An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon

In Science and engineering, so many nonlinear phenomena are presented as partial differential, ordinary differential and integral equation models. So in this article, our aim is to study on Hammerstein type nonlinear integral equation μS(σ)=−RTLn[∫abPS(σ∘)exp{−ϵ(σ,σ∘)+μS(σ∘)RT}dσ∘] Where R is the gas constant, T the temperature, the term ϵ(σ,σ∘) denotes the interaction energy expression for the segments with screening charge density σ and σ∘ respectively, the molecular interaction in solvent is PS(σ) and the chemical potential of the surface segments is described by μS(σ) that should be determined. This integral equation forms the basis for the conductor-like screening model for real solvents (COSMO-RS) which is appeared in chemical phenomena. Some of numerical methods usually use techniques based on a projection in terms of some basis functions or use some quadrature formulas, and the convergence rate of these methods are usually of polynomial order with respect to N, where N represents the number of terms of the expansion or the number of points of the quadrature formula. Also, in projection methods nonlinear Hammerstein integral equation is reduced to the nonlinear algebraic equations which solving them in large scales needs high memory capacity and CPU time and because of error propagation convergence of numerical technique may be at risk. So this paper presents a powerful numerical approach based on Sinc quadrature which has exponential type convergence rate to solve conductor-like screening model for real solvents (COSMO- RS). The approach is based on preparing an iterative method to recognize the Hammerstein integral equation for the determination of the chemical potential of a surface segment as a function of screening charge density.