A hybrid computational framework for fractional cancer diffusion models in uncertain settings.

Cancer tumor modeling is crucial for understanding tumor dynamics, treatment strategies, and the effects of therapies on progression. While extensive research exists on fractional cancer models, innovation is limited in incorporating fuzzy-fractional calculus into cancer diffusion models. This paper introduces a fuzzy-fractional diffusion cancer model solved using the He-Laplace algorithm in the Liouville-Caputo sense. A novel hybrid algorithm, combining homotopies, perturbation techniques, and Laplace transforms, is developed to solve this complex model. Our method employs triangular fuzzy numbers to capture uncertainty, enabling analysis of cancer diffusion across lower and upper bounds and extending beyond conventional crisp models, opening up an entirely new domain. This model explores two different scenarios for the killing rate of cancer cells, namely time-dependent killing rate, and space dependent killing rate. Numerical solutions are provided for both lower and upper bounds, with results varying across different fractional orders and residual errors calculated to validate the method's authenticity and applicability. 2D and 3D visualizations demonstrate the model's complexity, and solutions are analyzed in a fuzzy environment. Contour diagrams further enhance the accuracy of capturing diffusion models in the fuzzy-fractional context. The results demonstrate the method's efficiency and accuracy, providing valuable insights into cancer tumor dynamics. It effectively models tumor heterogeneity, improving understanding, prediction, and treatment optimization. Future work could involve applying real-life data to compare the simulation's accuracy in reflecting real-world tumor dynamics. This study emphasizes the method's effectiveness for solving complex models in scientific and biological fields.