Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T16:16:26.388Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  16 November 2021

BENJAMIN MALDON Open the ORCID record for BENJAMIN MALDON [Opens in a new window] ,
BISHNU PRASAD LAMICHHANE Open the ORCID record for BISHNU PRASAD LAMICHHANE [Opens in a new window]  and
NGAMTA THAMWATTANA Open the ORCID record for NGAMTA THAMWATTANA [Opens in a new window]
BENJAMIN MALDON*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW2308, Australia; e-mail: bishnu.lamichhane@newcastle.edu.au and natalie.thamwattana@newcastle.edu.au.
BISHNU PRASAD LAMICHHANE
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW2308, Australia; e-mail: bishnu.lamichhane@newcastle.edu.au and natalie.thamwattana@newcastle.edu.au.
NGAMTA THAMWATTANA
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW2308, Australia; e-mail: bishnu.lamichhane@newcastle.edu.au and natalie.thamwattana@newcastle.edu.au.
*
e-mail: benjamin.maldon@uon.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

dye-sensitized solar cellselectron densityefficiencyfractional diffusionfinite-difference methodsfinite-element methods

MSC classification

Primary: 35R11: Fractional partial differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 4 , October 2021 , pp. 420 - 433
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alberty, J., Carstensen, C. and Funken, S. A., “Remarks around 50 lines of Matlab: short finite element implementation”, Numer. Algorithms 20 (1999) 117–137; doi:10.1023/A:1019155918070.CrossRefGoogle Scholar
Anta, J. A., Casanueva, F. and Oskam, G., “A numerical model for charge transport and recombination in dye-sensitized solar cells”, J. Phys. Chem. B 110 (2006) 5372–5378; doi:10.1021/jp056493h.CrossRefGoogle ScholarPubMed
Anta, J. A., Idígoras, J., Guillén, E., Villanueva-Cab, J., Mandujano-Ramirez, H. J., Oskam, G., Pellejá, L. and Palomares, E., “A continuity equation for the simulation of the current–voltage curve and the time-dependent properties of dye-sensitized solar cells”, Phys. Chem. Chem. Phys. 14 (2012) 10285–10299; doi:10.1039/c2cp40719a.CrossRefGoogle ScholarPubMed
Baeumer, B., Kovács, M., Meerschaert, M. and Sankaranarayanan, H., “Reprint of: Boundary conditions for fractional diffusion”, J. Comput. Appl. Math. 339 (2018) 414–430; doi:10.1016/j.cam.2018.03.007.CrossRefGoogle Scholar
Benkstein, K. D., Kopidakis, N., van de Lagemaat, J. and Frank, A. J., “Influence of the percolation network geometry on electron transport in dye-sensitized titanium dioxide solar cells”, J. Phys. Chem. B 107 (2003) 7759–7767; doi:10.1021/jp0226811.CrossRefGoogle Scholar
Campos, R. G. and Huet, A., “Numerical inversion of the Laplace transform and its application to fractional diffusion”, Appl. Math. Comput. 327 (2018) 70–78; doi:10.1016/j.amc.2018.01.026.Google Scholar
Cao, F., Oskam, G., Meyer, G. J. and Searson, P. C., “Electron transport in porous nanocrystalline $\mathsf{TiO}_{\mathsf{2}}$ photoelectrochemical cells”, J. Phys. Chem. 100 (1996) 17021–17027; doi:10.1021/jp9616573.CrossRefGoogle Scholar
Caputo, M., “Linear models of dissipation whose Q is almost frequency independent—II”, Geophys. J. Int. 13 (1967) 529–539; doi:10.1111/j.1365-246X.1967.tb02303.x.CrossRefGoogle Scholar
Chen, Z. Q., Meerschaert, M. M. and Nane, E., “Space–time fractional diffusion on bounded domains”, J. Math. Anal. Appl. 393 (2012) 479–488; doi:10.1016/j.jmaa.2012.04.032.CrossRefGoogle Scholar
