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In this work, we study a diffusive predator–prey model with mutual interference among the predators while searching for food. We prove that the model exhibits bistability, which indicates that there is no patterns for our model. When proper cross fractional diffusion terms are introduced in the model, the Turing pattern emerges when cross fractional diffusion coefficients fall into some domain. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear dispersive structure namely the modified Benjamin–Bona–Mahony (BBM) model is studied by employing three different approaches to develop some new analytical and semi-analytical solutions. The modified extended t a n hyperbolic function method, the improved F -expansion method and the Adomian decomposition method are implemented to obtain various interesting wave structures. The comparison among various types of solutions is analyzed to study the corresponding absolute error. Furthermore, the stability of the results along with the modulation instability are successfully deliberated. The 3D, contour and 2D graphs are visualized for several fascinating exact solutions to comprehend their behavior. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the soliton solutions of the time–space fractional order nonclassical Sobolev-type (TSFNST) equation. Sobolev equations are used in thermodynamics, the flow of fluid through fractured rock, and other fields. There are certain significant partial differential equations having a third-order mixed derivative with regard to time and space that are now being used in real-world applications. The study aims to analyze the TSFNST equation analytically by employing the ϕ 6 -model expansion and modified G ′ / G 2 expansion techniques respectively. The solution is obtained successfully in the Jacobi elliptic function solutions that will prove to us the hyperbolic, trigonometric, and rational solutions. Additionally, we select some unique physical problems of the solution. Lastly, the 3-dimensional and their corresponding contour plots are drawn by choosing the different values of constants [ABSTRACT FROM AUTHOR]

This article examines the coupled KdV equations and the coupled system of variant Boussinesq equations with beta time derivative and explores their travelling wave solutions. This work explains the evolution of waves with fractional parameter. The simple ansatz approach produces a variety of novel solutions in terms of hyperbolic and periodic functions. Graphical representation of solutions are also presented. [ABSTRACT FROM AUTHOR]

This study uses the Hirota bilinear method and Maple, a symbolic computation program, to derive lump solutions for a new integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations. Furthermore, lump solutions with free parameters have been constructed using the dimensionally reduced new form of the (3 + 1)-dimensional Boussinesq equation. The derived lump solutions show it has two trough positions and one crest position. The amplitudes and shapes of lump waves don't vary during propagation but they change their positions. By making three-dimensional, two-dimensional, and density plots for specific values of the relevant free parameters, the propagations of the obtained lump wave solutions are displayed. They also demonstrate how the trough and crest positions of a lump wave change over time with constant velocity. The phase shifts, propagation directions, and energy distributions can be seen from the graphical outputs that the free parameters of the model play a significant role in changing the shapes and amplitudes of the waves. The resulting solutions and their physical characteristics may help to understand how the waves propagate in shallow water in oceanography. [ABSTRACT FROM AUTHOR]

The modified auxiliary equation (MAE) approach and the generalized projective Riccati equation (GPRE) method are used for the first time to solve Zoomeron problem which provided different types of exact traveling wave solutions, including some new dynamical behaviors. The governing model covers unique examples of solitons with distinct properties that emerge in a variety of physical situations, including laser physics, fluid dynamics and nonlinear optics. The achieved exact traveling wave solutions include solitary wave, periodic wave, bright, dark peakon, and kink-type wave solutions. Earned results are given as hyperbolic and trigonometric functions. Moreover, the dynamical features of obtained results are demonstrated through interesting plots. • The modified auxiliary equation and the generalized projective Riccati equation methods are used to solve Zoomeron problem. • The solitary wave, periodic wave, bright, dark peakon, and kink-type wave solutions are given for the model problem. • The dynamical features of obtained results are demonstrated through interesting plots. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Drinfel'd–Sokolov–Wilson equation (DSWE) and Boiti Leon Pempinelli equation (BLPE) in the conformable sense. The sine–cosine method is utilized to achieve various traveling wave solutions to the suggested nonlinear systems. It is an easy approach to use and does not require sophisticated mathematical software or a knowledgeable coder. It can also be used for various linear and nonlinear fractional issues, making it pervasive. The obtained solutions in the form of solitons emerge with the necessary constraints to ensure their existence. The obtained results hold significant role in elucidating some important nonlinear problems in applied sciences and engineering. • The DSWE and BLPE in the conformable sense are considered. • The sine–cosine method is utilized to achieve various solutions to the suggested models. • The obtained soliton solutions emerge with the necessary constraints to ensure their existence. • Physical explanations are discussed for the obtained solutions. [ABSTRACT FROM AUTHOR]

Every cell contains deoxyribonucleic acid (DNA), which serves as the fundamental information storage medium, preserving the genetic information required by organisms in order to survive and reproduce themselves. The analytical solution of the (3+1) double chain Deoxyribonucleic acid dynamical model is achieved using the Lie transformation approach. The equation is transformed into an ordinary differential equation (ODE) using three sequential symmetry reductions. Two examples are investigated, the results of which include some solitary wave solutions. The solutions found may be classified into three types: one-soliton, multi-soliton, and periodic wave solutions. Some of the answers that were discovered have been visually explored. The nonlinear dynamics of the system manifests itself in the solitary and anti-solitary waves of the DNA strands, which are both seen. The collected results may be beneficial to evaluate different applications for further scientific conclusions. [ABSTRACT FROM AUTHOR]

In this study, we propose a cancer-obesity-treatment model under the control parameters: IL-2 therapy, ACI therapy, and nutritional diet. We analyze the proposed model and examine the existence and stability of the equilibrium points for the cases: with and without treatment cases. Further, to reduce the number of cancer cells and minimize the toxicity effect of the drug dose on other healthy cells, we consider an optimal control problem over a finite time interval under the treatment control parameters. To understand the treatment effect, we present simulation results for our proposed model considering different treatment strategies: no treatment; only IL-2 therapy; a combination of IL-2 therapy and ACI therapy; and a combination of IL-2 therapy, ACI therapy, and nutritional diet. Our results demonstrate that we could achieve an optimal treatment schedule for cancer management by controlling all three treatment parameters. • A cancer obesity-treatment model under some control parameters: IL-2 therapy, ACI therapy, and nutritional diet is proposed. • Discussed the proposed model without treatment. • Further, to reduce the number of cancer cells and minimize the toxicity effect of the drug dose on other healthy cells, we consider an optimal control problem over a finite time interval under the treatment control parameters. • Finally, to understand the treatment effect, we present simulation results for our proposed model. • Our results demonstrate that we could achieve an optimal treatment schedule for cancer management by controlling all three treatment parameters. [ABSTRACT FROM AUTHOR]