Mathematics

Most existing depth-averaged granular flow theories assume that dry, cohesionless granular materials are incompressible, with the void ratio among grains remaining spatially and temporally invariant. However, recent large-scale experiments showed that the pore space among grains varies both spatially and temporally. This study, therefore, incorporates the effects of granular dilatancy to perform analytical and numerical investigations of granular flows down inclined planes. A high-resolution shock-capturing scheme is employed to numerically solve the compressible depth-averaged equations for temporal and spatial evolution of the flow thickness and depth-averaged velocity, as well as depth-averaged volume fraction. Additionally, a traveling wave solution is constructed. The comparison between analytical and numerical solutions confirms the accuracy of the numerical solution and also reveals that the gradient of the solids volume fraction, induced by granular dilatancy, results in a gentler slope of the granular front, in agreement with experimental observations. Furthermore, this numerical framework is applied to investigate granular flows transitioning from an inclined plane onto a horizontal run-out pad. The numerical solution shows that the incorporation of granular dilatancy causes the shock wave to propagate upstream more rapidly. As a result, the position and morphology of the mass deposit exhibit closer alignment with experimental data.

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework.

The paper investigates the oscillation, zero-convergence, and solutions of second-order neutral delay difference equations containing three nonlinear delayed terms with different growth rates. By using positivity and monotonicity conditions on an auxiliary function along with divergence-type conditions on the coefficient sequences of the neutral and delayed terms, the paper establishes new criteria that guarantee oscillation or convergence of all solutions. These novel findings extend and enhance several of the existing oscillation criteria established by the literature. Suggestions for further investigation are included with illustrative examples.

This paper introduces the concept of P-curvature inheritance in generalized recurrent Finsler spaces and establishes various types of curvature inheritance tensors in such spaces. We prove that the fundamental function of the Finsler space is given by $F=\sqrt{{\displaystyle \frac{y^i{y}^r{\delta }_r^k{g}_{ik}}{n}}}$. Moreover, we infer that the Lie derivative of the curvature scalar R is equal to the Lie derivative of the curvature scalar K, and the Lie derivative of the recurrence vector field ${\mu }_m$ vanishes. Additionally, we establish new mathematical formulas for the scalar function $\alpha \left(x\right)$ and the scalar form of the metric tensor $g_{ij}$ that admit P-curvature inheritance. A tensor $\left({\delta }_l^k{P}_{kh}^i\right)$ and the R-Ricci tensor possess an inheritance property in the generalized $\mathfrak{B}P$-recurrent Finsler space. In the same vein, we obtain conditions under which the Lie derivative and the Berwald covariant derivative of the curvature scalar P commute. Finally, we provide practical examples that illustrate the understanding of the obtained results.