National Research University Higher School of Economics (HSE) сообщил, that mathematician Ivan Remizov from Nizhny Novgorod discovered a way to solve second-order differential equations with variable coefficients in a relatively simple way. For nearly two centuries, such equations were considered unsolvable. Meanwhile, they play a key role in mathematics and the natural sciences, as they are used to describe dynamic processes.
Historically, this limitation stems from the work of the French mathematician Joseph Liouville, who demonstrated back in 1834 that solutions to such equations cannot be expressed through a finite number of standard operations and elementary functions. Because of this, mathematicians were forced to either seek particular solutions or use approximations, which precluded a universal method and greatly complicated the calculations. In other words, there was no general formula into which one could simply plug in the numbers and obtain a solution.
Ivan Remizov proposed a new approach, expanding the class of admissible mathematical operations. He didn't argue with Liouville, but simply added another mathematical tool to the equations—finding the limit of a sequence. To do this, the mathematician used Chernoff theory and the Laplace transform. This allowed him to construct a universal formula that formally provides a solution to any equation in the "unsolvable" class, circumventing the classical limitations of the theory.
"The idea is to break a complex, constantly changing process down into an infinite number of simple steps. For each such section, an approximation is constructed—an elementary fragment that describes the system's behavior at a specific point. Individually, these fragments provide only a simplified picture, but as their number approaches infinity, they seamlessly merge into a perfectly precise solution graph," explains the HSE press release.
"Second-order differential equations are used not only to model real-world events but also to define new functions that cannot be defined otherwise. These include, for example, the so-called Mathieu and Hill special functions, which are critical for understanding the motion of satellites in orbit or protons in the Large Hadron Collider."
The discovery can be described in slightly more complex mathematical language. read the on the HSE website. The work is in English. published in full in the Vladikavkaz Mathematical Journal.
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Source: 3dnews.ru
