Search Results (62)

Accurate modeling of multiphase flow in porous media remains challenging due to the nonlinear transport and sharp displacement fronts described by the Buckley–Leverett (B-L) equation. Although Fourier Neural Operators (FNOs) have recently emerged as powerful surrogates for parametric partial differential equations, they exhibit limited robustness when extrapolating beyond the training regime, particularly for shock-dominated fractional flows. This study aims to enhance the extrapolative performance of FNOs for one-dimensional B-L displacement. Analytical solutions were generated using Welge’s graphical method, and datasets were constructed across a range of mobility ratios. A baseline FNO was trained to predict water saturation profiles and evaluated under both interpolation and extrapolation conditions. While the standard FNO accurately reconstructs saturation profiles within the training window, it misestimates shock positions and saturation jumps when extended to longer times or higher mobility ratios. To address these limitations, we develop Physics-Informed FNOs (PI-FNOs), which embed PDE residuals and boundary constraints, and Transfer-Learned FNOs (TL-FNOs), which adapt pretrained operators to new regimes using limited data. Comparative analyses show that both approaches markedly improve extrapolation accuracy, with PI-FNOs achieving the most consistent and physically reliable performance. These findings demonstrate the potential of combining physics constraints and knowledge transfer for robust operator learning in multiphase flow systems. Full article

Rod-pumping systems represent complex nonlinear systems. Traditional soft-sensing methods used for efficiency prediction in such systems typically rely on complicated fractional-order partial differential equations, severely limiting the real-time capability of efficiency estimation. To address this limitation, we propose an approximate efficiency prediction model for nonlinear fractional-order differential systems based on selective heterogeneous ensemble learning. This method integrates electrical power time-series data with fundamental operational parameters to enhance real-time predictive capability. Initially, we extract critical parameters influencing system efficiency using statistical principles. These primary influencing factors are identified through Pearson correlation coefficients and validated using p-value significance analysis. Subsequently, we introduce three foundational approximate system efficiency models: Convolutional Neural Network-Echo State Network-Bidirectional Long Short-Term Memory (CNN-ESN-BiLSTM), Bidirectional Long Short-Term Memory-Bidirectional Gated Recurrent Unit-Transformer (BiLSTM-BiGRU-Transformer), and Convolutional Neural Network-Echo State Network-Bidirectional Gated Recurrent Unit (CNN-ESN-BiGRU). Finally, to balance diversity among basic approximation models and predictive accuracy, we develop a selective heterogeneous ensemble-based approximate efficiency model for nonlinear fractional-order differential systems. Experimental validation utilizing actual oil-well parameters demonstrates that the proposed approach effectively and accurately predicts the efficiency of rod-pumping systems. Full article

In this study, we explore the soliton solutions of the truncated M-fractional fifth-order Korteweg–de Vries (KdV) equation by applying the improved modified extended tanh function method (IMETM). Novel analytical solutions are obtained for the proposed system, such as brigh soliton, dark soliton, hyperbolic, exponential, Weierstrass, singular periodic, and Jacobi elliptic periodic solutions. To validate these results, we present detailed graphical representations of selected solutions, demonstrating both their mathematical structure and physical behavior. Furthermore, we conduct a comprehensive linear stability analysis to investigate the stability of these solutions. Our findings reveal that the fractional derivative significantly affects the amplitude, width, and velocity of the solitons, offering new insights into the control and manipulation of soliton dynamics in fractional systems. The novelty of this work lies in extending the IMETM approach to the truncated M-fractional fifth-order KdV equation for the first time, yielding a wide spectrum of exact analytical soliton solutions together with a rigorous stability analysis. This research contributes to the broader understanding of fractional differential equations and their applications in various scientific fields. Full article

Rod pump systems are complex nonlinear processes, and conventional efficiency prediction methods for such systems typically rely on high-order fractional partial differential equations, which severely constrain real-time inference. Motivated by the increasing availability of measured electrical power data, this paper introduces a series of prediction models for nonlinear fractional-order PDE systems efficiency based on multimodal feature fusion. First, three single-model predictions—Asymptotic Cross-Fusion, Adaptive-Weight Late-Fusion, and Two-Stage Progressive Feature Fusion—are presented; next, two ensemble approaches—one based on a Parallel-Cascaded Ensemble strategy and the other on Data Envelopment Analysis—are developed; finally, by balancing base-learner diversity with predictive accuracy, a multi-strategy ensemble prediction model is devised for online rod pump system efficiency estimation. Comprehensive experiments and ablation studies on data from 3938 oil wells demonstrate that the proposed methods deliver high predictive accuracy while meeting real-time performance requirements. Full article

