On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

In this research collection,we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional q-difference equation having the given form c D^(ζ)_(q) v(t)−h(t,v(t))=0,0≤t≤1,α_(1)v(0)+β_(1)D_(q)v(0)=v(η1),α_(2)v(1)−β_(2)D_(q)v(1)=v(η2),where 1<ζ≤2,(η1,η2)∈(0,1)^(2),α_(i),β_(i)∈R(i=1,2),h∈C([0,1]×R,R)and c Dζq represents the Caputo-type nonclassical q-derivative of orderζ.We use well-known principal of Banach contraction,and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem.Regarding the applications,some examples are solved to justify our outcomes.

A mathematical study of a tuberculosis model with fractional derivatives

In this work,we use a Predictor–Corrector method to implement and derive an iterative solution of an existing Tuberculosis(TB)model with two fractional derivatives,namely,Caputo–Fabrizio fractional derivative and the new generalized Caputo fractional derivative.We begin by recalling some existing results such as the basic reproduction number R0 and the equilibrium points of the model.Then,we study the global asymptotic stability of disease-free equilibrium of the fractional models.We also prove,for each fractional model,the existence and uniqueness of solutions.An iterative solution of the two models is computed using the Predictor–Corrector method.Using realistic parameter values,we perform numerical simulations for different values of the fractional order.Simulation results permit to conclude that the new generalized Caputo fractional derivative provides a more realistic analysis than the Caputo–Fabrizio fractional derivative and the classical integer-order TB model.

A malaria model with Caputo-Fabrizio and Atangana-Baleanu derivatives

In this paper,we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense,in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered.Using Lyapunov theory,we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model,and the fractional models,whenever the basic reproduction number R0 is greater than one.By using fixed point theory,we prove existence,and conditions of the uniqueness of solutions,as well as the stability and convergence of numerical schemes.Numerical simulations for both models,using fractional Euler method and Adams–Bashforth method,respectively,are provided to confirm the effectiveness of used approximation methods for different values of the fractional-orderγ.

Prediction studies of the epidemic peak of coronavirus disease in Japan: From Caputo derivatives to Atangana-Baleanu derivatives

New atypical pneumonia caused by a virus called Coronavirus(COVID-19)appeared in Wuhan,China in December 2019.Unlike previous epidemics due to the severe acute respiratory syndrome(SARS)and the Middle East respiratory syndrome coronavirus(MERS-CoV),COVID-19 has the particularity that it is more contagious than the other previous ones.In this paper,we try to predict the COVID-19 epidemic peak in Japan with the help of real-time data from January 15 to February 29,2020 with the uses of fractional derivatives,namely,Caputo derivatives,the Caputo–Fabrizio derivatives,and Atangana–Baleanu derivatives in the Caputo sense.The fixed point theory and Picard–Lindel of approach used in this study provide the proof for the existence and uniqueness analysis of the solutions to the noninteger-order models under the investi-gations.For each fractional model,we propose a numerical scheme as well as prove its stability.Using parameter values estimated from the Japan COVID-19 epidemic real data,we perform numerical simulations to confirm the effectiveness of used approxima-tion methods by numerical simulations for different values of the fractional-orderγ,and to give the predictions of COVID-19 epidemic peaks in Japan in a specific range of time intervals.

Lassa hemorrhagic fever model using new generalized Caputo-type fractional derivative operator

In some of the previous decades, we have observed that mathematical modeling hasbecome one of the most interesting research fields and has attracted many researchers.In this regard, thousands of researchers have proposed different varieties of mathematicalmodels to study the dynamics of a number of real-world problems. This research workis framed to analyzing the structure of the well-known Lassa hemorrhagic epidemic;adangerous epidemic for pregnant women, via new generalized Caputo type nonintegerorder derivative with the help of a modified Predictor–Corrector scheme. Lassa hemorrhagic disease is an epidemical and biocidal fever, whose negative impacts were initiallyrecognized in the countries of Africa. This virus has killed many pregnant women ascompared to the Ebola epidemic. It was noticed that Lassa virus was isolated in Verocell cultures from a blood pattern, and after 12 days it was ejective, after the climb ofthe sickness. In this research study, necessary theorems and lemmas are reminded toprove the existence of a unique solution and stability of given fractional approximationscheme. All necessary results are reminded to confirm the effectiveness of the proposedapproximation algorithm by graphical observations for various fractional-order values.In our practical calculations, we plotted the graphs for two different values of naturaldeath rate along with various values of given fractional-order operator. Our major target is to show the importance of the proposed modified version of the Predictor–Correctoralgorithm in epidemic studies by exploring the given Lassa hemorrhagic fever dynamics.