We derive an explicit series representation for the Dirichlet problem for Poisson’s equation in a right-isosceles triangle. Linearity is used to decompose the boundary data into three harmonic subproblems, each carrying the prescribed values on a single side, and a fourth Poisson subproblem with homogeneous Dirichlet conditions. The two leg-driven harmonic fields admit simple sine-hyperbolic expansions in the original coordinates, while the hypotenuse-driven field is obtained after a 45∘ rotation and scaling that maps the domain to a symmetric triangle. The remaining forced problem is treated by an eigenfunction expansion of the Dirichlet Laplacian, with coefficients obtained by orthogonal projection. The resulting formula provides a convenient analytic benchmark for assessing numerical solvers on polygonal domains.
This study applies the quantile connectedness approach to evaluate the linkages between ESG and sectoral stock markets in China within a network system. The findings reveal that the system exhibits strong connectedness under the impact of specific events. Under moderate market conditions, ESG300, Financial, Consumer Discretionary, and Materials predominantly act as net transmitters, while Consumer Staples, Energy, Real Estate, Healthcare, and Utilities persist as net recipients. Industrial and Telecom undergo connectedness shifts due to pivotal events. Moreover, under extreme market conditions, markets alternate between net receiving and net transmitting states.
The mold filling process can create small defects being trapped within the material, which compromise the strength of the cast piece. Here, we address the inverse problem of identifying the locations of these flaws using boundary measurements. To achieve this, we develop a numerical algorithm based on topological gradient optimization method.
This paper investigates a coupled system of nonlinear q-fractional differential inclusions involving nested p-Laplacian operators and nonlocal boundary conditions. The problem is formulated in a multivalued setting, where the nonlinear inclusion depends on the state vector and its Jackson q-derivative. Using tools from q-fractional calculus, Green-function representations, and multivalued fixed-point theory, we derive sufficient conditions for the existence of at least one solution in the associated product function space. The analysis combines the sequential structure of the Caputo and Riemann–Liouville q-fractional operators with compact convex multifunction values and nonlocal boundary constraints. An aerospace vibration-modeling illustration is also presented to show how a Caputo q-fractional damping term can represent memory-dependent behavior in a base-excited inertial-measurement-unit mounting model. The numerical comparison is reported against rigid mounting and a classical viscous isolator and is interpreted as an illustration of q-fractional memory modeling. The main mathematical contribution of the paper is the existence result for the proposed nonlinear q-fractional differential inclusion system.
This paper investigates the complex dynamics of Social Media Addiction (SMA) through the angle of fractional calculus. We employ a five-compartment deterministic model as Susceptible (S), Exposed (E), Addicted (A), Recovered (R), and Quit (Q) which map the transmission pathways of this behavioral addiction. The model is formulated using the Caputo fractional derivative to effectively capture the memory and hereditary properties inherent in human behavioral patterns, which integer-order models often overlook. We conduct a comprehensive qualitative analysis, establishing the existence, uniqueness, non-negativity, and boundedness of the solutions. The stability of the model’s equilibria as addiction-free and endemic states is rigorously examined through local and global stability analyses based on the basic reproduction number (R_0). To approximate the solution of the fractional system, we implement the robust Adams-Bashforth-Moulton predictor-corrector algorithm. Numerical simulations and a detailed sensitivity analysis are presented to illustrate the theoretical findings, demonstrating the influence of key parameters and the fractional order α on the system’s dynamics. The results confirm that fractional-order models offer a more nuanced and realistic framework for understanding the propagation of social media addiction. The main findings show that the fractional-order model provides slower convergence and reduced addiction peaks compared to the classical integer-order case, highlighting the significant role of memory effects. The analysis confirms that the basic reproduction number governs the persistence of addiction, while sensitivity results identify recovery and quitting rates as the most effective control parameters. These results demonstrate the importance of incorporating memory in accurately describing and controlling social media addiction dynamics.
