(192m) Application of the Extended Correction Function Method to Solve the Poisson Boltzmann Equation Under Non-Isothermal Conditions

Many technological applications need of differential models either for their microscopic or macroscopic description and, frequently, they involve non-linear equations. There is a class of differential models, i.e. the diffusive-reaction type models that have a very large number of applications in different bio-engineering problems. These include transport and reaction in biological processes, heat and mass transfer in more than one phase, and electrostatic potentials in many current molecular processes, just to name a few. Solutions to these models are either by linear approximations of the non-linear source, by other approximations of such non-linearity, or by numerical methods that frequently are heavily dependent of the mesh size for a successful convergence. In this contribution, an analytical-logarithmic approach, proposed by Oyanader and Arce, is revisited to extend its application to more complex modelling of bio-systems for different geometries. Due to the lack of general methods to derive analytical solutions for second order differential models with non-linear sources and constant, this study has focused on developing an efficient and economical procedure to obtain a formal analytical solution for such models that is used in a simple predictor-corrector approach to predict the correct solution. The proposed extended method involves the use of a recursive function, Æ’_AO, of the nonlinear dependable variable that work as a corrector function. The procedure converts the nonlinear ordinary differential equation in a simpler pseudo-linear ordinary differential equation whose analytical solution is later modified by means of the Æ’_AO correction function to obtain the correct solution. Several numerical examples will be presented to illustrate the method.

(*) Oyanader, M., Arce, P., “A New and Simpler Approach for the Solution of the Electrostatic Potential Differential Equation. Enhanced Solution for Planar, Cylindrical and Annular Geometries,” Journal of Colloid and Interface Science, 2005, 284, 315.