A novel approach for solving linear and nonlinear time-fractional Schrödinger equations. (September 2022) - Liaqat, Muhammad Imran; Akgül, Ali
Abstract: There is significant literature on Schrödinger differential equation (SDE) solutions, where the fractional derivatives are stated in terms of Caputo derivative (CD). There is hardly any work on analytical and numerical SDE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the SDE in the form of CFD. The main goal of this research is to offer a novel combined computational approach by using conformable natural transform (CNT) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable Schrödinger equation (TFCSE) with zero and nonzero trapping potential. We call it the conformable natural transform homotopy perturbation method (CNTHPM). The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the CNTHPM. The error analysis has confirmed the higher degree of accuracy and convergence rates, which indicates the effectiveness and reliability of the suggested method. Furthermore, 2D and 3D graphs compare the exact and approximate solutions. The procedure is quick, precise, and easy to implement, and it yields outstanding results. In addition, numerical results are also compared with other methods such as the differential transform method (DTM), split-step finite difference method (SSFDM), homotopy analysis method (HAM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), and two-dimensional differential transform method (TDDTM). The comparison shows excellent agreement with these methods, which means that CNTHPM is a suitable alternative tool to the methods based on CD for the solutions of the time-fractional SDE. Moreover, we can conclude that the CFD is a suitable alternative to the CD in the modeling of time-fractional SDE. The Banach fixed point theory was also used to test the uniqueness of the solution, convergence, and error analysis. Highlights: We consider the Schrödinger differential equation with fractional derivatives. We offer a novel combined computational approach by using conformable natural transform and the homotopy perturbation method for extracting analytical and numerical solutions of the time-fractional conformable Schrödinger equation with zero and non-zero trapping potential. We use the Banach fixed point theory to test the uniqueness of the solution, convergence, and error analysis. We demonstrate the numerical simulations.