Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative that exists. In the present article, Λ-Fractional Derivative is used to describe the phenomenon of viscoelasticity, while the whole methodology is demonstrated meticulously. The fractional viscoelastic Zener model is studied, for relaxation as well as for creep. Interesting results are extracted and compared to other methodologies showing the value of the pre-mentioned method.

Projectile motion is studied using fractional calculus. Specifically, a newly defined fractional derivative (the Leibniz L-derivative) and its successor (Λ-fractional derivative) are used to describe the motion of the projectile. Experimental data were analyzed in this study, and conclusions were made. The results of well-established fractional derivatives were also compared with those of L-derivative and Λ-fractional derivative, showing the many advantages of these new derivatives.

Based on the need for radiobiological databases, in this work, we mined experimental ionizing radiation data of human cells treated with X-rays, γ-rays, carbon ions, protons and α-particles, by manually searching the relevant literature in PubMed from 1980 until 2024. In order to calculate normal and tumor cell survival α and β coefficients of the linear quadratic (LQ) established model, as well as the initial values of the double-strand breaks (DSBs) in DNA, we used WebPlotDigitizer and Python programming language. We also produced complex DNA damage results through the fast Monte Carlo code MCDS in order to complete any missing data. The calculated α/β values are in good agreement with those valued reported in the literature, where α shows a relatively good association with linear energy transfer (LET), but not β. In general, a positive correlation between DSBs and LET was observed as far as the experimental values are concerned. Furthermore, we developed a biophysical prediction model by using machine learning, which showed a good performance for α, while it underscored LET as the most important feature for its prediction. In this study, we designed and developed the novel radiobiological ‘RadPhysBio’ database for the prediction of irradiated cell survival (α and β coefficients of the LQ model). The incorporation of machine learning and repair models increases the applicability of our results and the spectrum of potential users.

Fractional derivatives can express anomalous diffusion in brain tissue. Various brain diseases such as Alzheimer’s disease, multiple sclerosis, and Parkinson’s disease are attributed to the accumulation of proteins in axons. Discrete swellings along the axons cause other neuro diseases. To model the propagation of voltage in axons with all those causes, a fractional cable geometry has been adopted. Although a fractional cable model has already been presented, the non-existence of fractional differential geometry based on the well-known fractional derivatives raises questions. These minute parts of the human neural system are modeled as cables that function with a non-uniform cross-section in the fractional realm based upon the Λ-fractional derivative (Λ-FD). That derivative is considered the unique fractional derivative generating differential geometry. Examples are presented so that fruitful conclusions can be made. The present work is going to help medical and bioengineering scientists in controlling various brain diseases.