Partial differential equations

Differential equations have been around almost as long as derivatives. Shortly after people began computing derivatives and thinking about physical, practical problems in terms of derivatives, they realized how incredibly useful it would be to invert the process. Instead of needing to find the derivative of a known function, we often know equations the derivatives must satisfy and need to find the function.

So-called ordinary differential equations (ODEs) involve functions of one variable. Often that variable is time. Partial differential equations (PDEs) are more general, involving functions of several variables, such as several spatial variables or functions of space and time.

finite element grid

Boundary value problems

The hard part in working with differential equations, especially partial differential equations, is the boundary conditions. The differential equation itself is derived by examining what happens inside some system. Boundary conditions specify how that system interfaces with the rest of the world. With ODEs the boundary may simply be a single point, time zero, in which case the boundary conditions are called initial conditions. With ODEs, the boundary conditions may specify the value of the solution or some of its derivatives on the boundary of some region of space, and may involve initial conditions as well.

My background

My specialization in graduate school was PDEs, specifically nonlinear parabolic PDEs, from abstract theory (e.g. existence and uniqueness theorems) down to numerical computation (e.g. finite elements and finite differences). I earned a PhD at University of Texas and completed a postdoc at Vanderbilt University doing research into PDEs.

Finding parameters with filtering

One thing that bothered me at the time was wondering where the parameters in physical models come from. This introduces a different kind of problem. Not only is the solution to the differential equation initially unknown, but the exact form of the differential equation itself is unknown. Parameters are determined by a finite amount of empirical data, and so there is always some residual uncertainty regarding the exact values of the parameters. The parameters may also vary over time, in which case you may have two kinds of simultaneous evolution: the PDE solution evolving over time while the parameters of the PDE itself are also evolving over time. In practice techniques such as Kalman filters or particle filters are used to update our knowledge of parameters as the differential equations evolve.

My first career was working with PDEs. My second was software development, and my third was statistics, especially Bayesian statistics. Filtering brings these three areas together: Bayesian methods, used to determine PDE parameters, implemented in software.

Resources and help

If you’d like to learn more about differential equations, you might want to follow my Twitter account @diff_eq or read some of my blog posts on differential equations.

If you’d like for me to help your company with differential equations, particle filtering, or other aspects of mathematical modeling, please call or email to discuss your project.

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