Abstract

This paper presents an accurate application of the Lie symmetry method to find (and solve) the time-fractional Hamilton-Jacobi-Bellman (TFHJB) equation, which is a family of fractional differential equations that describe various nonlinear dynamical systems. The analysis commences with the detailed determination of the infinitesimal generators the equation possesses, which then serves as a basis for finding its respective Lie symmetries. Once these symmetries are recognized, the similarity reduction method will be applied to obtain exact solutions by reducing the original fractional equation into simpler cases. Additionally, the research goes beyond just determining the solutions and also includes deriving conservations laws associated with the TFHJB equation via a modified version of Ibragimov's method. These conservation laws yield fundamental information about the qualitative behavior, stability and integrability of the system.