On January 27, the High School of Economics (HSE) in Moscow announced a shocking scientific achievement. Russian mathematician Ivan Remizov has successfully built a universal formula to solve problems in the field of differential equations. This is something that mathematicians once considered impossible to do with traditional analytical methods for nearly 2 centuries.
This breakthrough is assessed to fundamentally change the understanding of one of the oldest fields of mathematics. It plays an essential role in basic physical models and economic calculations.
Ivan Remizov is currently a senior research fellow at HSE and the Institute of Information Transmission Issues at the Russian Academy of Sciences (RAS). He explained his invention with an intuitive metaphor of art.
Imagine the equation as a large picture. Considering the entire picture at once is very difficult," Mr. Remizov said. Instead of having to guess what the entire picture looks like, the new physics allows scientists to recreate its appearance by "quickly filming" its formation process through each small frame.
In fact, two-level differential equations are widely used to describe processes that change over time. However, from 1834, French mathematician Joseph Liouville proved that their solutions cannot be expressed through simple elementary functions.
Because of this theoretical barrier, finding an analytical solution has been considered hopeless. The problem has been almost "forgotten" by mathematicians for the past 190 years because they believe that there is no such simple formula as solving the common second-order equation.
Ivan Remizov broke this deadlock by pointing out that a complex process can be subdivided into countless simple steps. Each step is roughly calculated to describe the behavior of the system at a specific point.
If stood alone, these pieces only give a rudimentary picture. However, as their number advances to infinity, they connect seamlessly into a perfect precise graph.
In particular, by applying the Laplace transformation to these steps, the complex differential equation is translated into the common algebraic language. This allows for the desired result to be drawn quickly.
In the future, this approach will increase the calculation speed for equations being used in physics. At the same time, it helps mathematicians find and study new functions much more effectively than before.
