A Hybrid Approach to Approximate and Exact Solutions for Linear and Nonlinear Fractional-Order Schrödinger Equations with Conformable Fractional Derivatives

Fractional-order Schrödinger differential equations extend the classical Schrödinger equation by incorporating fractional calculus to describe more complex physical phenomena. The Schrödinger equations are solved using fractional derivatives expressed through the Caputo derivative. However, there is limited research on exact and approximate solutions involving conformable fractional derivatives. This study aims to address this gap by employing a hybrid approach that combines the Elzaki transform with the decomposition technique to solve the Schrödinger equation with conformable fractional derivatives, considering both zero and nonzero trapping potentials. The efficiency of this approach is evaluated through the analysis of relative and absolute errors, confirming its accuracy. Our method serves as a viable alternative to Caputo-based approaches for solving time-fractional Schrödinger equations. Moreover, we conclude that the conformable derivative is a suitable alternative to the Caputo derivative in modeling such systems.