In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter $\alpha$. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shall have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development.

Human immunodeficiency virus (HIV) infection is said to be a dangerous, extremely severe, and potentially fatal disease. After weakening a person's immune system, it impedes the body's ability to fend off future illnesses. In this study, we use the Caputo operator's fractional framework to describe how antiretroviral therapy (ART) affects the HIV/AIDS transmission process. Using fractional calculus, the basic characteristics of the formulated system are found. In addition, we investigated the steady states of the fractional system and determined the reproduction parameter $ \mathcal{R}_0 $. A fractional framework has been used for stability analysis. The optimal value of the Caputo fractional-order $ \alpha $ is around $ 9.437300e-01 $, and the system parameters are determined using the least-squares method, demonstrating that the system matches real HIV/AIDS data the best. Average and maximum absolute relative percentile errors have been used to draw comparisons between the classical and Caputo systems. The case summaries obtained demonstrate the Caputo operator's supremacy. A sensitivity test is run using the partial rank correlation coefficient (PRCC) technique to clearly see which parameters are crucial to the formulation of $ \mathcal{R}_0 $. Lastly, a powerful numerical technique is used to run simulations that show the HIV/AIDS time series in the Caputo fractional framework.

In this work, a new optimal iterative algorithm is presented with fourth-order accuracy for root-finding of real functions. It uses only function as well as derivative evaluation. The algorithm is obtained as a combination of existing third-order methods by specifying a parameter involved. The algorithm is based on local and semilocal analysis and has been specifically designed to improve efficiency and accuracy. The proposed algorithm represents a significant improvement over existing iterative algorithms. In particular, it is tested on a range of polynomial functions and was found to produce accurate and efficient results, with improved performance over existing algorithms in terms of both speed and accuracy. The results demonstrate the effectiveness of the proposed algorithm and suggest that it has great potential for use in a wide range of applications in polynomiography and other areas of mathematical analysis.