A Russian mathematician, Ivan Remizov, has successfully solved a second-order differential equation problem that had been deemed unsolvable for 190 years. The breakthrough, announced by HSE University where Remizov teaches, promises to fundamentally change a long-standing field of mathematics.
The research findings, published in the Vladikavkaz Mathematical Journal, are considered groundbreaking by HSE University and Russian news agency Tass. This work is expected to alter the understanding of one of the oldest areas of mathematics, which is crucial for fundamental physics and economics.
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Ivan Remizov. Photo: Sputnik. |
Second-order differential equations are widely used in economics and physics to describe processes that change over time, such as the motion of a pendulum or power grid signals related to planetary movement. However, as early as 1834, French mathematician Joseph Liouville demonstrated that the solutions to these equations could not be expressed through simple coefficients, basic operations, or elementary functions. Consequently, the search for analytical solutions was considered "hopeless" and effectively "abandoned" for nearly two centuries.
Remizov's innovative solution is based on Chernoff approximation theory. His approach involves breaking down a complex, continuously changing process into an infinite number of simple steps. For each such segment, Remizov constructs an approximation. As the number of these steps approaches infinity, they merge into an exact solution graph. The rate at which these approximations converge to the precise solution can be calculated using estimates that Ivan Remizov and his colleague Oleg Galkin obtained last year.
Furthermore, Remizov's new paper demonstrates that when the Laplace transform—a technique for converting complex change-related problems into ordinary algebraic calculations—is applied to these individual steps, they converge precisely to the final result, which scientists refer to as the solution function.
Beyond standard mathematical operations, Remizov introduced an additional operation: finding the limit of a sequence. This innovation enabled him to formulate an equation where coefficients a, b, c, and g can be substituted into the form ay'' + b'' + cy = g, thereby allowing for the derivation of its solution, the function y.
Ivan Remizov serves as a senior researcher at HSE University and the Institute for Problems in Mathematical and Information Transmission, Russian Academy of Sciences. He earned his Doctor of Philosophy degree in 2018 from Moscow State University and has made significant contributions to the research on Chernoff approximation of one-parameter operator semigroups.
Khanh Linh (According to HSE, Tass, Sputnik)
