On mathematical studies of the Mpemba effect Since the days of Aristotle and Descartes, it has been known that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named after a student who rediscovered the effect in the sixties. Several theories have been proposed to explain the effect, but so far none of them seem to be generally accepted, see, e.g., this discussion on Physics SE. In 2012, the Royal Society of Chemistry offered £1000 to the person or team producing the best and most creative explanation of the phenomenon, the Mpemba competition (Wayback Machine). One problem is that many factors might play a role. The theories that try to explain the effect involve, for example, evaporation, convection, gas dissolved in the water, or interactions on molecular level, and it is difficult to design experiments that allow to isolate these factors.
Are there any mathematical studies (exact solutions for special cases, numerical analysis, simulations, etc.) based on the equations proposed to describe or explain the Mpemba effect? Do they allow to isolate different influences and to compare them with experiments, e.g., by simulating heat flow with convection and/or evaporation?
Does anybody here know of any such work? Or does anybody have a reference on simulations of similarly complex thermodynamical systems like a heat flow with convection and/or evaporation?
PS: I discovered two papers by a group of Chinese chemical physicists, see O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox and Mpemba Paradox Revisited — Numerical Reinforcement. The second uses a finite element method to solve a one-dimensional model. I am not an expert in numerical analysis, but I believe modern mathematics should be able to go further than this.
PPS: I changed the formulation of the second paragraph, following Theo Johnson-Freyd's remark.
 A: Since this question is still open, I take the liberty of pointing to a recent survey of the status of the Mpemba effect, Pathological Water Science -- Four Examples and What They Have in Common, which draws the following conclusion:
If confounding factors (such as evaporation, dissolved gases, mixing by convective currents, inefficient thermal contacts) are removed, hot water may indeed freeze earlier than cold water, but not because it cools off more quickly: the hot water remains warmer than the cold water during the cooling process.
Nucleation sites in the container may lower the freezing temperature to anywhere between $0$ and $-45^\circ$ C and it may happen that the container filled with the cold water needs to be supercooled to a much lower temperature before the water freezes than the container filled with the hot water. The freezing temperature is a reproducible but unpredictable property of the container. If it would be possible to have two containers with identical nucleation sites, then cold water would freeze earlier than hot water.
A: Try this reference:
O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox, by Xi Zhang Yongli Huang, Zengsheng Ma and Chang Q Sun https://arxiv.org/abs/1310.6514
P.S. I see you have already found this reference. Some useful information about Mpemba effect can be found here https://math.ucr.edu/home/baez/physics/General/hot_water.html
By the way it seems the competition already has a winner: http://www.rsc.org/mpemba-competition/mpemba-winner.asp (Wayback Machine)
A: Try:
X. Zhang, Y. Huang, Z. Ma, Y. Zhou, J. Zhou, W. Zheng, Q. Jiang, and C.Q. Sun, Hydrogen-bond memory and water-skin supersolidity resolving the Mpemba paradox. PCCP, 2014. 16(42): 22995-23002.
X. Zhang, Y. Huang, Z. Ma, Y. Zhou, W. Zheng, J. Zhou, and C.Q. Sun, A common supersolid skin covering both water and ice. PCCP, 2014. 16(42): 22987-22994.
