Mathematics and Computational Sciences

Enhanced Milne-Simpson s methods for autonomous and singular differential equations

Document Type : Original Article

Authors

Ajimot Folasade Adebisi 1
Saheed Aremu 1
Muideen Odunayo Ogunniran 1
Kamiludeen Tijani 1
Adewole M Ajileye 2

1 Department of Mathematical Sciences, Osun State University Osogbo, Nigeria.

2 Department of Mathematical Sciences, University of Ilesa, Nigeria, adewole.

https://doi.org/10.30511/mcs.2025.2035400.1205
Abstract
Solving autonomous and singular differential equations remains a persistent challenge for traditional numerical methods due to the presence of critical points and singularities that degrade solution accuracy. This paper introduces a novel hybrid framework that uniquely integrates the classical Milne-Simpson’s method with a neural network-based refinement strategy to address these challenges. While Milne-Simpson’s method provides an efficient initial approximation, its accuracy deteriorates near singular behaviors. To overcome this, we propose a deep learning-based post-processing stage specifically designed to refine the coarse numerical solutions. Unlike previous works that either apply neural networks as standalone solvers or generic correctors, our approach explicitly tailors the neural architecture to learn correction functions that complement the structural dynamics of Milne-Simpson’s output. The neural network is trained on synthetic datasets generated to highlight the failure modes of classical methods, particularly focusing on complex autonomous and singular behavior. Experimental evaluations demonstrate that our hybrid approach significantly improves solution accuracy in problematic regions without compromising computational efficiency, thus offering a robust and scalable method for solving challenging differential equations.

Keywords

Radial basis functions
Autonomous Differential Equations
Singular Differential Equations
Deep Learning
Root mean square error

Subjects

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Mathematics and Computational Sciences
Volume 6, Issue 2
Spring 2025
Pages 125-140

  • Receive Date 13 July 2024
  • Revise Date 23 June 2025
  • Accept Date 29 June 2025

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