We study a mixed initial–boundary value problem on the half-line for a class of first-order nonhomogeneous quasilinear strictly hyperbolic systems with a non-autonomous source term F(t,x). Under the Lax linear degeneracy assumption, we prove the global existence and uniqueness of C^1 classical solutions. The explicit dependence of F on both t and x produces additional terms in the wave decomposition and in the subsequent a priori estimates. To control these terms, we adapt John’s wave decomposition method to the non-autonomous mixed initial–boundary setting and combine it with two basic L^1 estimates. The result allows the sup norms of the derivatives of the initial and boundary data to be large, while smallness is imposed only on several integral quantities. This extends the global C^1 existence theory for linearly degenerate quasilinear hyperbolic systems to the case of non-autonomous source terms.
This study focuses on multiplicative Fourier analysis. In the main results of our work, we first introduce multiplicative Riemann integrals. In this section, we construct multiplicative Riemann sums, prove the fundamental definitions and theorems, and establish the generalized integrals utilized in obtaining Fourier integrals. Subsequently, we present the complex form of the multiplicative Fourier series, which we then leverage to derive multiplicative Fourier integrals. Finally, building upon all these established definitions and theorems, we deduce the multiplicative Fourier transform and its inverse.
We study the inverse identification of a time-dependent reaction coefficient in a space-time fractional diffusion equation from integral-in-space measurements. We establish well-posedness of the direct problem, identifiability of the coefficient under a mild non-degeneracy condition, and a local conditional stability estimate. Two reconstruction methods, a conjugate gradient method and a physics-informed neural network, are developed and compared. Numerical experiments with smooth and nonsmooth coefficients under two noise levels confirm stable recovery and highlight the complementary strengths of both approaches.
Freshwater ecosystems are increasingly threatened by pollutants that disrupt species interactions and ecosystem stability. This study develops and analyzes a nonlinear eco-toxicological mathematical model that couples phytoplankton, two competing zooplankton species, and an environmental toxicant. The phytoplankton growth is modeled using a Gompertz function to capture stress-driven asymmetry, while predator–prey interactions follow a Beddington–DeAngelis functional response. Toxicant input, decay, and phytoplankton uptake are incorporated into the mathematical system. We establish positivity, boundedness, and the existence of five biologically feasible equilibria, followed by a detailed stability analysis using Jacobian eigenvalues and Routh–Hurwitz criteria. Sensitivity indices show that parameters linked to the zooplankton (predator) populations have the strongest influence on equilibrium levels, while toxicant-related parameters govern the stability transitions. Numerical simulations confirm the stability of non-trivial equilibria and reveal that increases in toxicant input rate and phytoplankton mortality due to toxicity can trigger subcritical Hopf bifurcations, leading to oscillatory instabilities. The results underscore the ecological importance of toxicant thresholds in freshwater communities and provide insight into how pollution influences coexistence, resilience, and the risk of ecosystem destabilization.
Using nested level domains and their normalized conformal maps, we create fuzzy geometric constraints. The initial coefficients and the Fekete–Szegő functional in the corresponding starlike and convex classes are sharply bounded for right-half-plane dominants by the first two Taylor coefficients. We derive explicit sharp growth, derivative-distortion, and covering estimates for half-plane levels, with extremal functions achieving the bounds. Additionally, we provide quantitative perturbation estimates for geometric envelopes and coefficient constants; the latter exhibit uniform convergence on compact subdisks. The findings show that while crisp membership recovers the classical classes, the membership profile regulates both extremal constants and image geometry. In this study, when target boundaries or tolerance levels are uncertain, normalized analytic maps of the unit disk are taken into consideration as geometric models. The radius r tracks the location where growth is measured, the profile beta establishes the order of the half-plane constraint, and the dimensionless membership parameter alpha records the confidence level. Epsilon measures profile perturbation, lambda chooses a coefficient functional, and the Taylor coefficients characterize the local map shape. Level selection, geometric classification, local and global estimates, and sensitivity analysis are all supported by these parameters, respectively.
