Compartmental model is the classic and widely used approach in pharmacokinetics. Recently, fractional calculus is introduced to describe the time course of drugs in human body which follow the anomalous diffusion mechanism. To consider the different fractional order transmission process, the two compartmental fractional model has been proposed and studied. And, it will be of great significance to find out a simple and efficient numerical method to solve the model and estimate model parameters. This work investigates two numerical methods of the two compartmental fractional model using finite difference schemes, which are based on the shifted Grünwald-Letnikov approximate formula and L1 formula, respectively. The hybrid Nelder-Mead simplex search and particle swarm optimization, denoted as NMSS-PSO, is provided to estimate the fractional model parameters. Comparison between the numerical solution and the solution by the numerical inverse Laplace transform method establishes the validity of the developed numerical algorithms. Then, the influence of the order of fractional derivative on the drug amount in human body is also discussed. Finally, the two compartmental fractional model is applied to fit the amiodarone pharmacokinetics data based on the NMSS-PSO algorithms. This work will be of importance for the development of fractional pharmacokinetics.

In this paper, a class of nonautonomous SEIRS epidemic models with vaccination and nonlinear incidence is investigated. Under some quite weak assumptions, a couple of new threshold values in the form of integral, i.e., $R_1$, $R^*_1$, $R_2$ and $R^*_2$ on the extinction and permanence of disease for the model are established. As special cases of our model, the autonomous, periodic and almost periodic circumstances are discussed respectively. The nearly necessary and sufficient criteria of threshold on the extinction and permanence of disease for above cases are obtained as well. Numerical examples and simulations are presented to illustrate the analytic results.

This paper is concerned with a 2-dimensional Klein-Gordon-Schrödinger system subject to two types of locally distributed damping on a compact Riemannian manifold $\mathcal{M}$ without boundary. Making use of unique continuation property, the observability inequalities, and the smoothing effect due to Aloui, we obtain exponential stability results.

Many autoimmune diseases are chronic in nature, so that in general patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a non-linear system of integro-differential equations with quadratic terms that describe the interactions between self-antigen presenting cells, self-reactive T cells and immunosuppressive cells. We consider a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well-posedness of the solution, and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.

In this paper, we employ three integration algorithms namely, the well known Kudryashove method, the new Kudryashov method and the unified Riccati equation expansion method to extract optical soliton solutions for the generalized Kudryashov's equation with power nonlinearities. Straddled soliton, bright solitons, dark solitons and singular solitons have been found

There are a few purely periodic phenomena in nature, which allows one to consider several other generalizations, such as almost automorphic and measure pseudo almost automorphic oscillations. In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed-point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general type reflection integro-differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe.

In this research study, fuzzy fractional differential equations in presence of the Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing Kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions; analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

In this article, we consider the following p-q-Laplacian system with singular and critical nonlinearity \begin{equation*} \left \{ \begin{array}{lllll} -\Delta_{p}u-\Delta_{q}u=\frac{h_{1}(x)}{u^{r}}+\lambda\frac{\alpha}{\alpha+\beta}u^{\alpha-1}v^{\beta} \ \ in\ \Omega ,\\ -\Delta_{p}v-\Delta_{q}v=\frac{h_{2}(x)}{v^{r}}+\lambda\frac{\beta}{\alpha+\beta}u^{\alpha}v^{\beta-1} \ \ in\ \Omega, \\ u,v>0 \ \ \ \ \ \ in \ \Omega, \ \ \ \ \ u=v=0 \ \ \ \ \ \ \ on \ \partial\Omega, \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial\Omega$. $11,\lambda\in(0,\Lambda_{*})$ is parameter with $\Lambda _{*}$ is a positive constant and $h_{1}(x),h_{2}(x)\in L^{\infty},h_{1}(x),h_{2}(x)>0$. We show the existence and multiplicity of weak solution of equation above for suitable range of $\lambda$.

In this study, a generalized nonlinear local fractional Lighthill-Whitham-Richards (LFLWR) model has been developed. The local fractional variational iteration method (LFVIM) solves and analyzes the proposed model. Numerous works have been described in past to address linear LWR and linear LFLWR models. This research highlighted on generalized nonlinear LFLWR model and LFVIM is employed to derive non-differentiable solutions of the suggested model. The existence and uniqueness of the resolution of LFLWR model have also been established. Furthermore, several exemplary instances are discussed to demonstrate the success of implementing LFVIM to the proposed model. The numerical simulations for each of the cases have also been shown. Additionally, the obtained solutions of the suggested model have been compared with the solutions of the classical LWR model with non-differentiable conditions in few examples. The study demonstrates that the employed iterative scheme is quite efficient and can be utilized for obtaining the non-differentiable solution to proposed generalized nonlinear LFLWR model of traffic flow.

