Question

Produce a 2/3 page explanation on

Applications of PDEs in Engineering,Science and Economics.

In your explanation you should give three detailed examples. One ex-

ample should involve a PDE from an area of Engineering, one example

from Science and one from Economics. In at least one example you

should discuss the solution to your PDE and how this has practical

implications

Answer 1

Partial differential equation models in the socio-economic sciences:-

Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges

Partial differential equations (PDEs) have been used since the times of Newton and Leibniz to model physical phenomena. Famous examples are Maxwell's formulation of the electrodynamical laws, the Boltzmann equation for rarified gases, Einstein's general relativity theory and Schrödinger's formulation of quantum mechanics. In the middle of the last century, PDE modelling began to be applied to certain biological processes, but only very recently was it realized that many socio-economic processes can be very successfully modelled by nonlinear PDEs.Where the PDE-based approaches are now a days quite standard as pricing models in finance and insurance, always strongly related to stochastic differential equations as in the famous Black–Scholes equation. Instead PDE modelling of socio-economic processes has developed significantly in the last decade, not only by delivering new insights into qualitative and quantitative analysis of socio-economics but also by opening up a whole new range of fascinating mathematical problems, which require the development of new mathematical tools.

(1)Kinetic models with non-physical interactions:-

Recently, ideas from statistical physics have started to permeate the socio-economic sciences, leading to mean-field and kinetic PDEs as the main tools for qualitative and quantitative research. Their applicability relies on the fact that socio-economic processes are often governed—similar to statistical physics—by the interaction of large ‘particle’ systems, where the particles are human beings (inter)acting in socio-economic scenarios. The natural step is to go from those particle models to kinetic and further macroscopic PDE models. In those, several novel issues appear due to the obvious differences compared to classical models in physics. In particular, the interactions in socio-economics can have missing conservation properties or be of unusual long-range nature, which leads to novel mathematical challenges, e.g. non-standard collision invariants to be determined.

(2)Spatial pattern formation by consensus and herding:-

In the last few decades, motion of socially interacting individuals has received a lot of attention. This ranges from animals (insect swarms, fish schools and bird flocks) to human crowds. Various interesting mathematical models and questions arise from the fact that those animals adapt their spatial motion due to some consensus reached with other individuals locally around them, e.g. birds adapt their velocity to follow the flock without explicitly knowing a determined group velocity. Hence, this is a natural paradigm of self-organized formation of macroscopic patterns from microscopic interactions. The mathematical analysis of flocking has become popular after Cucker & Smale introduced a simple model that is accessible to rigorous arguments. In the macroscopic setting, flocking means that—at least in some spatial region—the density looks like a Dirac delta distribution in velocity space, hence analysis of flocking is naturally linked to blow-up phenomena for PDEs

(3)Games and optimal control:-

Equilibria of optimization problems and games have a long tradition in socio-economic modelling .This has also driven some developments in differential games, related to ordinary differential equations in the time-continuous setting. The standard approach is based on considering a few representative rational agents and to derive models for those.Clearly, the basic assumptions of equal and highly rational agents are highly questionable in processes involving the behaviour of a large number of humans (often with unequal access to information); moreover, models based on those often fail to explain effects such as high volatilities observed in reality. Owing to this issue, models of large numbers of interacting agents have recently been considered, leading to optimization problems or games for large numbers of interacting individuals.

The most prominent examples are the related fields of optimal transport, mean-field games and mean-field optimal control. In particular, the idea of mean-field games introduced in the seminal paper by Lasry & Lions [35] and in parallel in the engineering literature has gained enormous interest in the last few years, both with respect to applications as well as due to the new kinds of mathematical and computational challenges that such games are introduced.