Numerical solutions for nonlinear partial differential equations arising from modelling dye-sensitized solar cells

Dye-sensitized solar cells have generated diverse research directions, which include a mathematical model based on the diffusion of electrons in the conduction band of a nano-porous semiconductor (traditionally TiO\(_2\)). We solve the nonlinear diffusion equation under its boundary conditions, as stated by Anta et al. [J. Phys. Chem. B 110 (2006) pp 5372--5378]. We employ a standard finite difference method, a fourth order finite difference method scheme and a Runge--Kutta scheme. We calculate errors and evaluate the utility of each scheme as it applies to this boundary value problem. References J. A. Anta, F. Casanueva, and G. Oskam. A numerical model for charge transport and recombination in dye-sensitized solar cells. J. Phys. Chem. B, 110(11):5372–5378, 2006. doi:10.1021/jp056493h. F. Cao, G. Oskam, G. J. Meyer, and P. C. Searson. Electron transport in porous nanocrystalline TiO\(_2\) photoelectrochemical cells. J. Phys. Chem., 100(42):17021–17027, 1996. doi:10.1021/jp9616573. A. J. Frank, N. Kopidakis, and J. van de Lagemaat. Electrons in nanostructured TiO\(_2\) solar cells: transport, recombination and photovoltaic properties. Coordin. Chem. Rev., 248:1165–1179, 2004. doi:10.1016/j.ccr.2004.03.015. Y. Gacemi, A. Cheknane, and H. S. Hilal. Simulation and modelling of charge transport in dye-sensitized solar cells based on carbon nano-tube electrodes. Phys. Scripta, 87(3):035703–035714, 2013. doi:10.1088/0031-8949/87/03/035703. B. O'Regan and M. Gratzel. A low-cost, high-efficiency solar cell based on dye-sensitized colloidal TiO\(_2\) films. Nature, 353:737–740, 1991. doi:10.1038/353737a0. S. Sodergren, A. Hagfeldt, J. Olsson, and S. Lindquist. Theoretical models for the action spectrum and the current-voltage characteristics of microporous semiconductor films in photoelectrochemical cells. J. Phys. Chem., 98:5552–5556, 1994. doi:10.1021/j100072a023