Shifted Chebyshev Neural Network Method for Solving Differential Equations

Sharmeen I Hassan; Salisu Ibrahim

doi:10.23918/eajse.v12i1p2

Authors

Sharmeen I Hassan Department of mathematics education, Tishk International University, Erbil, Iraq https://orcid.org/0000-0001-9763-0364
Salisu Ibrahim Department of information Technology, Tishk International University, Iraq https://orcid.org/0000-0002-1467-5426

DOI:

https://doi.org/10.23918/eajse.v12i1p2

Keywords:

Chebyshev Polynomials, Shifted Chebyshev Polynomial, Ordinary Differential Equation, Nonlinear Equations, Numerical Aprroximation

Abstract

The Lane–Emden equations, a class of nonlinear ordinary differential equations, are fundamental in modeling various physical phenomena, particularly in astrophysics for characterizing polytropic star structures. In this study, we provide a neural network-based model by incorporating Shifted Chebyshev Polynomials (SCPs) for solving Lane-Emden type equations. The universal approximation capacity of neural networks is harnessed in this model alongside the spectral accuracy of SCPs to efficiently handle the nonlinear and singular nature of these equations. Shifted Chebyshev Polynomials are embedded into the neural network structure to better capture solution behavior over the semi-infinite domain while naturally satisfying the required boundary conditions. A physics-informed loss
function, constructed from the residuals of the governing differential equations, is minimized during training. Numerical experiments on classical Lane–Emden problems validate the proposed method, demonstrating high accuracy and convergence compared to existing analytical and numerical techniques. The results confirm that the neural network-SCP framework is a robust, effective , and flexible tool for solving complex nonlinear differential equations.

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References

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