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A new tuberculosis model with health education and early therapy is introduced. The early therapy is available for both detected latent and infective individuals. The basic reproduction number R0$R_0$ is derived by the next generation matrix. Mathematical analyses show that the disease free equilibrium is globally asymptotically stable if R0<1$R_0<1$, and the endemic equilibrium is globally asymptotically stable if R0>1$R_0>1$. Numerical simulations are also carried out to illustrate our analytical results. Our results show that both health education and early therapy have the positive impact in reducing burden of tuberculosis.

Inputs can change timings of spikes in neurons. But it is still not clear how input’s parameters for example injecting time of inputs affect timings of neurons. HR neurons receiving both weak and strong inputs are considered. How pulse inputs affecting neurons is studied by using the phase-resetting curve technique. For a single neuron, weak pulse inputs may advance or delay the next spike, while strong pulse inputs may induce subthreshold oscillations depending on parameters such as injecting timings of inputs. The behavior of synchronization in a network with or without coupling delays can be predicted by analysis in a single neuron. Our results can be used to predict the effects of inputs on other spiking neurons.

We investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment u1(t)$u_{1}(t)$ plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically.

The kinetic flux vector splitting method has been introduced for two-dimensional system of shallow water equations with horizontal temperature gradients. The scheme preserves positivity conditions and resolves different regions of shock waves, rarefaction waves and contact discontinuity with negligible oscillations. The scheme is based on splitting of flux functions of the Ripa model. Moreover Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. The numerical example is taken from already published article. The obtained results reveal the accuracy and robustness of the proposed method.

The problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguishes these systems from the systems without gyroscopic forces, where the trajectories always cover the possible motion area. We also present some numerical and analytical results on the same matter for the Kovalevskaya case.

This article concerns with the construction of the analytical traveling wave solutions for the model of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave and the higher-order nonlinear Schrödinger equation by Riccati–Bernoulli sub-ODE method. We give the exact solutions for these two equations. The proposed method is effective tool to solve many other nonlinear partial differential equations. Moreover, this method can give a new infinite sequence of solutions. These solutions are expressed by hyperbolic, trigonometric and rational functions. Finally, with the aid of Matlab release 15, some graphical simulations were designed to see the behavior of these solutions.

Practical stability of a fractional order discrete-time linear state-space systems was put up in recent years. It is usually checked by the eigenvalues of the state matrix, some methods have been given during these years. But if the size of the state matrix is large, the computations of eigenvalues can be very onerous. In this paper, some new conditions on practical stability for positive fractional discrete-time linear system are presented. Numerically checking method of practical stability is presented based on the new conditions given in this paper. It is illustrated by the numerical examples that our checking method is effective and true. Compared to the now existing methods, numerically checking method is an attractive method because it’s easily implemented.

In recent years, there has been an incredible enthusiasm towards fractional order partial differential equations because of their incessant presence alongside different fields. Fractional derivatives offer an in-depth and precise analysis of the models of the systems. Particularly, fractional order telegraph equations (FOTE) have been taken into consideration and solved by plenty of researchers, using different techniques. In this paper, we present a novel approach and technique to solve fractional telegraph equation by fusion of cubic radial basis function and Chebyshev polynomials with the aid of Kronecker product. The numerical examples have been considered to verify the accuracy and also to demonstrate the performance of the new approach.

In this paper, the dynamical behavior of a discrete SIR epidemic model with fractional-order with non-monotonic incidence rate is discussed. The sufficient conditions of the locally asymptotic stability and bifurcation analysis of the equilibrium points are also discussed. The numerical simulations come to illustrate the dynamical behaviors of the model such as flip bifurcation, Hopf bifurcation and chaos phenomenon. The results of numerical simulation verify our theoretical results.

Referring to incommensurate fractional-order systems, this paper proposes a new type of chaos synchronization by combining full state hybrid function projective synchronization (FSHFPS) and inverse full state hybrid function projective synchronization (IFSHFPS). In particular, based on stability theory of linear integer-order systems and stability theory of linear fractional-order systems, the co-existence of FSHFPS and IFSHFPS between incommensurate fractional chaotic (hyperchaotic) systems is proved. To illustrate the capabilities of the novel approach proposed herein, numerical and simulation results are given.

In this paper, by using variational methods and a critical point theorem due to Bonanno and Marano, the existence of at least three weak solutions is obtained for a class of p -Laplacian type nonlinear fractional coupled systems depending on two parameters. Two examples are given to illustrate the applications of our main results.

Different kinematic mathematical models of nonlinear dynamics of a contact interaction of two microbeams are derived and studied. Dynamics of one of the microbeams is governed by kinematic hypotheses of the first, second, and third approximation orders. The second beam is excited through a contact interaction with the first beam and is described by the kinematic hypothesis of the second-order approximation in both geometric linear and nonlinear frameworks. The derived nonlinear partial differential equations (PDEs) are transformed to the counterpart system of nonlinear ordinary differential equations (ODEs) by the finite difference method. Nonlinear contact interaction dynamics of the microbeam structure is analyzed with the help of time series (signals), Fourier spectra, and wavelet spectra based on various mother wavelets, Morlet wavelet spectra employed to study synchronization phenomena, Poincaré maps, phase portraits, and the Lyapunov exponents estimated with the Wolf, Kantz, and Rosenstein algorithms. We have illustrated that neglecting the shear function (Euler–Bernoulli model) yields erroneous numerical results. We have shown that the geometric nonlinearity cannot be neglected in the analysis even for small two-layer microbeam deflection. In addition, we have detected that the contact between two microbeams takes place in the vicinity of x≈0.2x \approx 0.2 and x≈0.8x \approx 0.8 instead of the beams central points.

