A numerical solution for the fractional ideal equation of thermoelectric coolers

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.