## 1 Introduction  
Modern physics was born after classical mechanics failed in explanation of various physical phenomena, including microscopic scales, like photoelectric effect, black body radiation, and the stability of the atoms depending on the fact that all physical quantities of a bound system are restricted to discrete values quantization. Quantum mechanics can successfully describe different modern physics phenomena in atomic and nuclear physics and other modern physics branches, where electrons' behavior in atomic physics and nucleons in nuclear physics can be described quantum mechanically by the Schrodinger equation [1]. The discovery of the quantum mechanical path integral approach can be used as an alternative of the Schrodinger wave equation and Heisenberg matrix dynamics that originated to find what corresponds in quantum theory to the Lagrangian method of classical mechanics. [PERSON] searched in the role which classical mechanics fundamentals like Lagrangian and least-action principle, which effect quantum mechanics. Furthermore, [PERSON] developed this idea and called it path integral. Since then, the path integral approach and the perturbation technique based on it, Feynman diagrams, became powerful tools in quantum mechanics and quantum field theory, solid-state and quantum liquid theory, equilibrium and non-equilibrium statistical physics, theory of turbulence and chaotic phenomena, theory of random processes and polymer physics, mathematics, chemistry, and economic studies. Several powerful analytical strategies are established and developed in the literature to solve the Schrodinger equation. For example, an analytic, accurate solution beyond adiabatic approximation by transferring the 1D Schrodinger equation into the Ricatti equation has been obtained in [2]. In [3], the position-dependent mass Schrodinger equation has been studied and solved by using an effective advanced analytical technique. While the analytic solution of the 3D stationary Schrodinger equation for any arbitrary potential was proposed in [4] by transforming the Schrodinger equation to the generalized Ricatti equation. The fractional Schrodinger equation is a new fundamental equation of quantum physics, and it includes the space derivative of fractional order instead of a second-order space derivative in the standard Schrodinger equation. In this regard, the fractional Schrodinger equation is the fractional differential equation in accordance to modern terminology. In the limit of integer case, the fundamental equation of fractional quantum mechanics is transformed into the well-known equation of standard quantum mechanics [5]. Many elegant and fascinating models in quantum mechanics, fluid dynamics, and other physical fields were formulated by partial differential equations in a fractional sense, for instance, the nonlinear fractional Schrodinger equation, fractional wave equation, fractional fluid flow models, and fractional Boussinesq-like equation. The solution of partial differential equations (PDEs), in the sense of fractional, is very significant to describe the dynamic behavior of both linear and nonlinear physical systems such as the dynamics of nuclear reactors, randomized analytical models, fluid mechanics, viscoelastic damping, electromagnetism, and electrochemistry [6; 7; 8; 9; 10; 11]. However, only a little research has been done on nonlinear systems, and few analytic-numeric techniques have been presented for solving the nonlinear PDE problems for fractional models. This gives us the motivation to look for numerical solutions for these systems.