We extend the well-known Halanay inequality to the fractional order case in presence of distributed delays and delays of neutral type (in the fractional derivative). Both the discrete and distributed neutral delays are investigated. It is proved that solutions decay toward zero in a Mittag-Leffler manner under some rather general conditions. Some large classes of kernels and examples satisfying our assumptions are provided. We apply our findings to prove Mittag-Leffler stability for solutions of fractional neutral network systems of Cohen-Grossberg type.

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the corresponding convolution, inverse and Plancherel formulas, and Hausdorff-Young inequality. We show that this generalized Fourier transform is useful in the study of fractional partial differential equations, by solving the fractional heat equation on the real line.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.

Quantum Fisher information (QFI) and geometric phase have recently been performed different tasks in quantum information technology. We investigate the QFI and entanglement of a three-level atom in $\Lambda $ configuration interacting with a quantized field mode by using linear entropy. We study the dynamical behavior of the geometric phase based on the engineering of a three-level atomic configuration. We analyze the effect of energy dissipation of the dynamical properties of the geometric phase and the QFI as an entanglement quantifier between the three-level atom and field. We explore the correlation between the engineering geometric phase and QFI in the absence and presence of energy dissipation effect. We have found that the $\mathrm{QFI}$ is very sensitive to the effect of the time dependent coupling and energy dissipation.

In this article, by using the notion of newly defined $q_{1}q_{2}$-derivatives and integrals some new Simpson’s type inequalities for co-ordinated convex functions are shown. The outcomes raised in this paper are extensions and generalizations of the comparable results in the literature on Simpson’s inequalities for co-ordinated convex functions.

In this paper we investigate the global existence of small data solutions for the following structurally damped σ-evolution model with nonlinear memory term % \[u_{tt}+(-\Delta)^\sigma u+\mu(-\Delta)^{\frac{\sigma}{2}}u_t=\int_0^t (1+\tau)^{-\gamma}|u_t(\tau,\cdot)|^p\,d\tau,\] % with σ>0. In particular, for $\gamma\in ((n-\sigma)/n,1)$ we find the sharp critical exponent, under the assumption of small data in~$L^1$. Dropping the~$L^1$ smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration~$1-\gamma$ in the nonlinear memory term, and the assumption that initial data are small in~$L^m$, for some~$m>1$.

Bertoin, Roynette et Yor \cite{bertion} described new connections between the class $\Bd$ of L\’evy-Laplace exponents $\Psi$ (also called the class (sub)critical branching mechanism) and the class of Bernstein functions ($\BF$) which are internal, i.e. those Bernstein functions $\phi$ s.t. $\Psi \circ \phi$ remains a Bernstein function for every $\Psi$. We complete their work and illustrate how the class f internal function is rich from the stochastic point of view. It is well known that every $\phi \in \BF$ corresponds univocally to: (i) a subordinator ${(X_t)}_{t\geq 0}$ (or equivalently to transition semigroups ${\big(\pr(X_t\in dx)\big)}_{t\geq 0}$; (ii) a L\’evy measure $\mu$ (which controls the jumps of the subordinator). It is also known that, on $\oi$, the measure $\pr(X_t \in dx)/t$ converges vaguely to $\dd \delta_0(dx)+ \mu(dx)$ as $t\to 0$, where $\dd$ is the drift term, but rare are the situations where we can compare the transition semigroups with the L\’evy measure. Our extensive investigations on the composition of L\’evy-Laplace exponents $\Psi$ with Bernstein functions show, for instance, this remarkable facts: $\phi$ is internal is equivalent to: (a) $\phi^2 \in \BF$ or to (b) $t\mu(dx) - \pr(X_t\in dx)$ is a positive measure on $\oi$. We also provide conditions on $\mu$ insuring that $\phi$ is internal. We also show L\’evy-Laplace exponents are closely connected to the class of Thorin Bernstein function and provide conditions on $\mu$ insuring that $\phi$ is internal.

In this work, we analyse the controllability for certain classes of impulsive integro - differential equations(IIDE) of fractional order via Atangana Baleanu derivative involving finite delay with initial and nonlocal conditions using Banach fixed point theorem.

The present paper is devoted to study the space identification problem for the elliptic-telegraph differential equation in Hilbert spaces with the self-adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptictelegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic-telegraph differential equations are established.