CONVERGENCE ANALYSIS OF NEWTON-RAPHSON METHOD FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH SINGULARITIES
DOI:
https://doi.org/10.71146/kjmr932Keywords:
fractional differential equations, Newton-Raphson method, Convergence analysis, Singularities, Numerical methodsAbstract
This study focuses on the convergence of the Newton-Raphson method for the solutions of nonlinear fractional differential equations containing singularities, which are commonly used in the modeling of complex physical and engineering system problems. The fractional differential equations are of nonlocal and memory dependent type, and are challenging analytically and computation. The research is based on a theoretical and numerical approach to studying the impact of singular behaviour on convergence, stability and accuracy of iterative methods. A modified Newton-Raphson method is derived, having a better robustness in singular regions and addressing non-smoothness and unbounded derivatives problems. The results show that if there are no singularities, the classical rule of quadratic convergence is preserved, but if there are singularities, then the convergence efficiency is around 78-90%, depending on the strength of the singularity. Proposed method enhanced convergence performance of almost 96% and over 20 % computational efficiency over traditional methods. The result of error reduction analysis indicates that the iterative accuracy can be reached as high as 95% in the latter stages, which validates the method. A stability analysis also shows that the convergence behavior can be improved considerably by suitable initial approximations and by using regularization methods. The study helps closing the gap between classical numerical methods and the fractional calculus, by introducing the systematic convergence framework for singular problems. The results are significant for the solution of complex systems from applied mathematics, engineering and scientific computing where fractional models and singularities abound.
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Copyright (c) 2026 Imtiaz Hussain, Naila Afzal (Author)
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