This paper presents a novel hybrid approach for solving high-dimensional non-linear partial differential equations (PDEs) by integrating fractional calculus concepts with metaheuristic optimization algorithms. Specifically, we employ the Caputo fractional derivative to model the PDE and then utilize a modified Particle Swarm Optimization (PSO) algorithm to minimize the error functional associated with the fractional PDE. The proposed method addresses the challenges posed by high dimensionality and non-linearity, which often render traditional numerical techniques computationally infeasible or inaccurate. We demonstrate the efficacy and accuracy of our approach through several benchmark problems, comparing our results with those obtained by existing methods. The convergence analysis and computational efficiency of the hybrid algorithm are also investigated. The results demonstrate that the proposed method offers a promising alternative for solving complex fractional PDEs in various scientific and engineering applications.
Predicting chaotic time series remains a significant challenge due to their inherent sensitivity to initial conditions and complex nonlinear dynamics. This paper introduces a novel hybrid approach that combines fractal analysis techniques with machine learning models to improve prediction accuracy. Specifically, we leverage fractal dimension estimation, recurrence plot analysis, and the Hurst exponent to extract key features from chaotic time series. These features are then used as inputs to a Support Vector Regression (SVR) model. The efficacy of this hybrid method is demonstrated through extensive experimentation on benchmark chaotic time series datasets, including the Lorenz attractor, Rossler attractor, and Mackey-Glass equation. Results indicate that the proposed approach significantly outperforms traditional time series prediction methods, offering a robust and accurate framework for forecasting chaotic dynamics. This hybrid strategy effectively captures both the local and global characteristics of chaotic systems, leading to enhanced predictive performance.
This paper introduces a novel hybrid approach for time series forecasting of chaotic systems, integrating the strengths of fractional calculus and deep learning. Chaotic systems, characterized by their sensitive dependence on initial conditions, pose significant challenges for accurate prediction. While deep learning models, particularly Long Short-Term Memory (LSTM) networks, have shown promise in capturing complex temporal dependencies, they often struggle with long-range dependencies and noise inherent in chaotic data. We propose a hybrid model that leverages fractional derivatives to enhance the representation of past states, thereby improving the LSTM network's ability to learn and forecast chaotic time series. The fractional derivative captures non-local dependencies more effectively than traditional integer-order derivatives, providing richer information for the deep learning component. We evaluate the performance of our proposed model on benchmark chaotic systems, including the Lorenz attractor and the Rossler system. The results demonstrate that our hybrid approach significantly outperforms traditional LSTM networks and other established forecasting methods in terms of prediction accuracy, especially over longer forecasting horizons. This work provides a valuable contribution to the field of time series forecasting for chaotic systems, offering a powerful tool for modeling and predicting complex dynamical behaviors.
This paper presents a novel hybrid metaheuristic algorithm designed to address the challenges posed by high-dimensional global optimization problems. The algorithm synergistically combines the strengths of Particle Swarm Optimization (PSO) and Differential Evolution (DE) with an adaptive control mechanism to dynamically balance exploration and exploitation. The hybrid approach leverages PSO's efficient global search capability and DE's effective local refinement to achieve enhanced performance. The adaptive control mechanism adjusts the contributions of PSO and DE based on the search progress, promoting exploration in the early stages and intensifying exploitation as the search converges. The performance of the proposed algorithm is evaluated on a suite of benchmark functions, including unimodal, multimodal, and composite functions, and compared against established metaheuristic algorithms. The results demonstrate the superior performance of the hybrid algorithm in terms of solution accuracy, convergence rate, and robustness, particularly in high-dimensional spaces.
This paper presents a novel and efficient numerical method for solving Stochastic Differential Equations with Jumps (SDEJs). We introduce an adaptive time-stepping scheme coupled with a high-order Milstein approximation to enhance the accuracy and stability of the solution. The adaptive time-stepping is designed to dynamically adjust the step size based on the local behavior of the solution, thereby improving computational efficiency. The high-order Milstein scheme is tailored to handle the jump component effectively, particularly when the jump sizes are significant. We provide a rigorous convergence analysis of the proposed method and demonstrate its superior performance through numerical experiments. The results show that the adaptive Milstein scheme offers a significant improvement in accuracy and efficiency compared to traditional fixed-step methods, especially for SDEJs with large jump intensities.