In this article, we analyze the analytical result of fractional-order Jaulent-Miodek equations with the help of two novel methods, namely, -Laplace decomposition method and -Laplace variational iteration method. The achieved results are shown in a series form, which is rapidly converging. The approximate simulations were performed in absolute error to ensure that the suggested methods are accurate and reliable. The achieved results are graphically presented to confirm the validity and accuracy of the techniques. The study results reveal that the -Laplace decomposition method is computationally very effective and accurate compared to -Laplace variational iteration method to analyze the nonlinear system of fractional-order Jaulent-Miodek equations.

This paper presents an analysis based on a mixture of the Laplace transform and the new iteration method to obtain new approximate results of the fractional-order Klein-Gordon equations in the Caputo-Fabrizio sense. So, a general system to investigate the approximate results of the fractional-order Klein-Gordon equations is obtained. This technique’s effectiveness is demonstrated by comparing the actual results of the fractional-order equations suggested with the results achieved.

In this research work, a ratio‐dependent prey–predator system is investigated for bifurcation and stability analysis. The unique existence of the solution, boundedness, and positivity of the temporal model is derived. Stability analysis of positive steady states is analyzed. Stable and unstable regions for both equilibria are drawn in different parametric spaces. The parametric plots for the different initial conditions are drawn. The diffusive prey–predator model is analyzed for the bifurcation and stability regions. It is observed that the stability regions under the effect of diffusion are increased. To gain the numerical solution of the spatially extended prey–predator model, the operator splitting scheme is derived. The scheme preserves the positivity and other important features of the continuous model. Lastly, a test problem is considered to visualize the accuracy of our scheme. To show the physical behavior of the scheme, 3D and 2D plots are drawn, which clearly show the dynamics of the scheme. Theoretical results are supported by simulations.

We examine a heat-absorbing viscous fluid’s electrically conducting boundary layer flow over a semi-infinite permeable plate in a porous medium inclined at an angle α. Nonlinear partial differential equations are solved using perturbation methods, and graphical analysis is used to determine how parameters impact concentration, temperature, and velocity profiles. Buoyancy forces increase fluid velocity with the increased Grashof number. However, the presence of magnetic (Lorentz) and rotational (Coriolis) effects introduces resistance, leading to a reduction in velocity. A direct relationship is observed between the Grashof number and skin friction, while the radiation parameter inversely affects the Nusselt number. An increased Schmidt number lowers the Sherwood number. We also investigate the impact of rotation on unsteady magnetohydrodynamic slip flow using an Artificial Neural Network (ANN) model employing Levenberg–Marquardt Backpropagation. The ANN accurately predicts flow dynamics and heat transfer using numerical simulation data. Model accuracy is validated through mean squared error graphs, regression analysis, and error histograms, demonstrating reliable fluid dynamics predictions under varying conditions.

This paper focuses on approximate solution of the Jaulent-Miodek system, which is a nonlinear-coupledpartial differential equations. Physics-Informed Neural Network (PINN) architecture is implemented tofind an approximate solution against the independent variables. Loss function is defined using PDEs andinitial conditions. The PINN is set up to meet both the governing PDEs and the starting conditions.It does this by using automated differentiation to enforce the system’s residuals in the loss function.The results is compared with the exact solution using graphical representations. The model is tested forspecific values of ψ. The error is illustrated in table, which shows the model efficiency and accuracy.

This study examines the fractional order formulations of the nonlinear coupled Korteweg-de Vries type systems in the wave interaction phenomena, especially the Kersten-Krasil'shchik KdV-mKdV system and a two-component coupled KdV model. The φ-Caputo fractional derivative is used to describe the time evolution of the systems and offers a versatile framework for integrating memory and nonlocal time effects. Two effective semi-analytical methods, the φ-Caputo fractional q-homotopy analysis method, and the φ-Caputo fractional variational iteration method, are proposed to derive analytical approximations. Clear series solutions are developed and demonstrated to converge very fast when appropriate auxiliary parameters are chosen. The effect of the fractional order on the dynamical behavior of the solutions is explored in detail using two and three dimensional graphs. It has been shown that as the fractional order is lowered the wave profile becomes smoother with a reduction in amplitude, which reflects an increase in the effect of memory. The close similarity of the approximate solutions obtained and the exact solutions in the case of integer-order validates the precision and accuracy of the proposed methods. The fractional models in this paper provide a better understanding of the nonlinear propagation of waves in complex media where classical integer-order models might fail.