Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

Abstract

This study regards the time-fractional nonlinear Schrödinger equations (TFNLSEs). The TFNLSE is used to depict nonlocal quantum phenomena in quantum physics and investigate the quantum behaviors of either long-range relations or time-dependent processes with many scales. Basic definitions of fractional derivatives are described in the Caputo fractional derivative (CaFD) sense. We utilize the residual power series method (RPSM) to achieve the approximate solutions of the TFNLSE. The RPSM is a numeric analytic technique for solving various ordinary, partial, and fuzzy differential and integrodifferential equations of fractional order. It is a practical optimization approach, supplying solutions in the closed form of known functions. The primary benefit of this technique is the clarity in calculating the coefficients of terms of the series solution by employing only the differential operators and not the other well-known analytic approaches that require integration operators, which are challenging in the fractional case. Our main aim in this paper is to find approximate solutions using RPSM and compare these approximate solutions with exact solutions. We discuss three applications, showing the validity, precision, and efficiency of the RPSM. The outcomes are presented in graphs and tables, which display the method's effectiveness, quality, and strength. The RPSM is entirely compatible with the complexity of this problem, and the accepted outcomes are highly favorable. The offered technique can be easily used in the spaces of higher-dimension solutions without limiting the nature of the equation or the category style. Also, this method can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics.