Analytical solutions refer to mathematical solutions obtained through a systematic and step-by-step process that involves algebraic manipulations, calculus, and other mathematical techniques. These solutions are typically exact and provide precise answers to mathematical problems or equations. Analytical solutions are often used in various fields such as physics, engineering, and financial modeling to study and predict the behavior of systems or phenomena. Researchers in this area focus on developing methods and techniques to derive analytical solutions for complex mathematical problems in order to gain insights and understanding of the underlying processes.