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Abstract We introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus.

We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in $ \mathbb{R}_+^2 $. It is shown that for all $ a $, $ b $, $ c $ and $ p $, model has boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ if $ p > \frac{b c}{a} $. Further, local dynamical characteristics with topological classifications for the fixed points $ P_{0y}(0, \frac{p}{c}) $ and $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $. Moreover, it is also explored that for the fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $, laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period $ -2 $, $ -3 $, $ -4 $, $ -5 $, $ -8 $ and $ -9 $. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the L 2 $L_{2}$ error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

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.

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(−Θ(q))-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.