# On the numerical solution of nonlinear integral equation arising in conductor like screening model for realistic solvents

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## Abstract

The aim of this work is to introduce an efficient algorithm for the numerical solution of nonlinear integral equation arising from chemical phenomenon which is a famous equation in chemistry engineering. A procedure is described for transforming the nonlinear integral equation by using Chebyshev polynomials to nonlinear system of algebraic equations. Also, we present a convergence analysis and error bound for presented method. In addition, some numerical results are reported to evaluate the validity and applicability of the method and also comparison has been done with existing results.

## Keywords

Hammerestein integral equation Chemical phenomenon Chebyshev polynomials Numerical treatment## Introduction

*f*,

*k*and

*G*are known functions with

*G*(

*s*,

*u*) nonlinear in

*u*, and

*u*(

*t*) is a solution to be determined. These types of equations can be obtained by reformulation of some boundary value problems for ordinary differential equations, also mixed type of these equations can be seen in chemical reactor as the dynamic model (see e.g., [2, 3, 7, 8, 10, 12, 13, 14]). In this article, we study the numerical solution for a special kind of (1.1)

*R*and

*T*are the gas constant and the temperature, respectively, the interaction energy expression for the segments with screening charge density \(\delta\) and \(\delta _{1}\) has been explained by \(E_{\mathrm{int}}(\delta ,\delta _{1})\), \(P_{s}(\delta )\) is the molecular interaction in solvent and \(\mu _{s}(\delta )\) denotes the chemical potential of the surface segments that should be determined.

In 1995, A. Klamt achieved an approach that made it possible to compute the details of molecules quantum mechanically and he applied obtained results in an approximate statistical mechanics procedure which is called COSMO-RS. COSMO-RS is a quantum chemistry-based equilibrium thermodynamics method and it can show the link between the world of chemical quantum mechanics and engineering thermodynamics well. For the prediction of activity coefficients in multicomponent liquid mixtures, COSMO-RS uses conductor screening charges on molecular surface panels. Detailed description of these models may be found in [17].

The numerical solvability of nonlinear integral equations has been studied by several authors. Ghoreishi et al. [6] have been concerned with the numerical solution of Volterra–Hammerstein integral equations via the operational Tau method. Maleknejad et. al. in [9, 11] applied the computational method for Hammerstein integral equations based on Bernstein operational matrices and operational matrices of hybrid functions. In [12], the Sinc-quadrature formula are applied to approximate solution of (1.4). In [5], numerical solution of nonlinear Hammerstein integral equations via collocation method based on double exponential transformation is considered. Authors of [15] applied operational matrices of Chebyshev polynomials for solving singular Volterra integral equations. Our approach is to investigate how numerically one can solve a COSMO-RS integral equation with the Chebyshev polynomials.

To clarify the concepts of the Chebyshev polynomials, in “Preliminaries and notations” section we describe some notations and basic definitions of Chebyshev polynomials. We will apply Chebyshev polynomials for approximating the solution of nonlinear integral Eq. (1.2), then convergence analysis and error bound is proved in “Main result” section . Finally in “Numerical” section , some numerical experiments are reported to illustrate the efficiency and accuracy of our algorithm.

## Preliminaries and notations

In this section, we recall some necessary mathematical preliminaries and definitions of the Chebyshev polynomials, which are used further in this paper [1, 4].

### Definition 1

*i*are defined as follows

### Theorem 1

*The Chebyshev polynomials*\(T_{i}(t)\)

*satisfy the recurrence relation*

### Proposition 1

*Chebyshev polynomials are orthogonal functions with respect to a weighting function of*\((1-t^{2})^{-1/2}\),

*where*

## Main result

In this section for the sake of clarity, we prefer to split our presentation into two parts. The first part concerns with numerical scheme of nonlinear integral Eq. (1.2) based on the Chebyshev polynomials. The second part deals with convergence analysis of the presented method.

