TITLE:
Numerical Analysis of Approximate Solutions and Linear Growth in a Glial Cell Dynamics Model
AUTHORS:
Somayyeh Azizi
KEYWORDS:
Nonlinear Equation, Glial Cells, Numerical Methods, Fractional Diffusion Equation, Approximate Solution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.12,
December
24,
2025
ABSTRACT: In this study, we investigate a mathematical model that describes the growth dynamics of glial cells in glioma, formulated as a nonlinear partial differential equation with a treatment-dependent source term. To approximate the solution of this model, we employ three semi-analytical techniques: the Homotopy Analysis Method (HAM), the Homotopy Perturbation Method (HPM), and the Reduced Differential Transform Method (RDTM). A comparative analysis shows that while all three methods produce accurate results, RDTM exhibits rapid stabilization across various time points, outperforming HAM and HPM in terms of convergence speed and computational efficiency. To incorporate memory effects commonly observed in biological systems, we extend the model to a fractional-order framework. Within this extension, we apply HPM, the Fractional Reduced Differential Transform Method (FRDTM), and RDTM to construct higher-order approximations and examine their convergence behavior. We also conduct detailed convergence and error analysis for the resulting series of solutions, providing theoretical validation of their accuracy and reliability. The simulation results reveal a steady decline in glial cell concentration over time, eventually approaching negligible levels, indicating effective suppression of glioma growth under the modeled treatment. Notably, smaller values of the fractional-order parameter accelerate this decline, highlighting the significant influence of fractional dynamics on treatment outcomes. Finally, we establish the existence, uniqueness, and stability of the solution using the sectorial operator framework and the Mittag-Leffler function representation, reinforcing the mathematical soundness of the proposed model. These findings underscore the potential of fractional modeling and semi-analytical methods in capturing the complex behavior of glioma progression and enhancing therapeutic strategies.