ABSTRACT This investigation explores the integrable Akbota‐Gudekli‐Kairat‐Zhaidary (AGKZ) system, characterizing its nonlinear dynamics through the well‐known analytical approach, namely, the modified extended direct algebraic scheme (MEDAS). We establish novel exact solutions—including some soliton solutions like bright, dark, and singular, and some other solutions' forms such as singular periodic, exponential, Jacobi elliptic (JE), and Weierstrass elliptic doubly periodic. A comprehensive stability analysis reveals the solution branches' robustness, while bifurcation theory uncovers critical transition points in the parameter space where solution behavior qualitatively changes. The accompanying 2D, 3D, contour, and polar visualizations not only validate solution consistency but also demonstrate how stability properties evolve across different dynamical regimes. Our unified treatment, combining exact solution construction with stability characterization and bifurcation mapping, provides a complete theoretical framework for understanding this system's rich nonlinear phenomena and their potential applications in physical contexts.
ABSTRACT The relativistic Duffing oscillator presents a significant mathematical challenge due to its irrational Lorentz factor singularity, which causes traditional series‐expansion approximation techniques to severely diverge at velocities approaching the speed of light. To overcome this limitation, this article introduces a novel, non‐perturbative first‐harmonic analytical framework that accurately captures the system's dynamics across the entire physical velocity domain. The adopted methodology systematically clears the singularity by absorbing relativistic effects entirely into the restoring force term. Based on an assumed single‐harmonic cosine trial solution, a first‐order harmonic balance method via a Galerkin projection is utilized, and the resulting integrals are evaluated exactly in closed form using the Beta function and Gaussian hypergeometric functions. To overcome the implicit nature of the resulting transcendental equation, a novel quadratic parametric inversion framework is applied, explicitly mapping the physical amplitude to the nonlinear frequency without relying on iterative numerical root‐finding. Validation against high‐precision numerical integrations across a range of strong cubic nonlinearity parameters demonstrates exceptional accuracy. The main results show that the proposed approximation maintains relative errors below 1% all tested cubic nonlinearity parameters in weakly‐to‐moderately relativistic regimes (velocities up to half the speed of light). At the ultra‐relativistic limit, the relative error remains tightly bounded, peaking at approximately 5.7% for the strongest evaluated nonlinearity, which is a natural residual consequence of the single‐harmonic truncation. Comparative benchmarking confirms that while traditional first‐order approximations specifically the Taylor‐expanded Harmonic Balance Method and He's Energy Balance Method suffer severe divergence near the speed of light, the proposed hypergeometric framework remains physically robust.
ABSTRACT This article introduces an ‐dimensional weighted Fourier transform with respect to a function, extending the one‐dimensional framework proposed in [8]. A spectral theory is developed on weighted Hilbert spaces, establishing the Plancherel identity, the inversion formula, the convolution theorem, and a Heisenberg‐type uncertainty principle under geometric deformations. This framework is then used to define a weighted fractional Laplacian and to characterize its purely continuous spectrum. Finally, these tools are applied to solve the space‐time fractional diffusion‐wave equation. The fundamental solutions are expressed explicitly in terms of the Fox ‐function, establishing a direct connection with the weighted Mellin transform.
ABSTRACT This paper investigates the dispersion of transverse horizontal surface waves in a sandy layer overlying an inhomogeneous orthotropic half‐space with a parabolic interface irregularity. Two distinct gradient variations of the substrate, namely exponential and quadratic are considered to model different forms of material inhomogeneity. The study is motivated by the need to represent realistic geological structures such as sandy soil layers overlying heterogeneous rock formations commonly found in sedimentary basins, coastal regions, and seismic zones. The novelty of the present work lies in the unified treatment of sandy overlayers, orthotropic inhomogeneity, and parabolic interface irregularity, a combination scarcely addressed in existing literature. The governing equations are solved under appropriate boundary conditions at the interface and free surface, employing Fourier transforms and perturbation techniques to derive the dispersion relations. The formulation reduces to classical results in limiting cases, validating the model. Numerical computations elucidate the effects of the inhomogeneity parameter, sandy layer properties, and interface width ratio on phase velocity, while a comparative analysis highlights the contrasting influences of exponential and quadratic gradients. Benchmarking against the classical Love wave solution reveals a reduction in energy penetration into the substrate during propagation, with potential implications for seismic hazard mitigation.
ABSTRACT The inclusion of kernels in Hardy‐type operators is a significant research direction. However, deriving results for general kernels remains an open problem. Consequently, current studies focus on Hardy‐type operators with kernels from specific classes satisfying particular conditions. This work investigates discrete Hardy‐type inequalities involving iterated operators and . The aim of the article is to obtain necessary and sufficient conditions for the inequality in the case of parameters and . For the case , the boundedness criterion for quasilinear operators and was established for a general class of matrices, and for the cases and , the criterion was obtained for matrices belonging to wider classes. These results allow us to study generalized bilinear operators. In particular, the results on quasilinear inequalities with matrices provide alternative proofs for estimates of bilinear inequalities. In addition, the importance of the obtained results lies in their application to the study of difference equations and the embedding theory of weighted Sobolev spaces.
