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We analyze the nonlinear (2+1)-dimensional KdV–mKdV equation with Caputo fractal–fractional operator. Some theoretical features are demonstrated via fixed point results. The solution of the considered KdV–mKdV is studied by the composition of the J-transformation and decomposition method. For the validity and effectiveness of the considered method, two examples with suitable initial conditions are solved, where best agreements observed. The validity of the suggested approach is verified by convergence analysis and Picard stability. From the simulations of the obtained results, it is noted that fractional order and fractal dimension significantly affects the amplitude and shape of wave solutions.

The higher dimensional Fokas equation is the integrable expansion of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The Fokas model has an important role in wave theory, to describe the physical phenomena of waves on the surface and inside the water. This article deals with the (4+1)-dimensional conformable space-time fractional-order Fokas partial differential equation. Two efficient methods, namely the generalized exp(-ϕ(ξ))-expansion and improved F-expansion methods, are formulated for conformable fractional-order partial differential equation and new wave structures of fractional order Fokas model are constructed. The different kinds of new solitons are achieved such as bright soliton, dark soliton, Kink and anti-kink solitons, periodic solitary waves, and traveling waves. These new soliton waves are constructed at some values of fractional order α and using different parametric values of the methods by using the software package Mathematica. Newly obtained soliton solutions are compared with the available soliton solutions with different fractional derivatives in the literature. Some of the achieved results are explained 2D and 3D graphically. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. Furthermore, this paper gives an idea, how can reduce the conformable fractional order higher dimensional partial differential equation into an ODE of one variable to obtain the exact solutions. These results and methods can be help to investigate the other higher-dimensional conformable fractional-order models which appear in nonlinear wave theory such as optics, quantum gases, hydrodynamics, photonics, plasmas, and solid-state physics.

The physics of the (4 + 1)-dimensional Fokas equation follows necessarily from the physical nature of the Kadomtsev–Petviashvili and Davey-Stewartson equations in wave theory. In this article, we consider the nonlinear (4+1)-dimensional Fokas partial differential equation. Two recognizable methods namely improved F-expansion and generalized exp(-ϕ(ξ))-expansion methods are proposed to investigate the new wave structures of (4 + 1)-dimensional Fokas dynamical model which have never been constructed before. As a result, new solutions are in different solitons such as dark soliton, bright soliton, combined dark-bright soliton solutions, periodic, and solitary wave solutions. The new generalized solitary wave solutions can be constructed by assigning the specific values to those parameters involved in these methods. The achieved results are compared with existing results in the literature and we found that some of results are not available in literature. The derived results are explained graphically, to understand the phenomena of the proposed model. The constructed results rendering that the consider methods in this article are simple, effective, and easy to find the solution of many other nonlinear higher-dimensional models which arise in several areas of science and engineering.

In this paper, we propose a new SAIR model to depict the transmission dynamics of a novel coronavirus in China. We focus on the ability of asymptomatic COVID-19 patients to transmit and the potential impact of population movements on renewed outbreak transmission. Qualitative analysis of the model shows that when the basic productive number R0>1, the system will stabilize towards a unique endemic equilibrium and pass through a transcritical bifurcation around its disease-free equilibrium. Furthermore, by constructing an appropriate Lyapunov function, we prove that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable under appropriate parameter conditions. Finally, some important results have been verified by numerical simulations.

It is renowned fact that the study of nonlinear partial differential equations (NPDEs) play a vital rule in cosmic phenomena, atomic and matter physics, that is why researchers have a keen interest to study and find their solutions. Sharma–Tasso–Olever equation is such type of an important equation. In this article, we consider the fractional-order Sharma–Tasso–Olever equation with non-singular Atangana–Baleanu derivative and utilize three remarkable analytical methods, namely, the homotopy perturbation method (HPM), the homotopy perturbation method with Laplace transform (HPLM), and the Adomian decomposition method (ADM), for obtaining the rational solutions. To get a better understanding of this new derivative operator with these methods, the achieved results are compared with the existing results in the literature of Caputo derivative. The numerical results reveal the efficiency, reliability, significant features, and simple in computation with high accuracy by using Atangana–Baleanu derivative with these three approaches. These three methods can be used to find the analytical solutions of many other fractional order problems with Atangana–Baleanu derivative in science and engineering.