University of Arkansas, Fayetteville
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Abstract

It is difficult to underestimate the importance of differential equations in understanding the physical world. These equations, involving not just simple variables like temperature, speed or mass, but also the derivatives, i.e. the rate of change of these variables, are found in nearly every branch of science. Until the mid 20th century, all such equations were thought to be solvable. This was based on the discovery by Leonard Euler that certain differential equations, called ordinary differential equations (ODEs), are indeed always solvable. While ODEs deal with simple conditions, under which some quantity changes with some other quantity and its derivatives, there are more sophisticated differential equations known as Partial Differential Equations (PDEs), which describe how one quantity changes with respect to two or more other quantities and their derivatives. The hopes of an entire generation of mathematicians were dashed when it was discovered that there exist very simple linear PDEs that are unsolvable - and thus the worst objects that a mathematician could possibly face. It is the goal of this research to present one such example in a form accessible to anyone who has a basic knowledge of differential equations. Understanding of such equations is an extremely important step in developing numerical methods for estimating the extent to which PDEs may not be solvable, thus giving scientists valuable tools in unlocking the secrets of the physical world, many of which are hidden in Partial Differential Equations.

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