This paper investigates the analytical solution of the time fractional nonlinear Schrödinger equation (TFNLSE) using the homotopy analysis method (HAM). The TFNLSE is a fundamental equation in quantum mechanics and nonlinear optics, describing the evolution of wave functions in nonlinear media with fractional time derivatives. The HAM offers a powerful analytical technique for solving nonlinear differential equations, providing insights into the behavior of complex systems. Through a systematic application of the HAM, this study derives analytical expressions for the solution of the TFNLSE and explores its properties. The results contribute to the understanding of fractional calculus and its applications in quantum mechanics and nonlinear optics.

Keywords: Analytical Solution, Time Fractional Nonlinear Schrödinger Equation, Homotopy Analysis Method, Fractional Calculus, Quantum Mechanics, Nonlinear Optics.