ON THE APPLICATION OF ILL-POSED PROBLEMS OF EQUATIONS OF MATHEMATICAL PHYSICS

Abstract

Regarding the implementation of ill-posed problems in the context of mathematical physics equations, it is essential to acknowledge their significance. Ill-posed problems are characterized by their sensitivity to changes in initial or boundary conditions, leading to nonunique or unstable solutions. These challenges frequently arise in areas such as fluid dynamics, heat transfer, and wave propagation, where classical methods may struggle to provide reliable answers. Consequently, researchers seek innovative numerical techniques and regularization methods to stabilize these problems and obtain meaningful solutions. Among the approaches utilized are Tikhonov regularization, particle filtering, and machine learning methods, all aiming to mitigate the effects of ill-posedness. Addressing these issues is crucial, as they have profound implications for both theoretical understanding and practical applications in engineering and physics. It is clear that further advancements in this area will enhance the effectiveness of predictive models and simulation tools, ultimately contributing to the broader field of mathematical
physics. Through ongoing investigation, the mathematical community continues to refine strategies that accommodate the complexities posed by ill-posed problems, ensuring progress in this vital discipline.