The HIV infection of CD4+ T cells has been modeled by a system of first-order nonlinear differential equations.We applied the TSM, RK4, and ABM in this study. The model’s numerical solution has been found inthis work. The results show that, in comparison to the TSM approach and RK4, the relative error achieved bythe ABM is smaller. When solving systems of nonlinear differential equations, the ABM is highly accurate

This paper is based upon incomplete fractional calculus and with the help of this, derived the fractionalcalculus formula for the incomplete Mittag-Leffler function. The results obtained are found in the form ofincomplete Wright function and hypergeometric function.

The main of this article are presenting generalized Opial type inequalities which will be defined as theOpial-Jensen inequality for convex function. Further, new Opial type inequalities will be given for functionalsdefined with the help of the Opial inequalities.

The main concern of the this article is to study the non-linear Chen-Lee-Liu equation which describes themotion of waves in shallow water. For this purpose two analytical techniques namely, the Sardar-Subequationmethod and the new extended hyperbolic function method are utilized. Also, we established the idea ofthe construction of solitons solutions of non-linear evolution equations which are rising in fluid dynamics,nonlinear optics, mathematical biological models, mechanics, waves theory, quantum mechanics. Acquiredsolutions are demonstrated graphically to reveal the dynamics behavior of solitons solutions. It hoped thatthe established solutions can be used to enrich the dynamic behaviors of Chen-Lee-Liu equation. Further,these solutions disclose that our techniques are up-to-date, suitable and straightforward.

In the present work, the discrete homotopy analysis method is applied to solve nabla time-fractionalpartial difference equations. Fractional difference operator is considered in Caputo’s sense. We apply thediscrete homotopy analysis method to nabla fractional initial value problems. Obtained solutions involvean auxiliary parameter h, which we can determine. Thus, it may be concluded that the discrete homotopyanalysis method is a very powerful and successful analytical approach for fractional difference equations.

We study a class of fractional differential inclusions defined by Caputo-Katugampola fractional derivativeinvolving a nonconvex set-valued map in the presence of certain fractional integral boundary conditions.Using a technique developed by Filippov we establish an existence result for the problem considered underthe hypothesis that the set-valued map is Lipschitz in the state variable. Also, based on a result concerningthe arcwise connectedness of the fixed point set of a class of set-valued contractions, we prove the arcwiseconnectedness of the solution set of the problem considered. The paper is the first in literature which containssuch kind of results in the framework of the problem studied.

In this manuscript we have studied a five compartmental mathematical model of Ebola epidemic. Thesuggested mathematical model is classified into susceptible, incubation, infected, isolated infected and recoveredclasses. The Taylor series method (TSM) is used to achieve the approximate results for each compartment.The graphical presentation that corresponds to some real facts is given.

We introduce more general concepts of nabla Riemann-Liouville fractional integrals and derivatives ontime scales. Such generalizations on time scales help us to study relations between fractional differenceequations and fractional differential equations. Sufficient conditions for the existence and uniqueness of thesolution to an initial value problem are described by nabla derivatives on time scales. Some properties of thenew operator are proved and illustrated with examples.

The theory of fractional integral inequalities plays a pivotal role in approximation theory. It is very usefulin establishing the uniqueness of solutions for some fractional differential equations. Here, a generalizedfractional integral identity is established to deduce new estimates for Bullen type functional and of somerelated inequalities to provide some applications in probability and information theory for (s, p)−convexfunction by use of basic techniques of analysis.

In this paper, the existence of positive solutions of a class of nonlinear fractional boundary value problemsis considered. Two fixed point theorems are used, namely: Banach Contraction mapping principle andLeggett-Williams fixed point theorems. The former is used to prove the existence of a unique solution, whereasthe latter is used to prove the existence of at least three positive solutions to the problem. Some examples areprovided to illustrate the two results.