The book compiles works presented at a seminar aiming to attract global experts in differential equations, mathematical modeling, and integration methods. It covers classical and contemporary integration techniques for partial differential equations, including Monge and Darboux's approaches and their extensions. Additionally, it introduces a novel theoretical model for plane turbulent flows, presents gravitational equations derived from the principle of least action, and explores symmetry-preserving conservative finite-difference schemes for hydrodynamic-type equations. Analytical solutions for Maxwell's equations in incompressible viscoelastic mediums are examined, alongside theoretical-group analysis of wake mathematical models and reduction to ordinary differential equations. The book also delves into special classes of two-dimensional ideal fluid motion and advancements in discrete orthogonal polynomial theory, showcasing rapid decay properties near interval boundaries. In conclusion, this comprehensive collection is indispensable for researchers and practitioners in applied mathematics, fluid dynamics, and computational modeling, providing valuable insights into cutting-edge methods and solutions in the field.

Autorentext

Dr. Oleg Viktorovich Kaptsov, born on November 1, 1956, in Novokuznetsk, Russia, is a prominent figure in Mathematics and Hydrodynamics. Holding a Doctor of Sciences degree and the esteemed position of Professor, Dr. Kaptsov has dedicated his career to advancing our understanding of complex mathematical and hydrodynamic phenomena. Dr. Kaptsov began his academic journey at Novosibirsk University, where he pursued undergraduate studies in Mathematics and Mechanics from 1974 to 1979. He continued his education with postgraduate studies at the Lavrentjev Institute of Hydrodynamics in Novosibirsk from 1981 to 1984. Dr. Kaptsov's research career started as a junior researcher at the Institute of Theoretical and Applied Mechanics in Novosibirsk from 1985 to 1988, followed by a senior researcher position at the Computing Center of the Academy of Science in Krasnoyarsk from 1987 to 1994. Since 1994, he has been a leading researcher at the Institute of Computing Modelling, Academy of Science, Krasnoyarsk, and a Professor at Siberian Federal University (SFU, Krasnoyarsk). Dr. Kaptsov's research focuses on Differential Equations, Exact solutions of nonlinear partial differential equations, Continuous groups of transformations, and Inviscid Fluid dynamics, significantly contributing to advancing theoretical foundations in these areas. His recent publications since 2019 have gained significant attention, including investigations into Traveling Waves in Nondispersive Strongly Inhomogeneous Media, Dynamics of passive scalar in swirling turbulent far wakes, and General solutions of some linear equations with variable coefficients. Noteworthy works by Dr. Kaptsov include "Applications of Group-Theoretical Methods in Hydrodynamics" (a joint work with V. K. Andreev, V. V. Puckhnachov, and A. A. Rodionov) and "Integration Methods of Partial Differential Equations." Dr. Kaptsov's dedication to advancing knowledge in Mathematics and Hydrodynamics has earned him respect and admiration, making him a formidable force in scientific inquiry.

Dr. Evgeniy Igorevich Kaptsov is a researcher with expertise in computational mathematics and fluid dynamics. He graduated from Kuban State Technological University in 2008 with a Master's degree in Computer Science. Dr. Kaptsov has over a decade of industry experience as a software developer in prominent retail and game development companies in Russia. He completed his PhD in Computational Mathematics in 2020 at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, where he developed new methods for constructing invariant finite-difference schemes for one-dimensional gas dynamics and shallow water equations as a Research Assistant. Since 2018, he has been a researcher at the School of Mathematics, Suranaree University of Technology, Thailand, under the guidance of Professors V.A. Dorodnitsyn and S.V. Meleshko. Dr. Kaptsov has made significant contributions to the field, focusing on conservation laws, invariant solutions, and numerical implementations for various mathematical models, as evidenced by his numerous publications in esteemed journals.