Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming $exp(-\mathsf{\Theta }(q\left)\right)$-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics.

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains.

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved [ ]-expansion approach and the [ ]-expansion method. We use the definition of the Jumarie’s modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics.

This paper points out several exact travelling wave solutions for $(1+1)$-dimensional Ito integro-differential equation via the Riccati–Bernoulli sub-ODE approach. We also aim to develop a numerical solution of the respective equation using central finite difference formulas. The stability of the presented exact and numerical solutions is also deduced using the Hamiltonian system and Von Neumann's concept, respectively. Moreover, the numerical schemes are studied in terms of their accuracy. The relative error arising from executing the numerical method is exhibited. We compare our results with others published in some articles. The accomplished numerical solutions are successfully compared with the analytical ones. The used processes can be extended to solve more integrable problems as well as non-integrable ones.

The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations.