the molecular interaction in solvent is PS(σ)PS(σ) and the chemical potential of the surface segments is described by μS(σ)μS(σ) which is to be determined. This COSMO-RS integral equation has been solved by Bernstein collocation method, Haar wavelet method, and Sinc collocation method. These methods reduce the integral equation to nonlinear system of algebraic equations and then this algebraic system has been solved numerically by Newton’s method. Comparison has been done for these methods. Illustrative examples have been discussed to demonstrate the validity and applicability of Bernstein collocation method, Haar wavelet method and Sinc collocation method.