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Abstract. We study the isothermal flow of a dusty viscous incompressible conducting fluid between two types of boundary motions-oscillatory and non-oscillatory, under the influence of gravitational force. Within the frame work of some physically realistic approximations and suitable boundary conditions, closed form solutions were obtained for the velocity profiles and the skin friction of the particulate flow. These results show that for a constant pressure gradient, only the velocity profile of the fluid and the skin friction are unaffected by gravity, while magnetic field is seen to affect both the fluid, particle velocities and the skin friction. Thus, our results are extension of previous results in literature, and graphical demonstration of some these solutions have been presented.

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This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.

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