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Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement.

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A central goal of scientists and engineers is obtaining solutions of the differential equations that govern their physical systems.This can be done numerically for large and/or complex systems using finite-difference methods, finite-element methods, or spectral methods.This chapter gives an introduction and the formal basis for these methods, with particular emphasis on finite-difference methods.Second-order partial differential equations are classified as elliptic, parabolic, or hyperbolic, and the numerical methods developed for such equations must be faithful to their mathematical properties.

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NUMERICAL SOLUTIONS TO A FRACTIONAL DIFFUSION EQUATION USED IN MODELLING DYE-SENSITIZED SOLAR CELLS

7 - Numerical Methods for Differential Equations

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NUMERICAL SOLUTIONS TO A FRACTIONAL DIFFUSION EQUATION USED IN MODELLING DYE-SENSITIZED SOLAR CELLS

7 - Numerical Methods for Differential Equations

Summary