MOSCOW, January 27. /TASS/. Russian mathematician Ivan Remizov has derived a universal formula for solving problems in the field of differential equations, which had been considered unsolvable by analytical methods for over 190 years. This breakthrough radically changes the understanding of one of the oldest areas of mathematics, crucial to fundamental physics and economics, according to the press service of the National Research University Higher School of Economics (HSE).
"Imagine that the solution to the equation is a large painting. Considering it all at once is very difficult. Mathematics is excellent at describing processes that develop over time. Our theorem allows us to 'slice' this process into many small, simple frames. In simpler terms, instead of guessing what the whole picture looks like, the theorem enables us to reconstruct its appearance by quickly 'playing' the 'film' of its creation," explained Ivan Remizov, senior researcher at HSE in Nizhny Novgorod, as quoted by the university's press service.
As Remizov explains, second-order differential equations are widely used in economics and physics to describe various processes that change over time. As early as 1834, French mathematician Joseph Liouville demonstrated that solutions to these equations cannot be expressed through coefficients, simple operations, and elementary functions, similar to how quadratic equations are solved in school classes via discriminants.
For this reason, the search for analytical solutions to differential equations was considered hopelessly futile and remained essentially "abandoned" for the past 190 years. As a result, mathematicians stopped looking for a simple formula, similar to the high school solution of quadratic equations, for this class of problems. Ivan Remizov, who is also a senior researcher at the Institute of Problems of Information Transmission of the Russian Academy of Sciences (RAS), found a way to solve this centuries-old problem.
His analysis showed that a complex, constantly changing process can be broken down into an infinite number of simple steps. Each of these can be approximated to describe the system's behavior at a specific point. Individually, these pieces give only a primitive picture, but when their number tends to infinity, they seamlessly connect into a perfectly accurate graph of the solution.
Furthermore, applying another mathematical operation - the Laplace transform - to these steps makes it possible to "translate" the equation into the language of ordinary algebraic calculations, enabling a quick derivation of the desired result. Looking ahead, this approach will not only accelerate computations for differential equations already used in physics and other sciences, but also help mathematicians more rapidly find and study new functions, the scientist summarized.