AppliedMath

In high-dimensional optimization, particle swarm optimization (PSO) algorithms often suffer from premature convergence due to stagnation in certain dimensions. This study proposes an enhanced variant, ELPSO-C, which integrates dimension-wise convergence detection with adaptive exploration mechanisms. By applying agglomerative clustering to inter-particle velocity diversity, ELPSO-C identifies dimensions showing signs of stagnation and selectively reintroduces diversity through targeted mutation strategies. The algorithm preserves global search capability while reducing unnecessary perturbation in well-explored dimensions. Experimental results on a suite of 18 benchmark functions across various dimensions demonstrate that ELPSO-C consistently achieves superior performance compared to existing PSO variants, especially in high-dimensional and complex landscapes. These findings suggest that dimension-aware adaptation is an effective strategy for improving PSO’s robustness and convergence quality.

We introduce and analyze a subclass of analytic functions with negative coefficients, denoted by ${\mathcal{P}}_{q,\sigma }^{m,\ell ,p}(\alpha ,\eta )$, constructed through a generalized q-calculus operator in combination with a multiplier-type transformation. For this class, we obtain sharp coefficient bounds, growth and distortion estimates, and closure results. The radii of close-to-convexity, starlikeness, and convexity are determined, and further consequences, such as integral means inequalities and neighborhood characterizations, are derived. The results presented provide a broad framework that incorporates and extends several earlier families of analytic and geometric function classes.

The computation of solutions of Caputo fractional differential equations is paramount in modeling to establish its benefits over the corresponding integer order models. In the literature so far, in order to compute the solution of Caputo fractional differential equations, the solution is typically assumed to be a $C^n$ function, which is a sufficient condition for the Caputo derivative to exist. In this work, we assume the necessary condition for the Caputo derivative of order , to exist, which means that we assume it to be a $C^{nq}$ function. Recently, it has been established that the Caputo derivative of order $nq$ is sequential of order $q.$ As such, we assume the fractional initial conditions. In our work, we have obtained an analytical solution for the Caputo fractional differential equation of order $nq$ with fractional initial conditions by two different methods. Namely, the approximation method and the Laplace transform method. The application of our main results is illustrated with examples.