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The square-root dynamics of smoking model with cravings to smoke, in which square root of potential smokers and smokers is the interaction term, has been studied. We categorized net population in four different chambers: non-smokers/potential smokers, smokers/infected people, non-permanent smokers/temporary quitters and the permanent quitters. By dynamical systems approach, we analyzed our model. Moreover, for proving the unique equilibrium point to be globally stable, we took help of graph theoretic approach. The sensitivity analysis of the model is performed through the diseased classes effectively to design reliable, robust and stable control strategies. The model is designed like optimal control trouble to find out importance of various control actions on our system that are insisted by the sensitivity analysis. We have applied two controls, which are the awareness campaign through the media transmission to control the potential smokers and temporary quit smokers to become smokers and the treatment of smokers. Analytical and numerical methods are utilized for ensuring presence of these two control actions.

This manuscript scrutinizes the (2+1)-dimensions chiral nonlinear Schrodinger’s equation with constant coefficients by utilizing two creative mix strategies. The creative mix strategies are namely the functional variable method and first integral method. Solution and singular soliton solutions are successfully recovered through integration techniques. The intermittent particular arrangements have been investigated for the side-effect of creative mix strategies. In order to have an assurance of these solutions, the various imperative relations jumped out to provide necessary conditions for integrability. Moreover, the dynamical attributes of the obtained results have been underlined and depicted in terms of 3D and 2D graphical illustrations.

Pine wilt disease is one of the most serious conifer diseases: this is because pine trees contribute greatly to the economy and domestic wealth in Korea. Our model of this disease is based on the parametrisation of infectious pine trees in Korea for the period of 2010-2019. The model captures the growth in case onsets and the estimated results are almost compatible with the reported data. To control the spread of this disease to the whole pine tree community, we found a threshold parameter called 'basic reproduction number' using the nextgeneration matrix method. During the analysis of the model, equilibrium points were first computed: there are two points -one has no disease class and other has all the disease classes. For the global behaviour of the mathematical model of these two points, Lypunove functional theory was used for disease-free and endemic equilibrium. Sensitivity analysis was performed to observe the relative importance of these parameters to the transmission and prevalence of pine wilt disease. To control the dissemination of the disease, we formulated an optimal control problem. Strategies used to control this disease were based on the consequences of the significant effects of the estimated parameters on the basic reproduction number. We re-examined the mathematical system to determine the agreement between numerically and analytically calculated outcomes. After analysing the problem numerically, we can discern that the numerical findings support the results calculated analytically.

Transmission dynamics of swine influenza pandemic is analysed through a deterministic model. Qualitative analysis of the model includes global asymptotic stability of disease-free and endemic equilibria under a certain condition based on the reproduction number. Sensitivity analysis to ponder the effect of model parameters on the reproduction number is performed and control strategies are designed. It is also verified that the obtained numerical results are in good agreement with the analytical ones.

The qualitative study of a smoking model with parametric conditions for diseases controlling under the influence of smoking is investigated through rigorous mathematical study. The mathematical modeling of epidemiological smoking model having six compartments is traced out. Mathematical expressions for smoke-free and smoke-present equilibrium points have been developed. The strength of Lyapunov functional theory has been exploited to show that smoke-free equilibrium point is globally asymptotically stable whenever basic reproduction number $$R_\circ

In this paper, the velocity field and the time dependent shear stress of Maxwell fluid with Caputo fractional derivatives in an infinite long circular cylinder of radius R is discussed. The motion in the fluid is produced by the circular cylinder. The fluid is initially at rest and at time t = 0 + , cylinder begins to oscillate with the velocity fsin ω t , due to time dependent shear stress acting on the cylinder tangentially. The hybrid technique used in this paper for the solution of the problem has less computational efforts and time cost as compared to other commonly used methods. The obtained solutions are in transformed domain, which are expressed in terms of modified Bessel functions I 0 ( · ) and I 1 ( · ) . The inverse Laplace transformation has been calculated numerically by using MATLAB package. The semi analytical solutions for Maxwell fluid with fractional derivatives are reduced to the similar solutions for Newtonian and ordinary Maxwell fluids as limiting cases. In the end, numerical simulations have been performed to analyze the behavior of fractional parameter α , kinematic viscosity ν , relaxation time λ , radius of the circular cylinder R and dynamic viscosity μ on our obtained solutions of velocity field and shear stress.

