A Fractional Modeling of Tumor–Immune System Interaction Related to Lung Cancer with Real Data

In this study, we investigate a new fractional-order mathematical model which considers population dynamics among tumor cells-macrophage cells-active macrophage cells, and host cells involving the Caputo fractional derivative. Firstly, the stability of the positive steady state of the model is studied. Subsequently, the conditions for existence and uniqueness of the solutions are examined. Then, the least squares curve fitting method (LSCFM) which is one of the prominent methods for parameter estimation is used to fit the parameters of the model. It is aimed to fit the relevant parameters with the help of the tumor tissue samples which were collected from the patient with non-small cell lung cancer who had chemotherapy-naive hospitalized at Kayseri Erciyes University hospital in Turkey. A total of 12 parameters in the model are estimated using the data of lung tumor cells of this patient for 14 days. Moreover, the numerical simulations are given by considering the different fractional orders and different parameters for the model. So, it is achieved how the change in $$\alpha $$ α affects the dynamic behavior of the system. In the sequel, to point out the advantages of the fractional-order modeling, the memory trace and hereditary traits are taken into consideration. Finally, the interpretations in terms of biological science are provided in conclusion. We believe that this interdisciplinary study will open new doors for other similar studies and will shed light on the studies to be developed on the use of real data in the mathematical modeling of cancer.

Thermal characteristics of longitudinal fin with Fourier and non-Fourier heat transfer by Fourier sine transforms

The quest for high-performance of heat transfer components on the basis of accommodating shapes, smaller weights, lower costs and little volume has significantly diverted the industries for the enhancement of heat dissipation with variable thermal properties of fins. This manuscript proposes the fractional modeling of Fourier and non-Fourier heat transfer of longitudinal fin via non-singular fractional approach. The configuration of longitudinal fin in terms of one dimension is developed for the mathematical model of parabolic and hyperbolic heat transfer equations. By considering the Fourier and non-Fourier heat transfer from longitudinal fin, the mathematical techniques of Fourier sine and Laplace transforms have been invoked. An analytic approach is tackled for handling the governing equation through special functions for the fractionalized parabolic and hyperbolic heat transfer equations in longitudinal fin. For the sake of comparative analysis of parabolic verses hyperbolic heat conduction of fin temperature, we depicted the distinct graphical illustrations; for instance, 2-dimensional graph, bar chart, contour graphs, heat graph, 3-dimensional graphs and column graphs on for the variants of different rheological impacts of longitudinal fin.

Fractional Modeling and Characteristic Analysis of Hydro-Pneumatic Suspension for Construction Vehicles

The motion differential equation of hydro-pneumatic suspension is established to describe the vibration characteristics for a certain type of construction vehicle. The output force was deduced from the suspension parameters. Based on the suspension characteristics of a multi-phase medium, fractional calculus theory was introduced, and its fractional Bagley–Torvik equation was formed. The numerical computation by a low-pass filter of the Oustaloup algorithm was performed. The numerical solution of a nonlinear fractional equation was obtained to investigate the vibration characteristics of the suspension fractional system. Through the building of an equal-ratio test platform and simulation model, the fractional- integer-order model simulation and experimental data were compared. When the fractional order is 0.9, it better describes the motion characteristics of suspension system. The experiments show that the experimental data can fit the fractional-order system model well, and thereby prove the model on a hydro-pneumatic suspension system.