### Nonlinear function approximation and Chebyshev collocation method

*u*(

*t*) can be considered as

*a*,

*b*]. Owing to variety of orthogonal polynomials, in the following we choose Chebyshev polynomials. Now, we define an approximation function of the exact solution

*u*(

*t*) as follows

*u*(

*t*) will be obtained

### Error analysis

This section is concerned with the error estimation of the Chebyshev expansion for the nonlinear integral Eq. (1.2). The following theorem states and proves the main result of error analysis.

### Theorem 2

*Let*\(g(t)=\dfrac{1}{u(t)}\)

*be a second-order derivative square-integrable function defined on*\([-1, 1]\)

*with*\(|g^{''}(t)| \le L\)

*for some constant*

*L*,

*and let*\(g^{*}(t)\)

*be an approximation function of the exact solution*\(g(t)=\sum \nolimits _{i=0} ^{\infty }a_{i}T_{i}(t)\).

*Then, for sufficiently smooth functions*

*g*(

*t*)

*and*

*k*(

*t*,

*s*),

*with*\(|k(t,s)|\le M\),

*in*(1.2)

*we have*

*where*\(u^{*}(t)=\dfrac{1}{g^{*}(t)}\).

### Proof

## Numerical

In this section, three test problems will be considered to assess the accuracy and efficiency of the proposed method for solving nonlinear integral Eq. (1.2). Achieved nonlinear systems will be solved by Newton method and all calculations will be done by the Mathematics.

### *Example 1*

\(t_{i}\) | − 1.61 | − 1.11 | − 0.61 | − 0.11 | − 0.39 | − 0.89 | − 1.39 | − 1.89 |
---|---|---|---|---|---|---|---|---|

\(H(t_{i})\) | 0.052225 | − 0.237845 | − 0.201385 | 0.139676 | − 0.078774 | 0.129343 | − 0.044902 | − 0.000194 |

*u*(

*t*) will be obtained. Obtained numerical results can be seen in Figs. 1 and 2 with \(n=8, 16\), respectively. Example 1 was solved in [9, 16] by a method based on Sinc, Bernestein polynomials and Block pulse functions. In [16] reported results show that they can achieve good numerical results for \(n\ge 40,\) whereas reported results using proposed method show that we can achieve good numerical results for \(n\ge 8\). Figure 1 of the proposed method shows that high accuracy obtained for \(n=8\) in comparison with figure of Sinc method in [9].

### *Example 2*

Numerical results of Example 2 using proposed method

| Absolute error | ||
---|---|---|---|

Presented method (\(n=4\)) | Bernestein Method (n = 10) | Sinc Method (n = 10) | |

0 | 0 | 0 | \(7.77\times 10^{-3}\) |

0.1 | \(4.44 \times 10^{-16}\) | \(2.22\times 10^{-16}\) | \(2.85\times 10^{-3}\) |

0.2 | 0 | \(3.33\times 10^{-16}\) | \(1.04\times 10^{-3}\) |

0.3 | 0 | \(3.33\times 10^{-16}\) | \(3.86\times 10^{-4}\) |

0.4 | 0 | \(1.11\times 10^{-16}\) | \(1.42\times 10^{-4}\) |

0.5 | 0 | 0 | \(5.22\times 10^{-5}\) |

0.6 | 0 | \(1.11\times 10^{-16}\) | \(1.92\times 10^{-5}\) |

0.7 | 0 | \(1.11\times 10^{-16}\) | \(7.07\times 10^{-6}\) |

0.8 | \(1.11\times 10^{-16}\) | \(1.11\times 10^{-16}\) | \(2.60\times 10^{-6}\) |

0.9 | 0 | 0 | \(9.56\times 10^{-7}\) |

1 | 0 | 0 | \(3.52\times 10^{-7}\) |

### *Example 3*

## Conclusion

Being able to efficiently solve the problems in modeling chemical phenomenon that can be formulated as nonlinear integral equations is important. In this research, an efficient and accurate numerical scheme was presented for solving nonlinear integral equations and COSMO-RS integral equation which appeared in chemical phenomenon. By using the Chebyshev polynomials, the main problem was converted to a problem of solving a nonlinear system of algebraic equation. Comparing our numerical results and other numerical experiments, we may conclude the high accuracy of the presented method. With the availability of this methodology, it might be used to solve the other types of applied integral equations in other sciences.

## Notes

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