This study addresses the classical shortest path problem, a fundamental challenge in graph theory with critical applications in transportation networks. Using real inter-district distance data from southwestern Türkiye, particularly covering 13 districts of Muğla province, the region frequently affected by forest fires during summer, this study also emphasizes the importance of shortest path analyses for routing firefighting vehicles and emergency logistics. We compare the deterministic Dijkstra algorithm with four metaheuristic approaches: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Simulated Annealing (SA). The evaluation focuses on path length optimization, computational efficiency, and convergence behavior across multiple independent runs. Experimental findings confirm that Dijkstra consistently produces the optimal solution (200 km), while ACO achieves the same performance despite its stochastic nature. In contrast, GA (mean 621 km), PSO (mean 734 km), and SA (mean 759 km) exhibited significant deviations from the optimal solution, with varying convergence stability and computation times. These results highlight the robustness of Dijkstra in small-scale static problems, while also demonstrating the potential of ACO as a competitive metaheuristic for dynamic and large-scale applications. Furthermore, Dijkstra-Seeded (DS) hybrid variants of the three underperforming algorithms (DS-GA, DS-PSO, DS-SA) were developed to investigate whether informed initialization could improve solution quality. Experimental results revealed that Dijkstra-based seeding within a permutation-based encoding yielded mixed results, with DS-SA showing meaningful improvement (689 km mean, 9.22% better than original SA), while DS-GA (645 km mean, -3.86%) and DS-PSO (754 km mean, -2.72%) slightly deteriorated. These findings underscore the fundamental challenges of adapting continuous-space metaheuristics to discrete graph problems and highlight the importance of encoding-aware hybridization strategies.

This study investigates the generalized Emden-Fowler equation with logarithmic nonlinearity within the framework of complete integrability by means of the S-linearizability approach. Analytical proofs show that, for the fixed nonzero logarithmic source term, the quadratic-damping coefficient is the unique value within the family $a_2=k/\theta$ compatible with the integrability condition, and that, once this damping coefficient is fixed, every source term compatible with the integrability condition necessarily takes the form $\theta[\chi(t)+\Psi(t)\ln\theta]$, a family that contains source terms linear in $\theta$ as well as the logarithmic model investigated here. Under the parameter specialization that places the system in the scale-invariant regime, two functionally independent first integrals are constructed. These integrals are shown to generate equivalent closed-form general solutions through both a local point transformation and a generalized Sundman transformation, a result confirmed through symbolic substitution, internal consistency analysis, and independent numerical integration. The closed-form solution displays three distinct asymptotic regimes, determined by the sign of the discriminant of the characteristic equation. In particular, in contrast to the movable singularities observed in exponentially nonlinear equation classes, this family is shown to have no movable singularities; its only possible finite singular point is fixed at the origin. The S-linearizability framework provides a constructive coefficient-based route to the first integrals and the linearizing pair, consistent with the direct substitution $u=\ln\theta$. The conductivity relation provides structural motivation for possible heat-transfer applications, for which a physical interpretation requires a specified model.

Fins at large temperature differences lose heat by convection and radiation. With temperature-dependent conductivity, the nonlinear problem is usually treated numerically. Using $\lambda$-symmetries, the extended Prelle–Singer method and Lie symmetries, this study identifies the conductivity laws that make the problem linearizable. In this class, one relation gives the temperature for every heat-loss function, and the problem is well posed for heat loss with positive derivative, vanishing at ambient temperature. For power-law convection with Stefan–Boltzmann radiation, explicit solutions exist for purely convective or radiative loss and for constant heat transfer coefficients. They also exist for mixtures at negligible ambient temperature reducing to polynomials of degree at most four, including turbulent natural convection and nucleate boiling. Other mixtures have implicit exact solutions. The fin efficiency equals $\tanh \mu /\mu$ throughout the class, so the classical design relations hold. Tip losses and internal heat generation admit closed forms. For temperature-dependent heat loss, the class consists of the fin equations with maximal point symmetry $\mathfrak{sl}(3,\mathbb{R})$. Within the Rosseland and thin-fin approximations, its constant-coefficient member is realized by semitransparent media of high refractive index satisfying a matching condition. A perturbation theory gives the efficiency correction outside the class. The exact solutions provide benchmarks for numerical methods.