What are Analysis and Partial Differential Equations?

Partial differential equations have been developed as a mathematical toolbox for the description of natural and engineered phenomena, ranging from vibrating strings to weather prediction to the design of the Boeing 777. Partial differential equations provide a language for modern science and analysis provides a theoretical background for their study. As such, this area of research interacts with many branches of mathematics and physics, such as geometry, complex and harmonic analysis, mathematical physics, continuum dynamics, and scientific computation.

Most of the equations which arise in the sciences are nonlinear and do not yield to standard methods or exact solutions that were developed for linear equations. Another nonlinear effect arises when the PDEs have random coefficients, namely random (or stochastic) partial differential equations. Random coefficients occur in stochastic modeling of complex environments where the systems, described PDEs, operate. The techniques of using (analytical or probabilistic) estimates often succeeds where explicit calculation fails. Powerful numerical techniques are now available to study complex, nonlinear phenomena.

Numerical simulation is increasingly used to analyze the behavior of complex systems. Their proper interpretation requires a deep analytical understanding of the underlying phenomena, so the development of a strong program in numerical analysis/scientific computing requires a parallel development in analysis/partial differential equations. Moreover, computational resources are generally not sufficient to resolve multiple length scales in three-dimensional problems, and the essential role which must be assigned to theory and the design of algorithms of new nature becomes evident.

The importance of nonlinear analysis is seen in the new scientific discoveries to which scientists have been lead by mathematical analysis. In mathematical physics the focusing effects of the nonlinear effects in reentry problems for the space shuttle and the behavior of gravity waves are a few examples. Such phenomena are not present in the linearized version of the equations.