Document Type : Research Paper
Authors
1
Department of Basic Science, Faculty of Engineering, Horus University, New Damietta 34517, Egypt.
2
Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt.
3
1. Nanoelectronics Integrated Systems Center, Nile University, Giza, 12588, Egypt.\\ 2. Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt.
10.22034/cmde.2025.66899.3165
Abstract
This paper presents an enhanced numerical method for solving nonlinear fractional boundary value problems using the Genocchi quasilinearization method (G-QLM). Fractional differential equations (FDEs) are gaining significant attention due to their ability to model various real-world phenomena with memory and hereditary properties in fields such as mechanics, fluid dynamics, chemistry, and astrophysics. However, these equations often lack analytical solutions, necessitating the development of efficient numerical techniques. Traditional methods such as the spectral collocation method, finite difference, and various wavelet approaches have demonstrated limitations in accuracy and computational efficiency when applied to highly nonlinear problems. The proposed approach combines the quasilinearization technique with Genocchi polynomials to transform the nonlinear fractional models into a sequence of linearized subproblems. This iterative process involves solving the Bratu, Troesch, and Lane-Emden equations—three well-known mathematical models with applications in areas such as heat transfer, plasma confinement, and stellar structure. The paper details the formulation of these equations, the application of the Genocchi collocation method, and the generation of approximate solutions through iterative refinement. The accuracy and convergence of the G-QLM method are validated through error analysis in the weighted $L^2$-norm. Numerical experiments using MATLAB are conducted for various cases of fractional orders $\beta$ and different initial and boundary conditions. Comparative studies demonstrate that the G-QLM method outperforms existing methods, including Bessel-based techniques, reproducing kernel methods, and Taylor wavelets, in terms of reduced absolute errors and computational efficiency.
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