Discipline: Technology and Engineering
Subcategory: Civil/Mechanical/Manufacturing Engineering
Kin Li - California State University, Los Angeles
Fractional calculus, as opposed to the integer calculus (which is taught in most university STEM curricula), is a study of calculus in arbitrary orders. The history of fractional calculus is as old as integer calculus itself, when Leibniz wrote a letter to L’Hopital in 1695 discussing the meaning of the half derivative “the half derivative will lead to a paradox, but one day there will be useful consequences.” Until recently, much of the interest in fractional calculus has been drawn from a mathematician’s perspective, deriving formulations by pure mathematical manipulation with no application in mind (i.e., Euler, Laplace, Liouville attempted to derive expressions for the fractional derivative or integral). However, in recent years, fractional derivative started to receive much attention in modeling complex systems that would be difficult to model using traditional integer-calculus-based approaches. The research focus shifted from pure mathematics to application in various fields. In many cases – such as long memory materials, anomalous diffusion, heat transfer, among others – the fractional derivative has shown to be an excellent model for complex physical processes. As an example, in anomalous diffusion, experiments have shown that many materials do not obey the normal diffusion equation, and can be better modeled by a differential equation of fractional order. On the other hand, in heat transfer it has been demonstrated that the transfer of energy in a transient conductive system can be modeled by a fractional order differential equation, and that its solution was in good agreement with the complete analytical solution. The current project focuses on the application of fractional calculus to model efficient energy usage in buildings. To this end, we take small steps to develop the appropriate background by solving three simple problems related to the fractional heat equation for a plane wall. We solve the fractional transient heat conduction equation for three different boundary conditions, namely, Dirichlet, Neumann, and mixed boundary conditions. For each problem, the corresponding solution is obtained using both the Laplace transform and finite difference methods. The numerical solutions obtained from the finite difference method are found to provide good approximations to the analytical counterparts obtained by the Laplace Transform method, and with results recently reported in literature. The extension of this work is the application to the heat transfer in buildings, which is currently being investigated, and detailed results will be reported in the future.
Funder Acknowledgement(s): CeAS
Faculty Advisor: Arturo Pacheco-Vega, email@example.com
Role: I did the literature review, came up with the model and solved the equations analytically and numerically.