Well-posed and Ill-posed Problems

Abstract

This article examines the concepts of well-posed and ill-posed problems in mathematics and applied sciences. It provides a detailed analysis of the criteria that define these problems, including the existence, uniqueness, and stability of solutions. The paper also discusses common challenges associated with ill-posed problems and introduces regularization methods as a solution strategy. Examples from practical applications in physics, engineering, and data science are included to highlight the importance of these classifications.