In this article, the new Ishikawa iteration method is presented to find the approximate solution of an ordinary differential equation with an initial condition. Additionally, some numerical examples with initial conditions are given to show the properties of the iteration method. Furthermore, the results of absolute errors are compared with Euler, Runge-Kutta and Picard iteration methods. Finally, the present method, namely the new modified Ishikawa iteration method, is seen to be very effective and efficient in solving different type of the problem. MSC:65K15, 65L07, 65L06, 65L70.

The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear PDE. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the PDE is converted into an algebraic system of non-linear equations and this system has been solved using MAPLE computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computational for the non-linear or PDE.

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 research scrutinizes the abrupt behavioral shifts surrounding digital financial platforms in the Turkish sector, utilizing a multidimensional forecasting apparatus that fuses classical regression, seasonal econometrics, and tree-based predictive learners. Fifty-three quarters of aggregate usage logs (late-2011 to late-2024) were processed via a tiered methodology: foundational polynomial estimators mapped the continuous growth baseline, while complex seasonal autoregressive models (SARIMA) absorbed cyclic temporal echoes. To address erratic nonlinear surges, ensemble protocols—specifically Random Forest and XGBoost—were subsequently deployed alongside an innovative composite framework that mitigates structured econometric residuals via iterative gradient boosting. Our quantitative results isolate a violent, statistically verified inflection point triggered during the second quarter of 2020, an anomaly that persistently overwhelmed standard linear projection boundaries. Comparatively, the integrated SARIMA–XGBoost construct achieved unmatched tracking accuracy (RMSE: 214,586; MAE: 122,847; MAPE: 2.6%; 𝑅2: 0.995), dramatically eclipsing the precision of isolated ARIMA frameworks. Ultimately, this analysis corroborates the assertion that melding standard time-series inference with algorithmic error-correction produces extraordinarily resilient predictive tools, particularly when modeling markets destabilized by sudden macroeconomic shocks.

Finding the root is one of the most common problems in scientific disciplines. Due to their increasing importance in a wide variety of practical applications, nonlinear functions are utilized across the entire spectrum of various areas within mathematics, science, and engineering. Using computers to solve these functions is very important, especially in today's world. The most important things to think about when working out the roots of non-linear equations are how well the method works and how much it costs to do it. A better method is more effective and less expensive. Considering these facts, the main aim of this paper is to present a new method that doesn't involve derivatives and is more efficient. This paper presents an approach to solving nonlinear equations by extending the Lagrange method to find roots of polynomial curves. These equations can be solved using various iterative techniques, including the Secant, Newton-Raphson, Bisection, Regula Falsi, and others. The objective is to find the roots of such equations without considering any derivative operations. This is achieved by considering polynomial curves produced using Lagrange interpolation under specific initial circumstances. The effectiveness of the proposed approach is demonstrated by examining well-documented cases from the literature. It could also be used in mathematical modeling, optimization, and computational science, making it useful for solving many practical problems.