An in-depth examination of the fuzzy fractional cancer tumor model and its numerical solution by implicit finite difference method

The cancer tumor model serves a s a crucial instrument for understanding the behavior of different cancer tumors. Researchers have employed fractional differential equations to describe these models. In the context of time fractional cancer tumor models, there's a need to introduce fuzzy quantities instead of crisp quantities to accommodate the inherent uncertainty and imprecision in this model, giving rise to a formulation known as fuzzy time fractional cancer tumor models. In this study, we have developed an implicit finite difference method to solve a fuzzy time-fractional cancer tumor model. Instead of utilizing classical time derivatives in fuzzy cancer models, we have examined the effect of employing fuzzy time-fractional derivatives. To assess the stability of our proposed model, we applied the von Neumann method, considering the cancer cell killing rate as time-dependent and utilizing Caputo's derivative for the time-fractional derivative. Additionally, we conducted various numerical experiments to assess the viability of this new approach and explore relevant aspects. Furthermore, our study identified specific needs in researching the cancer tumor model with fuzzy fractional derivative, aiming to enhance our inclusive understanding of tumor behavior by considering diverse fuzzy cases for the model's initial conditions. It was found that the presented approach provides the ability to encompass all scenarios for the fuzzy time fractional cancer tumor model and handle all potential cases specifically focusing on scenarios where the net cell-killing rate is time-dependent.