International Journal of Differential Equations

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition for all , and . If , this definition coincides to the classical definition of the first order of the function .

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers , the occasional smokers , the chain smokers , the temporarily quit smokers , and the permanently quit smokers . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.

In this study, the spreading of the pandemic coronavirus disease (COVID-19) is formulated mathematically. The objective of this study is to stop or slow the spread of COVID-19. In fact, to stop the spread of COVID-19, the vaccine of the disease is needed. However, in the absence of the vaccine, people must have to obey curfew and social distancing and follow the media alert coverage rule. In order to maintain these alternative factors, we must obey the modeling rule. Therefore, the impact of curfew, media alert coverage, and social distance between the individuals on the outbreak of disease is considered. Five ordinary differential equations of the first-order are used to represent the model. The solution properties of the system are discussed. The equilibria and the basic reproduction number are computed. The local and global stabilities are studied. The occurrence of local bifurcation near the disease-free equilibrium point is investigated. Numerical simulation is carried out in applying the model to the sample of the Iraqi population through solving the model using the Runge–Kutta fourth-order method with the help of Matlab. It is observed that the complete application of the curfew and social distance makes the basic reproduction number less than one and hence prevents the outbreak of disease. However, increasing the media alert coverage does not prevent the outbreak of disease completely, instead of that it reduces the spread, which means the disease is under control, by reducing the basic reproduction number and making it an approachable one.

In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.

Breast cancer is a type of carcinoma with a high prevalence. The treatment of breast cancer through chemotherapy can cause a risk to healthy cells throughout the body. The neutrophil is one of the cells that is influenced by chemotherapy drugs. Chemotherapy-induced neutropenia is one of the most common toxic effects experienced by patients and often threatens chemotherapy to use efficiency. In this paper, we introduce an interaction model between blood components, i.e., neutrophil, lymphocytes, and albumin, with chemotherapy drugs. The model is important to understand the neutropenia effect due to chemotherapy in mathematical perspective and to calculate breast cancer patients’ survival level. Our model is a four-dimensional system of the first-order ODE with 13-dimensional parameter space. We focus our study for understanding the steady-state conditions and the bifurcations when the parameter values are varied. Here, we also study the role of albumin for reducing the neutropenia effects for breast cancer patients mathematically, where the results can be used as an alternative solution for treating neutropenia in a breast cancer case.