A. H. Tedjani, Mahmoud M. Abdelwahab, M. A. Abdelkawy

AN EFFICIENT SCHEME TO SOLVE FOURTH ORDER NONLINEAR TRIPLY SINGULAR FUNCTIONAL DIFFERENTIAL EQUATION

Introduction

An efficient scheme to solve fourth order nonlinear triply singular functional differential equation. Efficiently solve complex fourth-order nonlinear triply singular functional differential equations. A novel mathematical model, featuring delayed and multi-prediction terms, is numerically solved via spectral collocation.

Abstract

We present a novel mathematical model based on the fourth order multi-singular nonlinear functional differential equations. This designed nonlinear functional model has singularities at three points, making the model more complicated and harder in nature. The  delayed and multi-prediction terms in the model clearly represent the functionality of the model. Three different variants of the novel nonlinear triply singular functional differential model have been presented and the numerical results of each variant are obtained by using a well-known spectral collocation technique. For the perfection and excellence of the designed mathematical nonlinear model, the obtained numerical results of each variant have been compared with the exact solutions. Received: December 10, 2023Accepted: January 3, 2024

Review

The submitted work, titled "AN EFFICIENT SCHEME TO SOLVE FOURTH ORDER NONLINEAR TRIPLY SINGULAR FUNCTIONAL DIFFERENTIAL EQUATION," presents a numerical approach for a highly challenging class of mathematical problems. The abstract introduces a "novel mathematical model" that embodies a fourth-order nonlinear functional differential equation characterized by singularities at three distinct points, making its analytical and numerical treatment particularly intricate. The inclusion of delayed and multi-prediction terms further enhances the complexity and functional nature of the model. The authors state that they have explored three variants of this model, employing a spectral collocation technique to obtain numerical solutions, which are subsequently compared against exact solutions for validation. The proposed research tackles a problem of significant complexity and practical importance, as fourth-order nonlinear functional differential equations with multiple singularities are notoriously difficult to solve and often arise in various scientific and engineering applications. The use of a spectral collocation technique is a promising avenue for such problems, given its high-order accuracy and efficiency for well-behaved functions. The stated comparison with exact solutions is a crucial step in validating the accuracy and reliability of the numerical scheme. However, the abstract raises a few points for clarification. It refers to both a "novel mathematical model" and an "efficient scheme to solve" it; the precise nature of this novelty—whether it pertains to the formulation of the differential equation itself or to the specific adaptation of the spectral collocation method for this particular class of equations—needs to be elaborated. Furthermore, while the abstract claims "perfection and excellence" for the designed model, substantiation of such strong claims, particularly regarding efficiency and robustness across a broader range of parameters, would be essential in the full manuscript. Overall, this work appears to address a valuable and challenging problem in numerical analysis and differential equations. If the full paper comprehensively details the adaptation of the spectral collocation method to effectively handle the triple singularities, delays, and multi-prediction terms, and rigorously demonstrates the "efficiency" and "perfection" claimed, it could represent a significant contribution. The authors should ensure clear demarcation of the novelty, provide detailed methodological insights, and present thorough computational evidence, including convergence rates, computational costs, and stability analyses, to support their assertions. The potential impact of such a robust and efficient scheme for triply singular functional differential equations would be substantial for researchers in applied mathematics and related fields.

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