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The diffusion-based algorithm to produce density-equalizing maps interprets
diffusion as an advection process. This algorithm uses the dynamics of a flow that is defined
by an initial value problem that turns out to be very singular at the initial time. The singularities
appear when the initial density has line or angle discontinuities, which is always the
case, for example, in area cartogram maps. This singular initial value problem is analyzed
mathematically in this article and the conclusion is that despite these singularities, it has a
unique solution. This justifies the extensive numerical use of this algorithm in the recent years.
The techniques presented in this article use both partial and ordinary differential equations
CitacióAvinyó, A.; De Solà-Morales, J.; València, M. A singular initial value problem to construct density-equalizing maps. "Journal of dynamics and differential equations", Desembre 2011, vol. 24, núm. 1, p. 51-59.