Oumar MADAI, Germain KABORE, Bakari Abbo, Ousséni SO, Blaise SOME

THE SOME BLAISE ABBO (SBA) PLUS METHOD APPLIED TO FRACTIONAL NONLINEAR TIME SCHRÖDINGER EQUATIONS IN $d$ DIMENSION $(d = 1, 2,$ or $3)$ IN THE SENSE OF CAPUTO

Introduction

The some blaise abbo (sba) plus method applied to fractional nonlinear time schrÖdinger equations in $d$ dimension $(d = 1, 2,$ or $3)$ in the sense of caputo. Apply the SBA plus method to solve fractional nonlinear time Schrödinger equations (d=1, 2, 3) in the Caputo sense. This method offers rapid convergence to exact solutions.

Abstract

In this paper, we have solved some time fractional Schrödinger equations of order $\alpha$ with $0<\alpha \leq 1$ in dimension $1, 2$ or $3$ in the sense of Caputo by the SBA plus method. This method is based on two principles (successive approximations, and Picard) and the Adomian method. Secondly, it uses a process of rapid convergence in the functional space of the problem posed towards the exact solution, if it exists. Received: April 3, 2024Accepted: June 6, 2024

Review

This paper presents an application of the Blaise Abbo (SBA) plus method to solve fractional nonlinear time Schrödinger equations, a class of problems critical in various fields of physics and engineering. The study focuses on these equations in one, two, or three spatial dimensions, employing the Caputo definition for the fractional derivative of order $\alpha$ ($0 < \alpha \leq 1$). The chosen domain highlights the method's potential for handling complex, multi-dimensional systems where traditional integer-order models may fall short. This work is significant given the increasing interest in fractional calculus for modeling non-local and memory effects in physical systems. The methodology adopted, termed the SBA plus method, is described as a hybrid approach, drawing principles from successive approximations, Picard iteration, and the Adomian decomposition method. This synthesis suggests an attempt to combine the strengths of these established techniques to address the complexities of the target equations. A central claim made for the SBA plus method is its "rapid convergence" in the functional space of the problem towards the exact solution. While the abstract outlines the foundational elements, a deeper understanding of how these components are integrated and how the rapid convergence is theoretically or numerically substantiated would be essential for a comprehensive evaluation of the method's novelty and efficiency. In summary, this paper proposes an intriguing new application of the SBA plus method to a highly relevant and challenging area of mathematical physics. Should the detailed exposition in the full manuscript substantiate the claims of rapid convergence and superior performance compared to existing techniques, this work could offer a valuable semi-analytical tool for solving a complex class of fractional differential equations. Future scrutiny of the complete work should focus on the rigorous derivation of the SBA plus method, clear examples illustrating its implementation, a thorough analysis of its convergence properties, and a comparative study against other established methods to fully appreciate its practical advantages and limitations.

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