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Abstract
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The Schrödinger equation is one of the most fundamental equations in quantum mechanics, governing the behavior of quantum systems. Its nonlinear and fractional variants have attracted significant attention in recent years because they describe a wide range of complex phenomena in physics, including nonlinear optics, plasma physics, quantum field theory, anomalous diffusion, and condensed matter systems. The inclusion of fractional derivatives provides a powerful framework for modeling memory effects, nonlocal interactions, and anomalous transport phenomena, which cannot be captured by the classical Schrödinger equation.
Despite its importance, obtaining exact analytical solutions of the nonlinear fractional Schrödinger equation (NLFSE) is extremely challenging due to the combined presence of nonlinearity and fractional-order operators. In most cases, closed-form solutions are either unavailable or limited to very specific conditions. This creates a strong necessity for developing efficient and accurate numerical methods capable of handling such equations under general boundary and initial conditions.
Gaussian radial basis function (RBF) interpolation emerges as a promising tool for approximating solutions of fractional differential equations. Unlike traditional grid-based methods, RBF-based approaches are mesh-free, flexible, and highly accurate for multidimensional and irregular domains. The Gaussian RBF, in particular, offers excellent smoothness and approximation properties, making it well-suited for capturing the nonlocal and highly oscillatory behavior of solutions to the NLFSE. Moreover, it reduces the complexity of handling fractional derivatives, which often introduce singularities and long-range dependencies.
The importance of this research lies in advancing numerical strategies for solving one of the most challenging classes of equations in applied mathematics and physics. By applying Gaussian RBF interpolation to the NLFSE, this study contributes to the d
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