The existence and uniqueness of weak solutions to the 2-dimensional reaction diffusion system with superdiffusion and the optimal control of such model are investigated in this paper. Fractional function spaces, Galerkin approximation method and Gronwall inequality are used to obtain the existence and uniqueness of weak solutions. On this basis, an optimal control problem of such superdiffusive system is further considered by using the minimal sequence.
Category: All Issues
Maoan Han and Junling Ma
Dear Readers, Authors and Friends,
The publishers, Zhejiang Normal University, China, and Pacific Edilite Academic Inc, Canada, and the editorial board, are very proud to announce the inauguration of the Journal of Nonlinear Modeling and Analysis.
Mathematical modeling is a thriving field of research, with applications to engineering, computer science, physics, chemistry, earth science and geography, biology, medicine, public health, economics, management sciences, and many others. In fact, the list to too long to enumerate here. The research on modeling relies on, and drives the research of, almost all fields of mathematics and statistics, such as nonlinear analysis, differential equations and dynamical systems, optimization, operation research, probability theory, graph theory, combinatory, topology, experimental design, data analysis, parameter estimation, and many others. As we know, models widely used in different applications may resemble striking similarities, and can be studied with similar theories and methods.
There have been numerous journals on mathematical modeling and analysis, ranging from very theoretical to very applied aspects. However, there is a strong need for an easy-to-access journal for modelers and mathematicians to present their new models, powerful methods, and innovative analysis, so that researchers from different fields in both pure and applied mathematics can inspire and learn from each other. This journal aims to provide such a platform.
The editorial board welcome quality contributions in original research, reviews, and communications. We invite the readers, authors, and reviewers to work closely with the editors, to help us fulfill our vision.
Maoan Han and Junling Ma, Editors in Chief
On behalf of the editorial board
In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained.
Rebiha Benterki and Jaume Llibre
We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.
A twisted heteroclinic cycle was proved to exist more than twenty-five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their traveling wave moving frame. The result implies the existence of infinitely many traveling front waves and infinitely many traveling back waves for the system. However efforts to numerically render the twisted cycle were not fruitful for the main reason that such orbits are structurally unstable. Presented here is a bisectional search method for the primary types of traveling wave solutions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo equations represent. The algorithm converges at a geometric rate and the wave speed can be approximated to significant precision in principle. The method is then applied for a recently obtained axon model with the conclusion that twisted heteroclinic cycle maybe more of a theoretical artifact.
Shanshan Chen, Zonghao Liu and Junping Shi
In this paper, we investigate a diffusive predator-prey model with fear effect. It is shown that, for the linear predator functional response case, the positive constant steady state is globally asymptotically stable if it exists. On the other hand, for the Holling type II predator functional response case, it is proved that there exist no nonconstant positive steady states for large conversion rate. Our results limit the parameters range where complex spatiotemporal pattern formation can occur.
Weiwei Gao and Binxiang Dai
Recent manipulations on vertebrates showed that the fear of predators, caused by prey after they perceived predation risk, could reduce the prey’s reproduction greatly. And it’s known that predator-prey systems with fear effect exhibit very rich dynamics. On the other hand, incorporating the time delay into predator-prey models could also induce instability and oscillations via Hopf bifurcation. In this paper, we are interested in studying the combined effects of the fear effect and time delay on the dynamics of the classic Lotka-Volterra predator-prey model. It’s shown that the time delay can cause the stable equilibrium to become unstable, while the fear effect has a stabilizing effect on the equilibrium. In particular, the model loses stability when the delay varies and then regains its stability when the fear effect is stronger. At last, by using the normal form theory and center manifold argument, we derive explicit formulas which determine the stability and direction of periodic solutions bifurcating from Hopf bifurcation. Numerical simulations are carried to explain the mathematical conclusions.
Zhongkai Guo, Haifeng Huo, Qiuyan Ren and Hong Xiang
A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.
Jianbo Zhu and Xianlong Fu
In this paper, we first establish the separable $Hamiltonian$ system of rectangular cantilever thin plate bending problems by choosing proper dual vectors. Then using the characteristics of off-diagonal infinite-dimensional $Hamiltonian$ operator matrix, we derive the biorthogonal relationships of the eigenfunction systems and based on it we further obtain the complete biorthogonal expansion theorem. Finally, applying this theorem we obtain the general solutions of rectangular cantilever thin plate bending problems with two opposite edges slidingly supported.
Pei Yu, Maoan Han and Yuzhen Bai
In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.