The nonlinear wave behavior in the tropical and mid-latitude troposphere has been simulated using the space-time fractional Landau-Ginzburg-Higgs model. These waves are the consequence of interactions between equatorial and mid-latitude waves, fluid flow in dynamic systems, weak scattering, and extended linkages. The mEDAM method has been used to obtain new and extended closed-form solitary wave solutions of the previously published nonlinear fractional partial differential equation via the beta derivative. A wave transformation converts the fractional-order equation into an ordinary differential equation. Several soliton, single, kink, double, triple, anti-kink, and other soliton types are examples of known conventional wave shapes. The answers are displayed using the latest Python code, which enhances the usage of 2D and 3D plotlines, as well as contour plotlines, to emphasise the tangible utility of the solutions. The results of the study are clear, flexible, and easier to replicate.

This study presents an innovative application of Feedforward Neural Networks ‘FNNs’ to solve Variable-Order Fractional Partial Differential Equations ‘VO-FPDEs’ with time delays. Utilizing the Caputo definition, the variable-order fractional derivatives are approximated in terms of integer-order derivatives. The problem is reformulated as a system of partial differential equations with delay terms, which is then addressed using ‘FNNs’ to achieve explicit approximate solutions. Comprehensive error and convergence analyses validate the method’s precision and reliability. The effectiveness of the proposed approach is highlighted through numerical examples, with graphical and tabular representations showcasing minimal absolute errors and robust convergence. These results demonstrate the proposed method’s efficiency and simplicity, establishing it as a powerful tool for addressing complex fractional delay problems.

In this study, we present a unified symmetry-conservation solution analysis of a well-posed resonant nonlinear Schrödinger (NLS)-type equation incorporating spatio-temporal dispersion and inter-modal dispersion. Working within the truncated M-fractional derivative framework, we first construct exact traveling-wave solution families via the Kudryashov expansion method, together with the corresponding parameter constraints and limiting cases. We then determine the admitted Lie point symmetries and establish the associated Lie algebra, including the commutator structure, adjoint representation, and an optimal system of one-dimensional subalgebras for classification. Using the conservation theorem, we derive conserved vectors associated with the fundamental invariances of the model; in the NLS setting and under suitable conditions, these quantities can be interpreted as generalized power (mass), momentum, and energy-type invariants. Overall, the results provide explicit wave profiles and structural invariants that enhance the interpretability of the model and offer benchmark expressions useful for further qualitative, numerical, and stability investigations in nonlinear dispersive wave dynamics.

This study investigates consecutive reactions in chemical kinetics. Consecutive reactions are irreversible, meaning that the products formed do not return to the original reactants, and this makes their modeling more complex compared to other reaction types. Examples of such reactions include polymerization, radioactive decay, hydrocarbon chlorination, the reaction of unabsorbed ethanol in the human body, and thermal cracking. For these consecutive reactions, a conformable kinetic model is considered and analytically solved by means of the conformable Laplace transform. The effects of different conformable orders α on the reaction dynamics are examined in detail. The obtained solutions are presented graphically and interpreted in terms of the temporal evolution of each species. The results indicate that the conformable model provides an effective and flexible representation of the behavior of consecutive reactions.

This study presents an integrated analytical and geometric investigation of the Hamiltonian amplitude equation through Lie-point symmetry analysis, conservation-law construction, Hamiltonian–Poisson formulation, and soliton-based numerical verification, together with an exact-solution analysis of its M-truncated form. The mass, momentum, and energy conservation laws are derived using the multiplier method and are independently recovered through an extended Lagrangian formulation and Noether’s theorem, demonstrating the consistency of these approaches. The Lie-point symmetry generators associated with time and space translations, phase rotation, and Galilean invariance are explicitly constructed. The equation is subsequently formulated within a Hamiltonian–Poisson framework, showing that the conserved functionals generate the corresponding continuous symmetry flows through the Poisson bracket. In addition, the rational extended sinh-Gordon equation expansion method is applied to the M-truncated form of the equation to derive several families of exact hyperbolic-function solutions, including dark, singular, mixed bright–dark, and mixed singular soliton structures. The analysis establishes the Lie symmetry, conservation-law, Noether, and Hamiltonian–Poisson structures of the Hamiltonian amplitude equation, while extending its exact-solution treatment to the M-truncated form through distinct soliton families. The derived conservation laws are subsequently verified numerically using bright and dark soliton profiles of the Hamiltonian amplitude equation. The conserved quantities remain invariant to high numerical accuracy over long propagation intervals, with relative errors remaining below $$10^{-6}$$ . Two-dimensional, three-dimensional, and contour plots further illustrate the localized and shape-preserving profiles of the obtained solutions. These findings confirm the consistency between the analytical and numerical results and clarify the relationships among symmetry transformations, conserved quantities, Hamiltonian structures, and soliton propagation.

This study employs the conformable fractional q‐Shehu homotopy analysis transform approach to investigate the nonlinear conformable time‐fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation incorporating a proportional delay term. Error analysis was performed for the results obtained with this method and the analytical solution. Graphical representations of the computed numerical solutions were generated using Maple software. The findings demonstrate that the proposed method provides accurate and stable results, confirming its computational efficiency.