Optimization of parameters of the corrugated shell aims to achieve its minimum weight while keeping maximum stiffness ability. How an introduction of functionally graded corrugations resulted in improved efficiency of this thin-walled structure is demonstrated. The corrugations are graded varying their pitch. The effect of variation in pitch is studied. Homogenization approach gives explicit expressions to calculate the equivalent shell properties. Then well-elaborate methods of optimal design theory are used. The illustrative examples for hydrostatic load demonstrate a high efficiency of the used method.

Analysis and control of electrical and mechanical behavior of electrical machines devoted to capture the wing energy is a subject of huge interest nowadays, due to the need to respond to the actual demands related to green energy, more specific to the demand to convert the mechanical wind energy into electrical energy. The working regime of a permanent magnet synchronous generator associated with a wind turbine under a variable wind profile is investigated in this paper starting from the reality that the wind speed and especially its variation generate significant electrical and mechanical loads in the system, which could lead to various damages. A new and effective analytical technique, namely the Optimal Auxiliary Functions Method is employed to obtain approximate analytical solutions to the governing equations which are expressed in non-dimensional form. It is found that the analytical results obtained by means of our new method agree very well with numerical integration results, if an appropriate number of convergence-control parameters are employed in the construction of the auxiliary functions. A detailed analysis of the convergence of approximate analytical solutions is developed and it is concluded that the convergence of solutions greatly depends on the number of convergence-control parameters, whose optimal values are rigorously determined using various reliable procedures. Several numerical examples are developed in order to illustrate the capabilities of the proposed approach, which proves to be highly efficient and controls the convergence of the approximate solutions, ensuring a very fast convergence after only one iteration.

This paper is a Direct Numerical Simulation (DNS) of the temperature distribution in the flow with the forward and backward facing steps with buoyancy forces for different lengths of the bottom walls and flow regimes. The different lengths of the bottom walls effect were studied with buoyancy forces. A two-dimensional incompressible Navier–Stokes equation and equations for temperature transfer were used to describe this process. The obtained numerical solutions for the test problem, laminar flow with the backward facing step with buoyancy force, are compared with the numerical results of other authors. A verified numerical algorithm is applied to the problem with the forward and backward facing steps with buoyancy forces. The influence of changes in the bottom wall length on temperature distributions and velocity components is revealed. It was also revealed that the heat transfer characteristics of the bottom wall have strong changes in different lengths. In cases for the lengths of the bottom walls (Xe=12,16${X_e} = 12, \,16$) at Re =1,000, the rotational movement of the vortices along the channel wall revealed an increase in the exchange of cold and hot fluids, which leads to temperature fluctuations from the lower wall to the upper wall of the channel.

This paper presents sufficient conditions for the existence of at least one homoclinic solution for a nonlinear second-order difference equation with p -Laplacian. Our technical approach is based on variational methods. An example is offered to demonstrate the applicability of our main results.

The generalized nonlocal nonlinear Hirota (GNNH) equation has been widely concerned, it can be regarded as the generalization of the nonlocal Schrödinger equation, and can be reduced to a nonlocal Hirota equation. In this paper, we mainly study a GNNH equation and its determinant representation of the N -fold Darboux transformation. Then we derive some novel exact solutions including the breather wave solitons, bright solitons, some characteristics of solitary wave and interactions are considered. In particularly, the dynamic features of one-soliton, two-soliton solutions and the elastic interactions between the two solitons are displayed. We find that unlike the local case, the q ( x , t ) and q∗(−x,t)$q^{*}(-x,t)$ of the GNNH equation have some novel characteristics of solitary wave, which are different form the classical Hirota equation.

A new fractional Navier–Stokes equation is constructed based on the notion of fractional velocity recently introduced in the literature. Its implications in fluid mechanics were discussed. In particular, the Couette and the Poiseuille flows and some insights of fluid flow in microfilaments were addressed accordingly.

In this paper, we consider limit behaviors of Riemann solutions to the isentropic Euler equations for a non-ideal gas (i.e. van der Waals gas) as the pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for van der Waals gas is solved. Then it is proved that, as the pressure vanishes, any Riemann solution containing two shock waves to the isentropic Euler equation for van der Waals gas converges to the delta shock solution to the transport equations and any Riemann solution containing two rarefaction waves tends to the vacuum state solution to the transport equations. Finally, some numerical simulations completely coinciding with the theoretical analysis are demonstrated.

The main objective of this paper is to present sufficient conditions for controllability of Hilfer fractional nonlinear stochastic systems in finite dimensional space. The main results are obtained by using the Nussbaum fixed point theorem, stochastic analysis approach and generalized fractional calculus (Hilfer fractional derivative) which is universality of Riemann-Liouville and Caputo fractional derivative. Finally, a numerical example is provided to show the effectiveness of the obtained theoretical result. The obtained result is more generalized one than the existing results on fractional stochastic system in finite dimensional space.

In centrifugal pumps, it is important to select appropriate turbulence model for the numerical simulation in order to obtain reliable and accurate results. In this work, ten turbulence models in 3-D transient simulation for the centrifugal pump are chosen and compared. The pump performance is validated with experimental results. The numerical results reveal that the SST turbulence model was closer to the experimental results in predicting head. In addition, the pressure variation trend for the ten models is very similar which increases and then decreases from the inlet to outlet of the pump along the streamline. The SST k- ω model predicts the performance of the pump was more accurately than other turbulent models. Furthermore, the results also found that the error is the least at design operation condition 300(l/min), which is around 1.98 % for the SST model and 2.14 % and 2.38 % for the LES and transition omega model. Within 7.61 %, the errors at higher flow rate 350(l/min) for SST. The error for SST model is smaller as compared to different turbulent models. For the Realizable k- ɛ model, the errors fluctuate were more high than other models.