This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012. We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.
The generalized Caputo fractional derivative is a name attributed to the Caputo version of the generalized fractional derivative introduced in Jarad et al. (J. Nonlinear Sci. Appl. 10:2607-2619. Depending on the value of ρ in the limiting case, the generality of the derivative is that it gives birth to two different fractional derivatives. However, the existence and uniqueness of solutions to fractional differential equations with generalized Caputo fractional derivatives have not been proven. In this paper, Cauchy problems for differential equations with the above derivative in the space of continuously differentiable functions are studied. Nonlinear Volterra type integral equations of the second kind corresponding to the Cauchy problem are presented. Using Banach fixed point theorem, the existence and uniqueness of solution to the considered Cauchy problem is proven based on the results obtained.
This paper presents an analytical solution for transient natural convection heat and mass transfer flow in a vertical channel with Soret and Dufour effects. Due to the presence of these two effects, energy and concentration equations are coupled. The dimensionless governing equations for momentum, energy and concentration are first decoupled using perturbation method and then solved using Laplace Transform Technique (LTT) under relevant initial and boundary conditions. The expressions for temperature, concentration, velocity, rate of heat transfer, rate of mass transfer and skin-friction are obtained. Numerical solutions are also obtained using pdepe in MATLAB so as to validate the accuracy of the proposed analytical method. The effects of Soret parameter, Dufour parameter, Grashof number, modified Grashof number, Prandtl number, Schmidt number and dimensionless time are presented graphically and discussed. It is observed that the temperature and velocity increase with increase in Dufour number, while concentration decreases with increase of Dufour number. The Dufour effect is more significant on the temperature and velocity in comparison to concentration. Moreover, it is observed that the concentration and velocity increase with increase in Soret number while the impact of Soret number is just contrast on temperature variation.
This research presents an analytical solution of unsteady free convection and mass transfer flow past a vertical plate with Soret and Dufour effects. The dimensionless system of governing equations is solved analytically with appropriate initial and boundary conditions. The accuracy of the analytical method is ensured by obtaining numerical solutions with PDEPE of MATLAB and comparing with the analytical results. Perturbation method is first adopted to decouple the system of equations that arise as a result of coupling Soret and Dufour effects. Laplace Transform Technique is then applied to solve the system. The expressions for velocity, temperature, concentration, Skin-friction, Nuselt and Sherwood numbers are obtained. In the course of discussions, the effects of main parameters are described. It is observed that increase in Soret number reduces the temperature while increasing the velocity and concentration. Moreover, Soret effect is more significant on the concentration than on the temperature. Similarly, the Dufour parameter causes the temperature and velocity to increase while the concentration decreases and the effect is more significant on the temperature than on the concentration. However, there is no significant difference on the effects of Dufour and Soret parameters on the velocity. The velocity, temperature and concentration profiles are presented graphically for Pr = 0.71 and Sc = 0.78 as well as for arbitrary values of other parameters.
This work investigates unsteady free convection heat and mass transfer flow in a vertical channel in the presence of Soret and Dufour effects. The bounding walls and the specie are considered to have ramped temperature and ramped concentration respectively. Perturbation method is first used to decouple the governing equations that arise from the model due to the presence of combined Soret and Dufour effects. Laplace transform technique is then used to obtain an analytical solutions for the temperature, concentration and velocity. In order to cross check the accuracy of the proposed analytical method, numerical solutions are obtained using PDEPE in MATLAB. The influences of the two effects as well as the ramped boundary conditions on the fluid flow are graphically presented and discussed. The results show that these two effects and the imposed boundary conditions affect the fluid temperature, concentration, velocity, rate of heat transfer, rate of mass transfer and wall Skin friction. Moreover, it is found that the solutions obtained by the authors Jha and Gambo correspond to the results of the present work when Soret effect is absent.
Objectives: The main objectives of the study were to use data to corroborate the reported mysterious deaths being recorded in Kano state of Nigeria and find possible explanations to the causes. Study design: This was a cross-sectional study. Methods: With total lockdown in force at the time of conducting the survey, it was not possible to get data through one-on-one interview. Instead, an online survey form was developed and shared widely among residents in the state. The form captured quantitative data about symptoms and circumstances for the deceased and the respondents were only the state residents that witnessed the death cases in their respective communities. Results: A total of 260 responses from various local government areas (LGAs) were received within a period of four days beginning from April 21, 2020. About half of the respondents affirmed that the death toll in their communities within the last two weeks before the survey were in multiples and most of the deaths started to occur from April 13 2020. Moreover, since then, the rate of deaths reported did not decline during the period of the research. Conclusion: There was a sudden increase in the number of deaths due to acute illness suggestive of a disease. Highest number of deaths was recorded among senior adults, affecting males than females. The most affected area was Kano Municipal which was one of the populous LGA in the state. Although it was hard to point out the possible cause of death based on the data, however, the possibility of linking the situation to the current COVID-19 pandemic could not be ruled out. This was because the data strongly aligned with the pandemic in terms of symptoms, incubation period, age group affected and other demographic characteristics.
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