The velocity field and shear stress for unsteady flow of a second grade fluid with fractional derivatives through an infinite long circular cylinder are evaluated. The fluid is initially at rest and at t=0+, the fluid inside the cylinder starts to move longitudinally due to tangential shear stress. Semi analytical solutions are obtained with the help of Laplace transformation and modified Bessel equation. This hybrid technique, which we have used has less computational efforts and time cost as compared to other schemes that are commonly used. The solutions in the transformed domain, are presented in terms of modified Bessel functions I0(·) and I1(·) and satisfy all given conditions. Inverse Laplace transformations have been numerically calculated by using MATLAB. The semi analytical solutions for the motion of second grade fluid with fractional derivatives are reduced to the similar solutions for ordinary second grade and Newtonian fluid. Finally, the effect of different parameters of the flow is graphically examined. Keywords: Second grade fluid, Velocity field, Quadratic shear stress, Fractional derivatives, Laplace transformation, Modified Bessel function

In this work, a deterministic model is dedicatedly studied for the infection mechanism of pine wilt disease subject to varying sensitivity and optimality. We include time dependent controls into the pine wilt disease model and then analysed optimal conditions for the control of infection. Explicit form for the reproduction number has been obtained. Ultimate constant levels of infectious vectors and hosts have been discussed by employing the threshold condition. Two most effective techniques namely Lyapunov functional and graph theoretic have been used to find the final endemic level of population. The concept of full eradication of disease and reduction of constant level has been investigated through the utilization of two effective techniques. Using the concept of sensitivity analysis, control policies have been designed to control the disease. Additionally, the robustness of control plans has been shown graphically on the basis of data collected from open literature.

In the present research article, we investigated the slip flow of an unsteady incompressible Oldroyd-B fluid model. The electroosmosis and the pressure gradient have been seized for the stimulation of flow. The progress of the fluid is treated to be as passage composed by two micro-parallel plates. The potential difference alive between hard surface and the fluid is taken to be non symmetric. The governing equations are established for an Oldroyd-B fluid by using newly defined Caputo–Fabrizio fractional derivative. We used Laplace transform for converting the problem into spacial coordinates after proposing the geometry free variables. To find inverse Laplace transform, numerical Stehfest algorithm is used in place of promoting an analytical expression for it. In order to have the validity of our obtained results, the tabular comparison of two different algorithms (Stehfest and Tzou) and graphical simulation have been investigated. The conclusions are also illustrated in terms of graphs and carry the report of slip flow effect as well as some pertinent parameters on the velocity of the fluid.

This research work aims to provide Mathematical simulation of the flow of human blood through an arterial duct which has been narrowed by a cosine-shaped stenosis. This simulation provides a better understanding of this particular natural phenomena of blood flow. The flow has been taken similar to that of Newtonian flow and the Newtonian model is used. Regular perturbation method coupled with the Laplace transform is used for solving the model with appropriate initial and boundary conditions. Numerical technique-Stehfest’s algorithm is used for presenting the components of velocity, shear stress and temperature in graphs. The results conform to the observed theoretical descriptions.

In this paper, the generalized fifth-order (2[Formula: see text]+[Formula: see text]1)-dimensional KdV equation is scrutinized via the extended homoclinic test technique (EHTT) and extended transformed rational function (ETRF) method. With the aid of Hirota’s bilinear form, various exact solutions comprising, periodic solitary-wave, kinky-periodic solitary-wave, periodic soliton and complexiton solutions are constructed. Moreover, the mechanical features and dynamic characteristics of the obtained solutions are presented by three-dimensional plots.

In this note the velocity field and the adequate shear stress corresponding to the unsteady flow of a fractional Maxwell fluid due to a constantly accelerating circular cylinder have been determined by means of the Laplace and finite Hankel transforms. The obtained solutions satisfy all imposed initial and boundary conditions. They can easily be reduced to give similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.

In this note, the velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. Initially both cylinders and fluid are at rest and then the two cylinders suddenly start torsional oscillations around their common axis with simple harmonic motions having angular frequencies ω 1 and ω 2 . The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the Newtonian, second grade and generalized second grade fluids are shown graphically by plotting velocity profiles.

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.

The unsteady flow of a generalized second-grade fluid through an infinite straight circular cylinder is considered. The flow of the fluid is due to the longitudinal time dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the governing equation corresponding to a second-grade fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the finite Hankel and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is used. The corresponding solutions for ordinary second-grade and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the material constants and of the fractional parameter on the velocity and shear stress variations is underlined by graphical illustrations.

In this paper we determine the velocity field and the shear stress corresponding to the unsteady flow of a Maxwell fluid with fractional derivatives driven by an infinite circular cylinder that slides along its axes with a velocity Ata. The general solutions, obtained by means of integral transforms, satisfy all imposed initial and boundary conditions. They can be easily particularized to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the parameters a and on the fluid motion as well as a comparison between models is underlined by graphical illustrations.Keywords: Maxwell fluid with fractional derivatives, velocity field, shear stress, exact solutionsQuaestiones Mathematicae 37(2014), 139–156

In this note the velocity field and the adequate shear stress corresponding to the unsteady flow of a fractional Maxwell fluid due to a constantly accelerating circular cylinder have been determined by means of the Laplace and finite Hankel transforms. The obtained solutions satisfy all imposed initial and boundary conditions. They can easily be reduced to give similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.