The paper explores analytical solutions to the coupled Konno-Oono and Kadomtsev-Petviashvili (modified) extended wave (KP-MEW) equations in (2 +1)-dimensions, which have applications in the study of nonlinear wave propagation in shallow water, plasma, and optical systems. The method (Riccati Modified Extended Simple Equation Method) was applied to provide us with a wide range of exact traveling wave solutions as hyperbolic, trigonometric, exponential, and rational. The solutions obtained are highly nonlinear, including soliton pulses, periodic waves, kink-like structures and blow-up modes, which give an understanding of the interaction of waves and their stability. It is important to note that the rational-type solutions are a new category that has never been mentioned in other studies. The authors generalize the previous studies by providing a coherent structure in building different families of solutions, which reveal the usefulness and flexibility of the suggested approach in investigating the nonlinear wave dynamics in higher dimensions.

Abstract This work studies the tempered fractional (2+1)-dimensional Konopelchenko-Dubrovsky system. The tempered fractional derivative considered in the context of Caputo tempered derivative. We develop a new method based on the Laplace transform, with an improvement on the residual power series method, to handle the governing system and reveal its dynamics. We present a theoretical and experimental analysis to investigate the convergence of the developed method. Different initial conditions and values for the system parameters were used, and the resulting solutions were presented in 3D and 2D panels. A comparison of the discovered approximate solutions with the exact solutions is presented to investigate the efficiency and accuracy of the proposed approach. In addition, the effect of the tempered coefficient and fractional derivative on the behavior of the derived solutions is investigated. To connect the mathematical results with physical insights, a physical example is presented for modeling intrinsic perturbations in a highly coupled magneto plasma (tokamak plasma) to describe the propagation of nonlinear waves in this plasma. Using the discovered solutions for the governing system, the control of perturbations within the plasma and external factors is investigated using the tempered coefficient and fractional derivative. Our results demonstrate the efficiency and reliability of the proposed method for handling the governing system and confirm the possibility of describing complex physical phenomena in several domains using the tempered fractional (2+1)-dimensional Konoplychenko-Dubrovsky system.

We develop a finite-N matched-asymptotic theory for Smoluchowski coagulation. Starting from the infinite system of ODEs, we use conservation laws to obtain closed scalar evolution equations for suitable moments. For the constant kernel $a_{j,k}=K$, this leads to an exact early-time decay law for the cluster number and the linear coalescence-time scaling $T_{N}\asymp N$ when the volume scales as $V\sim N$. For the sum kernel $a_{j,k}=j+k$, the same reduction yields an exponential decay regime for the number of clusters and a logarithmic finite-N coalescence time. For the multiplicative kernel $a_{j,k}=jk$, we recover the classical finite-time blowup of the second moment and show that finite N produces a sharp gelation cutoff preceding the mean-field gelation time by a window of order $N^{-2}$. In all kernels considered, the late-time dynamics involve only finitely many clusters and contribute only lower-order corrections. The resulting structure closely parallels that of finite-N two-species annihilation: a conserved quantity reduces the dynamics to a scalar ODE, asymptotic matching at a characteristic time yields the finite-size scaling laws, and post-matching effects do not alter the leading behaviour.

Abstract The governing equations of the thermoelastic Timoshenko beam with microtemperatures are derived based on type III entropy balance. Our development of this theory is prompted by the increasing use of materials that exhibit thermal variations at the microstructure level. According to this theory, thermal and microthermal waves can propagate with energy dissipation at finite speeds. The existence and uniqueness of weak and classical solutions to the derived model is established by means of semigroup theory. Using the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is not analytic. Then, using the multiplier technique, we show the exponential stability result when (5) holds (when the wave speeds are equal) and the polynomial stability result when (5) does not hold. In the latter case, we prove using a result by Borichev and Tomilov that the solutions decay polynomially at an optimal rate of $t^{-1/2}$ t − 1 / 2 .

In this paper, we study the decay behavior of solutions to the biharmonic Schrödinger equation under the effect of internal fractional damping. By employing semigroup theory and energy methods, we establish well-posedness results and investigate the long-time behavior of solutions. In particular, we prove polynomial stability under suitable assumptions on the damping mechanism. Our results contribute to the understanding of how fractional damping influences the decay rate and stability of higher-order dispersive equations. These findings have potential applications in mathematical physics and control theory.

Global public health continues to be significantly threatened by infectious diseases, demanding the creation of reliable mathematical models for their investigation and management. Despite being on the verge of eradication, the polio virus is nevertheless a subject of worry owing to its possible return. In this study, the fractional differential equations with the Mittag-Leffler kernel are used as a modeling tool to investigate the polio virus epidemic disease. The non-local and memory-dependent aspects of the Polio virus transmission mechanism are captured by the Mittag-Leffler kernel. In comparison to conventional integer-order models, the resultant system of fractional differential equations offers a more realistic depiction of the dynamics of the epidemic. The results of this study shed light on the crucial elements driving the polio virus epidemic’s development and management, providing insightful information for public health measures. Analytical results establish the existence, uniqueness, and Hyers-Ulam stability of solutions, while numerical simulations confirm that fractional models outperform classical approaches in predicting epidemic trends. Furthermore, this research highlights the importance of fractional calculus in simulating and understanding the dynamics of infectious diseases, emphasizing the improved accuracy and practicality of fractional differential equations with the Mittag-Leffler kernel in epidemiological studies. These findings advance our knowledge of epidemic modeling and control, and they may also have implications for the control of other infectious diseases.