Volume 20, Issue 1 -

In this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the fixed point theorems and fractional cosine family combined with solutions operator theorems. Our results improve and generalize some classical results. Finally, an example is given to illustrate the main results.

The study of Rossby solitary waves are of great significance in physical oceanography, atmospheric physics, water conservancy project, military and communications engineering, etc. All the time, in the study of Rossby solitary waves, people have been focusing on integer order models. Recently, fractional calculus has become a new research hotspot, and it has opened a new door to research atmospheric and ocean. Thus, the fractional order model has the potential value in the study of Rossby solitary waves. In the present paper, according to the quasi-geostrophic potential vorticity equation with the complete Coriolis force, we get a new integer order mZK equation. Using the semi-inverse method and the fractional variational principle, the space-time fractional mZK(STFmZK) equation is obtained. To better understand the property of Rossby solitary waves, we study Lie symmetry analysis, nonlinear self-adjointness, similarity reduction by applying the STFmZK equation. In the end, the conservation and Caputo fractional derivative are discussed, respectively.

In this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

We are concerned with the following generalized quasilinear Schrödinger equations −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(u)+μk(x),x∈RN,$$\begin{align*}&-\text{div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+V(x)u=h(u)+\mu k(x), \\&\qquad\quad x\in \mathbb{R}^N,\end{align*}$$ where N≥3, g:R→R+$N\ge 3, \ g:\mathbb{R}\rightarrow\mathbb{R}^+$ is an even differentiable function satisfying limt→+∞g(t)tα−1=β/gt0$\displaystyle \lim_{t \rightarrow +\infty} \frac {g(t)}{t^{\alpha -1}} = \beta /gt0$ for some α≥1$\alpha \ge 1$, h is a nonlinear function covering the case h(t)=|t|p−2t (2\ltp<α2∗)$h(t)=|t|^{p-2}t\ (2\ltp\lt\alpha2^*)$, the potential V:RN→R$V:\mathbb{R}^N\rightarrow\mathbb{R}$ is positive and µ k ( x ) is a perturbation term with µ  > 0. Combining the change of variables and variational arguments, we show that the given problem has at least two positive solutions for some μ0>0$\mu_0\gt0$ and μ∈(0,μ0)$\mu\in(0,\mu_0)$.

This work describes design and implementation of a navigation and obstacle avoidance controller using fuzzy logic for four-wheel mobile robot. The main contribution of this paper can be summarized in the fact that single fuzzy logic controller can be used for navigation as well as obstacle avoidance (static, dynamic and both) for dynamic model of four-wheel mobile robot. The bond graph is used to develop the dynamic model of mobile robot and then it is converted into SIMULINK block by using ‘S-function’ directly from SYMBOLS Shakti bond graph software library. The four-wheel mobile robot used in this work is equipped with DC motors, three ultrasonic sensors to measure the distance from the obstacles and optical encoders to provide the current position and speed. The three input membership functions (distance from target, angle and distance from obstacles) and two output membership functions (left wheel voltage and right wheel voltage) are considered in fuzzy logic controller. One hundred and sixty-two sets of rules are considered for motion control of the mobile robot. The different case studies are considered and are simulated using MATLAB-SIMULINK software platform to evaluate the performance of the controller. Simulation results show the performances of the navigation and obstacle avoidance fuzzy controller in terms of minimum travelled path for various cases.

The problem of maximization of the horizontal coordinate of a mass-point moving in the vertical plane driven by gravity, non-linear viscous drag, and thrust is considered. The slope angle and the thrust are considered as a control variables. The problem is related to the modified brachistochrone problem. Principle maximum procedure allows to reduce the optimal control problem to the boundary value problem for a set of systems of two non-linear differential equations. The qualitative analysis of the trajectories of these systems is performed, and the characteristic features of the optimal solutions are determined. Thrust control depending on the velocity and slope angle is designed. Results obtained allow to construct quasi-optimal solutions for the more complex systems, where phase plane method is not applicable.

Here, we introduce a numerical solution by using the generalized Euler method for the (Caputo sense) fractional Susceptible-Infected-Recovered (SIR) model with a constant vaccination rate. We compare the obtained numerical solutions with those solutions by using the RK4. Hence, the obtained numerical results of the SIR model show the simplicity and the efficiency of the proposed method.

In this paper, we study the dynamics of the fractional-order chaotic system corresponding to the original Chua system with the same nonlinearity. We place bounds on the fractional order to guarantee a chaotic behavior. In addition, we propose a one-dimensional adaptive synchronization strategy, whereby we assume knowledge of one of the states and reconstruct the rest. The proposed synchronization scheme is put to the test in a secure communication scenario based on the antipodal chaos shift keying modulation scheme. Throughout the analysis and examples, numerical results are presented to affirm the validity of the findings.

We propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.