Dimitrov, Y., “A second order approximation for the Caputo fractional derivative”, J. Fract. Calc. Appl. 7 (2016) 175–195; https://www.researchgate.net/publication/272022891_A_second_ order_approximation_for_the_Caputo_fractional_derivative.Google Scholar
Duan, J. S., Fu, S. Z. and Wang, Z., “Fractional diffusion-wave equations on finite interval by Laplace transform”, Integral Transforms Spec. Funct. 25 (2014) 220–229; doi:10.1080/10652469.2013.838759.CrossRefGoogle Scholar
El Danaf, T. S., “Numerical solution for the linear time and space fractional diffusion equation”, J. Vib. Control 21 (2015) 1769–1777; doi:10.1177/1077546313500687.CrossRefGoogle Scholar
Esen, A., Ucar, Y., Yagmurlu, N. and Tasbozan, O., “A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations”, Math. Model. Anal. 18 (2013) 260–273; doi:10.3846/13926292.2013.783884.CrossRefGoogle Scholar
Gacemi, Y., Cheknane, A. and Hilal, H. S., “Simulation and modelling of charge transport in dye-sensitized solar cells based on carbon nano-tube electrodes”, Phys. Scr. 87 (2013) 035703–035714; doi:10.1088/0031-8949/87/03/035703.CrossRefGoogle Scholar
Garrappa, R., Kaslik, E. and Popolizio, M., “Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial”, Mathematics 7 (2019) 407–428; doi:10.3390/math7050407.CrossRefGoogle Scholar
Gómez, R. and Salvador, P., “Photovoltage dependence on film thickness and type of illumination in nanoporous thin film electrodes according to a simple diffusion model”, Solar Energy Mater. Solar Cells 88 (2005) 377–388; doi:10.1016/j.solmat.2004.11.008.CrossRefGoogle Scholar
Griffiths, G. W., Numerical analysis using R: solutions to ODEs and PDEs (Cambridge University Press, New York, 2016); doi:10.1017/CBO9781316336069.CrossRefGoogle Scholar
Henry, B. I. and Wearne, S. L., “Fractional reaction–diffusion”, Physica A 276 (2000) 448–455; doi:10.1016/S0378-4371(99)00469-0.CrossRefGoogle Scholar
Langlands, T. A. M. and Henry, B. I., “The accuracy and stability of an implicit solution method for the fractional diffusion equation”, J. Comput. Phys. 205 (2005) 719–736; doi:10.1016/j.jcp.2004.11.025.CrossRefGoogle Scholar
Langlands, T. A. M., Henry, B. I. and Wearne, S. L., “Anomalous subdiffusion with multispecies linear reaction dynamics”, Phys. Rev. E 77 (2008) 021111; doi:10.1103/PhysRevE.77.021111.CrossRefGoogle ScholarPubMed
Li, C. and Wang, Z., “The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: numerical analysis”, Appl. Numer. Math. 140 (2019) 1–22; doi:10.1016/j.apnum.2019.01.007.CrossRefGoogle Scholar
Li, C. and Zeng, F., “Finite difference methods for fractional differential equations”, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012) 1230014; doi:10.1142/S0218127412300145.CrossRefGoogle Scholar
Luchko, Y., “Initial value problems for the one-dimensional time-fractional diffusion equation”, Fract. Calc. Appl. Anal. 15 (2012) 141–160; doi:10.2478/s13540-012-0010-7.CrossRefGoogle Scholar
Lynch, V. E., Carreras, B. A., del-Castillo-Negrete, D., Ferreira-Mejias, K. M. and Hicks, H. R., “Numerical methods for the solution of partial differential equations of fractional order”, J. Comput. Phys. 192 (2003) 406–421; doi:10.1016/j.jcp.2003.07.008.CrossRefGoogle Scholar
Maldon, B., Lamichhane, B. P. and Thamwattana, N., “Numerical solutions for nonlinear partial differential equations arising from modelling dye-sensitized solar cells”, Proc. 18th Biennial Comput. Tech. Appl. Conf., CTAC-2018, (eds Lamichhane, B., Tran, T. and Bunder, J.), (2019), C231–C246; doi:10.21914/anziamj.v60i0.14053.Google Scholar
