This article studies a pharmacokinetics problem, which is the mathematical modeling of a drug concentration variation in human blood, starting from the injection time. Theories and applications of fractional calculus are the main tools through which we establish main results. The psi-Caputo fractional derivative plays a substantial role in the study. We prove the existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The application of the theoretical results on two data sets shows the following results. For the first data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields a mean square error (MSE) of 0.04065 . The second best is the simple fractional method whose MSE is 0.05814 ; finally, the classical approach is in the third position with an MSE of 0.07299 . For the second data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields an MSE of 0.03482 . The second best is the simple fractional method whose MSE is 0.04116 and, finally, the classical approach with an MSE of 0.048640 .
This article studies modeling of a population growth by logistic equation when the population carrying capacity K tends to infinity. Results are obtained using fractional calculus theories. A fractional derivative known as psi-Caputo plays a substantial role in the study. We proved existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The Chinese population, whose carrying capacity, K, tends to infinity, is used as evidence to prove that the proposed approach is appropriate and performs better than the usual logistic growth equation for a population with a large carrying capacity. A psi-Caputo logistic model with the kernel function x + 1 performed the best as it minimized the error rate to 3.20% with a fractional order of derivative α = 1.6455.
This work is dedicated to the study of the relationship between altitude and barometric atmospheric pressure. There is a consistent literature on this relationship, out of which an ordinary differential equation with initial value problems is often used for modeling. Here, we proposed a new modeling technique of the relationship using Caputo and Caputo–Fabrizio fractional differential equations. First, the proposed model is proven well-defined through existence and uniqueness of its solution. Caputo–Fabrizio fractional derivative is the main tool used throughout the proof. Then, follow experimental study using real world dataset. The experiment has revealed that the Caputo fractional derivative is the most appropriate tool for fitting the model, since it has produced the smallest error rate of 1.74% corresponding to the fractional order of derivative α = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative α = 1.042, and finally the classical method produced an error rate of 4.36%.
This paper studied the existence and uniqueness of the solution of the fractional logistic differential equation using Hadamard derivative and integral. Previous work has shown that there is not an exact solution to this fractional model. Hence several numerical approaches, such as generalized Euler's method (GEM), power series expansion (PSE) method, and Caputo-Fabrizio (CF) method, were used to compute the solution. The classical solution obtained from the first order non-linear differential equation was also considered to enable the comparison of error levels.
This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.
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