Recent Advances in Differential Equations and Applications

This volume is a collection of papers from the Congress on Differential Equations and Applications held in Cartagena in 2017. The editors say that they chose papers that represented the range of contributions presented at the congress. The book contains fourteen refereed papers on topics that include ordinary and partial differential equations, numerical analysis, control systems and optimization. Many of the papers have a strongly applied flavor. Beyond this there is no particular organizing theme.

 

To give a sense of what’s included here, summaries of three representative papers are presented below.

 

Apraiz’s “Applications of Observability Inequalities” considers an observability condition in control theory. Observability inequalities are derived for the heat equation in an application where a controller uses feedback to switch abruptly between two states. (Such a controller is known in the art as a bang-bang controller.) The author uses the inequalities to prove a bang-bang property for minimal time and minimal control problems.

 

“Numerical Simulation of Wear-Related Problems in a Blast Furnace Runner” by Barral et al. describes a numerical simulation of the wear suffered by refractory linings of blast furnace runners during one stage of steelmaking. The simulation includes both thermal-hydrodynamical and radiative transfer models. After formulation of the pertinent equations, shear stress, the driving factor for erosion, is computed using standard finite element software.

 

“Classical Symmetries for Two Special Cases of Unsteady Flow in Nanoporous Rock” by Garrido et al. examines two partial differential equations arising as models of fluid and gas flow in nanoporous rock, one for a weakly compressible fluid and the other for isothermic gas flow. Both models are special cases of the basic equation for unsteady flow in nanoporous rock. The authors apply the symmetries of Lie point theory to get a classification of solutions in different six different cases.

 

Because the book lacks a unifying theme and the conference papers have so many divergent topics, it is unlikely that any particular reader would find more than one or two papers of interest. I thought the book, by virtue of its title, might offer a broader overview of recent work in differential equations, but most of the papers have a narrow scope. This is a book clearly aimed at experts, and most of the papers are focused on applications that are also very specialized. Large parts of the book would likely be inaccessible to readers without a good background in partial differential equations and control theory. 

---