Maldon, B., Lamichhane, B. P. and Thamwattana, N., “A cubic B-spline collocation method for numerically solving fractional diffusion equations used in modelling dye-sensitized solar cells”, Proc. 19th Biennial Comput. Tech. Appl. Conf., CTAC-2020 (eds W. McLean and S. MacNamara), (2020).Google Scholar
Maldon, B. and Thamwattana, N., “A fractional diffusion model for dye-sensitized solar cells”, Molecules 25 (2020) 2966–2975; doi:10.3390/molecules25132966.CrossRefGoogle ScholarPubMed
Maldon, B., Thamwattana, N. and Edwards, M., “Exploring nonlinear diffusion equations for modelling dye-sensitized solar cells”, Entropy 22(2) (2020) Article ID 248; doi:10.3390/e22020248.CrossRefGoogle ScholarPubMed
Meerschaert, M. M. and Sikorskii, A., Stochastic models for fractional calculus, Volume 43 (De Gruyter, Berlin, 2012); doi:10.1515/9783110258165.Google Scholar
Méndez, V., Fedotov, S. and Horsthemke, W., Reaction-transport systems (Springer, Berlin, 2010); doi:10.1007/978-3-642-11443-4.CrossRefGoogle Scholar
Nelson, J., “Continuous-time random-walk model of electron transport in nanocrystalline $\mathrm{TiO}_{\mathrm{2}}$ electrodes”, Phys. Rev. B 59 (1999) 15374–15380; doi:10.1103/PhysRevB.59.15374.CrossRefGoogle Scholar
Ni, M., Leung, M. K. H., Leung, D. Y. C. and Sumathy, K., “An analytical study of the porosity effect on dye-sensitized solar cell performance”, Solar Energy Mater. Solar Cells 90 (2006) 1331–1344; doi:10.1016/j.solmat.2005.08.006.CrossRefGoogle Scholar
Nigmatullin, R., “The realization of the generalized transfer equation in a medium with fractal geometry”, Phys. Stat. Sol. B 133 (1986) 425–430; doi:10.1002/pssb.2221330150.CrossRefGoogle Scholar
O’Regan, B. and Grätzel, M., “A low-cost, high-efficiency solar cell based on dye-sensitized colloidal $\mathsf{TiO}_{\mathsf{2}}$ films”, Nature 353 (1991) 737–740; doi:10.1038/353737a0.CrossRefGoogle Scholar
Oldham, K. and Spanier, J., The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Volume 111 (Elsevier, New York, 1974); https://catalogue.nla.gov.au/Record/1869235/Copyright.Google Scholar
Södergren, S., Hagfeldt, A., Olsson, J. and Lindquist, S., “Theoretical models for the action spectrum and the current–voltage characteristics of microporous semiconductor films in photoelectro- chemical cells”, J. Phys. Chem. 98 (1994) 5552–5556; doi:10.1021/j100072a023.CrossRefGoogle Scholar
Thomée, V., Galerkin finite element methods for parabolic problems, Volume 25 of Springer Ser. Comput. Math. (Springer, Berlin–Heidelberg, 2006); doi:10.1007/3-540-33122-0.Google Scholar
Tomovski, Z. and Sandev, T., “Exact solutions for fractional diffusion equations in a bounded domain with different boundary conditions”, Nonlinear Dynam. 71 (2013) 671–683; doi:10.1007/s11071-012-0710-x.CrossRefGoogle Scholar
Yuan, Q. and Chen, H., “An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem”, Adv. Differential Equations 2018 (2018) Article ID 34; doi:10.1186/s13662-018-1483-4.CrossRefGoogle Scholar
Yuste, S. B. and Acedo, L., “An explicit finite difference method and a new von-Neumann-type stability analysis for fractional diffusion equations”, SIAM J. Numer. Anal. 42 (2005) 1862–1874; doi:10.1137/030602666.CrossRefGoogle Scholar
Zahra, W. K. and Elkholy, S. M., “The use of cubic splines in the numerical solution of fractional differential equations”, Int. J. Math. Math. Sci. 2012 (2012) 638026; doi:10.1155/2012/638026.CrossRefGoogle Scholar