ANALYSIS OF FRACTIONAL ORDER SCHR<inline-formula><tex-math id="M1">$\ddot{O}$</tex-math></inline-formula>DINGER EQUATION WITH SINGULAR AND NON-SINGULAR KERNEL DERIVATIVES VIA NOVEL HYBRID SCHEME

In the present study, a novel semi-analytic scheme is proposed to obtain exact and approximate series solutions for the time fractional linear and non-linear Schr$ \ddot{o} $dinger equation. This hybrid scheme employs the general bivariate transform followed by the homotopy perturbation method to formulate the recurrence relation. The recurrence relation leads to a system of linear differential equations that associates with the desired components of the series solution. To characterize the considered model with memory effects, the fractional temporal order is considered in the Caputo, Caputo-Fabrizio, and Atangana-Baleanu in Caputo senses. The adapted scheme appears efficient and competent in identifying a diverse collection of trigonometric, wave, and soliton solutions with the availability of initial data. Configurational variations in the governing phenomena with alterations in the fractional order are addressed through graphical illustrations. The potential of the developed regime is affirmed through the uniqueness and convergence analysis of the acquired results. Numerical results are found to be in accordance with existing results in terms of absolute error norms. The main highlight of the proposed scheme is its efficacy and simplicity in constructing a series solution that rapidly converges to the exact solution.