Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Fouman and Shaft Branch, Islamic Azad University, Fouman, Iran.

2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

Abstract

One of the fields studied in the science of heat physics is the thermoelectric phenomenon. This phenomenon is in fact the interaction between the current of electricity and the thermal properties of a system. In simpler terms, it is a phenomenon in which the direct conversion of a temperature difference to voltage occurs. In this paper, we introduced a method based on the finite difference technique for solving a fractional differential equation in the field of thermal physics which describes the thermoelectric phenomena, numerically. For this purpose, we used fractional order derivatives with the definitions of Caputo, finite differences with the second order central finite-difference approach, and the first order central finite-difference. By using this method, we translate the desired differential equation to a system of nonlinear differential equations which can be solved. Finally, some numerical are used to demonstrate the effective and accuracy of the scheme. The obtained numerical results show that our proposed method is highly accurate.

Keywords

Main Subjects

  1. Aminikhah, H., Sheikhani, A. H. R., & Rezazadeh, H. (2018). Approximate analytical solutions of distributed order fractional Riccati differential equation. Ain shams engineering journal9(4), 581-588. https://doi.org/10.1016/j.asej.2016.03.007
  2. Ansari, A., & Sheikhani, A. R. (2014). New identities for the Wright and the Mittag-Leffler functions using the Laplace transform. Asian-european journal of mathematics7(03), 1450038. https://doi.org/10.1142/S1793557114500387
  3. Mashoof, M., Sheikhani, A. R., & NAJA, H. S. (2018). Stability analysis of distributed order Hilfer-Prabhakar differential equations. Hacettepe journal of mathematics and statistics47(2), 299-315.
  4. Pirmohabbati, P., Sheikhani, A. R., Najafi, H. S., & Ziabari, A. A. (2020). Numerical solution of strongly nonlinear full fractional duffing equation. Journal of interdisciplinary mathematics23(8), 1531-1551. https://doi.org/10.1080/09720502.2020.1776934
  5. Zeng, F. (2015). Second-order stable finite difference schemes for the time-fractional diffusion-wave equation. Journal of scientific computing65(1), 411-430. https://doi.org/10.1007/s10915-014-9966-2
  6. Mahan, G. D. (2016). Introduction to thermoelectrics. APL materials4(10), 104806. https://doi.org/10.1063/1.4954055
  7. Lee, H. (2013). The Thomson effect and the ideal equation on thermoelectric coolers. Energy56, 61-69. https://doi.org/10.1016/j.energy.2013.04.049
  8. Ioffe, A. F., Stil'Bans, L. S., Iordanishvili, E. K., Stavitskaya, T. S., Gelbtuch, A., & Vineyard, G. (1959). Semiconductor thermoelements and thermoelectric cooling. Physics today12(5), 42. https://archive.org/details/A.F.IoffeSemiconductorThermoelementsAndThermoelectricCoolingInfosearch1957
  9. Lee H.S. (2010). Thermal design: heat sinks, thermo-electrics, heat pipes, compact heat exchangers, and solar cells. John Wiley & Sons, Inc., Hoboken, New Jersey, USA.
  10. Rowe, D. M. (1995). CRC handbook of thermoelectrics. CRC-Press.
  11. Aminikhah, H., Refahi Sheikhani, A., & Rezazadeh, H. (2013). Stability analysis of distributed order fractional Chen system. The scientific world journal2013. https://doi.org/10.1155/2013/645080
  12. Aminikhah, H., Sheikhani, A. H. R., Houlari, T., & Rezazadeh, H. (2017). Numerical solution of the distributed-order fractional Bagley-Torvik equation. IEEE/CAA journal of automatica Sinica6(3), 760-765. DOI:1109/JAS.2017.7510646
  13. Pirmohabbati, P., Sheikhani, A. H., & Ziabari, A. A. (2020). Numerical solution of nonlinear fractional Bratu equation with hybrid method. International journal of applied and computational mathematics6(6), 1-22. https://doi.org/10.1007/s40819-020-00911-5
  14. Pirmohabbati, P., Sheikhani, A. R., Najafi, H. S., & Ziabari, A. A. (2019). Numerical solution of fractional mathieu equations by using block-pulse wavelets. Journal of ocean engineering and science4(4), 299-307. https://doi.org/10.1016/j.joes.2019.05.005
  15. Pirmohabbati, P., Sheikhani, A. R., Najafi, H. S., & Ziabari, A. A. (2020). Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities. AIMS math5(2), 1621-1641.
  16. Milne-Thomson, L. M. (2000). The calculus of finite differences. American Mathematical Soc..
  17. Wilmott, P., Howson, S., Howison, S., & Dewynne, J. (1995). The mathematics of financial derivatives: a student introduction. Cambridge university press. https://doi.org/10.1017/CBO9780511812545
  18. Olver, P. J. (2014). Introduction to partial differential equations(pp. 182-184). Berlin: Springer. https://doi.org/10.1007/978-3-319-02099-0
  19. Chaudhry, M. H. (2008). Open-channel flow(Vol. 523). New York: Springer. https://doi.org/10.1007/978-3-030-96447-4     
  20. Rudin, W. (1976). Principles of mathematical analysis (Vol. 3). New York: McGraw-hill.