## 1 Introduction
In this analysis, the application of conformable residual power series (CRPS) technique is extended to construct the approximate solution of the following nonlinear time-fractional Schrodinger equation (FSE):  
\[iT_{t}^{a}u(x,t)+\xi u_{xx}(x,t)+\psi(x)u(x,t)+\eta|u(x,t)|^{2}u(x,t)=0, \tag{1}\]  
along with the following initial conditions  
\[u(x,0)=u_{0}(x), \tag{2}\]  
where \(\xi,\eta\in\mathbb{R}\); \(0<\alpha\leq 1\), \(i^{2}=-1\); \(T_{t}^{a}\) indicates conformable fractional derivative of order \(\alpha\); \(u(x,t)\) is unknown complex-valued function to be determined, \(x\in\mathbb{R}\); \(t\geq 0\), \(\psi(x)\) is the trapping potential; \(|u|\) represents the modulus of \(u\); and \(u_{0}(x)\) is given smooth function. For \(\alpha=1\), Equation (1) reduces to classical nonlinear Schrodinger equation. The linear case occurs when \(\eta=0\). Anyhow, the CRPS is applied for both linear and nonlinear cases with zero and nonzero trapping potential. In this light, it is assumed that fractional initial value problems (IVPs), Equations (1) and (2), have a unique and sufficiently smooth solution in the domain of interest. The fractional calculus theory is a powerful generalization of classical calculus theory dealing with differentiation and integration of arbitrary order. It has recently attracted the attention of numerous researchers for its considerable importance in many applications, such as fluid dynamics, signal processing, viscoelasticity, bioengineering, finance, Hamiltonian chaos, and vibrations, which can be used to describe the memory and transmissibility for multiple types of materials [12; 13; 14; 15; 16]. Moreover, it has successfully proved to be a valuable tool due to the realistic modeling that does not depend only on the instant time, but also on the history of the previous time, which can be achieved using fractional calculus [17; 18; 19; 20]. Unlike the classical calculus, which has unique definitions and clear geometrical and physical interpretations, there are numerous definitions for the operations of differentiation and integration of fractional order. Riemann-Liouville, Riesz, Grunwald-Letnikov, and Caputo are some examples of these definitions [21; 22; 23; 24]. In this light, a novel definition of fractional derivative, the conformable fractional concept, was proposed in 2014 by [PERSON] et al. [25]. Since then, several studies have appeared in the literature. For more details see [26; 27; 28; 29; 30; 31; 32]. The mathematical construction of most natural phenomena leads us to nonlinear differential equations of integer or non-integer order. Typically, there is no general theory for finding a closed-form solution of these nonlinear differential equations. To avoid such difficulty associated with the nonlinear terms, computer transform techniques, linearization process, or simplifying assumptions can be used to obtain solutions and consider linear approximations, and then attempt to reach geometric and arithmetic interpretations that enable us to understand these phenomena further. Indeed, such assumptions lead to significant errors in the solution. Therefore, nonlinear differential equations must be solved in a way that represents the real nonlinear problem, particularly in the fractional case. Anyhow, the difficulty in solving the model depends on the degree of nonlinearity [33; 34; 35]. Furthermore, the computation of solution through the power series while attractive from the theoretical point of view is difficult to implement since the terms of the power series may become very high, provoking machine overflow. In this orientation, a truncated conformable power series approximation can be used to compute the solution based on a simple algorithm, which is acceptable over the interval of interest and has a wide range of applications, particularly in simulating nonlinear problems in a fractional sense. This paper is structured as follow. The necessary definitions and properties of the conformable fractional derivative and fractional power series are shown in Section 2.