Qualitative Theory of Dynamical Systems最新文献

The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended ((frac{{G'}}{{{G^2}}}))-expansion method, two reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions used the definition of the truncated M-derivative (TMD) to understand its dynamical behaviour. The hyperbolic, periodic, and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, and singular solitary wave solitons are obtained. We observe the fractional parameter impact of the above derivative on the physical phenomena. Each set of travelling wave solutions have a symmetrical mathematical form. Last but not least, we employ Mathematica to produce 2D and 3D figures of the analytical soliton solutions to emphasize the influence of TMD on the behaviour and symmetry of the solutions for the proposed problem. The physical importance of the solutions found for particular values of the combination of parameters during the representation of graphs as well as understanding of physical incidents.

Employing some classical analysis methods, in this paper we establish the global boundedness of R-component of traveling wave solutions for a discrete diffusion susceptible-infected-recovered (SIR) epidemic model with delay. This result is a sufficient condition to obtain the limit behavior of traveling wave solutions at far fields. Meanwhile, the present results improve our recent work.

The objective of this article is to investigate the issue of existence results for Hilfer fractional stochastic differential inclusions of order (1<mu <2) in Hilbert spaces. Our discussion is based on fractional calculus, multivalued analysis, sine and cosine operators, and Bohnenblust–Karlin’s fixed point theorem. At first, we investigate the existence of a mild solution for the Hilfer fractional stochastic differential system of order (1<mu <2). After that, we developed our system with Sobolev-type, and we provided the existence results of a mild solution for the considered system. Then, the ideas of nonlocal conditions are applied in the Sobolev-type Hilfer fractional stochastic system. Finally, an example is offered in order to illustrate the effectiveness of the main theory.

We investigate the existence of solutions for coupled systems of fractional p-Laplacian quasilinear boundary value problems at resonance given by the Atangana–Baleanu–Caputo (shortly, ABC) derivatives formulations are based on the well-known Mittag-Leffler kernel utilizing Ge’s application of Mawhin’s continuation theorem. Examples are provided to demonstrate our findings.

In this paper, we prove a long time existence result for fractional Rayleigh-Stokes equations derived from a non-Newtonain fluid for a generalized second grade fluid with memory. More precisely, we discuss the existence, uniqueness, continuous dependence on initial value and asymptotic behavior of global solutions in Besov-Morrey spaces. The proof is based on real interpolation, resolvent operators and fixed point arguments. Our results are formulated that allows for a larger class in initial value than the previous works and the approach is also suitable for fractional diffusion cases.

We are concerned with the parametrized family of problems

where (Omega ) is a bounded domain of ({mathbb {R}}^N~(Nge 3)) with regular boundary (partial Omega ,~{mathcal {L}}) is a general second-order uniformly elliptic operator, (lambda ,~l>0), (a:{overline{Omega }}rightarrow {mathbb {R}}) is a continuous function which may change sign, (f:{mathbb {R}}^+rightarrow {mathbb {R}}) is subcritical and superlinear at infinity. Under some suitable conditions, we obtain there exists (lambda _0 > 0) such that (P) has positive solutions for all (0 < lambda le lambda _0 ) by topological degree argument and a priori estimates. In doing so, we require f to be of regular variation at infinity.

In this paper, we investigate the approximate controllability of mild solutions for second-order differential systems. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential systems under the assumption that the associated linear part of the system is approximately controllable. In addition, we extend our system with nonlocal conditions. Our research on approximate controllability was also extended by utilizing impulse systems. To demonstrate the theory of the primary outcomes, an application is shown.

In this paper, the solitary wave solutions, the periodic type, and single soliton solutions are acquired. Here, the Hirota bilinear operator is employed to investigate single soliton, periodic wave solutions and the asymptotic case of periodic wave solutions. By utilizing symbolic computation and the applied method, generalized (3+1)-dimensional shallow water wave (GSWW) equation is investigated. The variational principle scheme to case periodic forms is studied. The (3+1)-GSWW model exhibits travelling waves, as shown by the research in the current paper. Through three-dimensional design, contour design, density design, and two-dimensional design using Maple, the physical features of single soliton and periodic wave solutions are explained all right. The findings demonstrate the investigated model’s broad variety of explicit solutions. As a result, exact solitary wave solutions to the studied issues, including solitary, single soliton, and periodic wave solution, are found. The phase plane is quickly examined after establishing the Hamiltonian function. The effects of wave velocity and other free factors on the wave profile are also investigated. It is shown that the approach is practical and flexible in mathematical physics. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering.

We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity

where (sin (0,1)), (alpha >0) and (Omega subset mathbb R^N) is a bounded domain with smooth boundary (partial Omega ) and (N>2s.) Under some appropriate assumptions of (alpha , p, mu ) and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of ( C^{1,1}_{loc}cap L^{infty }) solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of ( C^{1,1}cap L^{infty }) solutions are obtained for star-shaped domain under a condition of K.