Robust epidemiological models are essential for managing COVID-19, especially in diverse urban settings. In this study, we present a fractional advection–diffusion–reaction model to analyze COVID-19 spread in three major Turkish cities: Ankara, Istanbul, and Izmir. The model employs a Caputo-type time-fractional derivative, with its order dynamically determined by the Hurst exponent, capturing the memory effects of disease transmission. A nonlinear reaction term models self-reinforcing viral spread, while a Gaussian forcing term simulates public health interventions with adjustable spatial and temporal parameters. We solve the resulting fractional PDE using an implicit finite difference scheme that ensures numerical stability. Calibration with weekly case data from February 2021 to March 2022 reveals that Ankara has a Hurst exponent of 0.4222, Istanbul 0.1932, and Izmir 0.6085, indicating varied persistence characteristics. Distribution fitting shows that a Weibull model best represents the data for Ankara and Istanbul, whereas a two-component normal mixture suits Izmir. Sensitivity analysis confirms that key parameters, including the fractional order and forcing duration, critically influence outcomes. These findings provide valuable insights for public health policy and urban planning, offering a tailored forecasting tool for epidemic management. Full article

We introduce a novel class of coherent states, termed ${\mathcal{W}}_{\alpha ,\beta ,\nu }$-coherent states, constructed using a deformed boson algebra based on the generalised factorial ${\left[n\right]}_{\alpha ,\beta ,\nu }!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and mass transfer modeling in unsteady Maxwell fluid within a cylinder. Governing equations within a cylinder involve a system of coupled, nonlinear fractional partial differential equations (PDEs). A machine learning technique based on the Levenberg–Marquardt scheme with a backpropagation neural network (LMS-BPNN) is employed to evaluate the predicted solution of governing flow equations up to the required level of accuracy. The numerical data sheet is obtained using series solution approach Homotopy perturbation methods. The data sheet is divided into three portions i.e., $80\%$ is used for training, $10\%$ for validation, and $10\%$ for testing. The mean-squared error (MSE), error histograms, correlation coefficient (R), and function fitting are computed to examine the effectiveness and consistency of the proposed machine learning technique i.e., LMS-BPNN. Moreover, additional error metrics, such as R-squared, residual plots, and confidence intervals, are incorporated to provide a more comprehensive evaluation of model accuracy. The comparison of predicted solutions with LMS-BPNN and an approximate series solution are compared and the goodness of fit is found. The momentum boundary layer became higher and higher as there was an enhancement in the value of Caputo, fractional order α = 0.5 to α = 0.9. Higher thermal boundary layer (TBL) profiles were observed with the rising value of the heat source. Full article

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved oxygen (DO) concentration. The coupling of these equations arises due to the chemical interactions between oxygen and pollutants, forming harmless chemicals. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for the given model. Also, the convergence of solutions is checked for efficacy and accuracy. The effect of longitudinal and transverse diffusion coefficients of pollutant and DO on the concentration of pollutant and DO is analyzed numerically and graphically. Also, we checked the effect of change in the river’s longitudinal and transverse seepage velocity on pollutant and DO concentration numerically and graphically. We analyzed the comparison of change in the value of half-saturated oxygen demand concentration for pollutant decay on pollutant and DO concentration numerically and graphically. Also, numerical and graphical analysis examined the effect of fractional parameters on pollution levels. Full article

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. Full article

This paper studies synchronization behaviors of two sorts of non-linear fractional-order complex spatio-temporal networks modeled by hyperbolic space-varying PDEs (FCSNHSPDEs), respectively, with time-invariant delays and time-varying delays, including one delayed coupling. One distributed controller with space-varying control gains is firstly designed. For time-invariant delayed cases, sufficient conditions for synchronization of FCSNHSPDEs are presented via LMIs, which have no relation to time delays. For time-varying delayed cases, synchronization conditions of FCSNHSPDEs are presented via spatial algebraic LMIs (SALMIs), which are related to time delay varying speeds. Finally, two examples show the validity of the control approaches. Full article

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms $E_2$ and $E_{\infty }$, and compared to existing methods. Full article

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer’s disease, or Parkinson’s disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients. Full article

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination. Full article

The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both $\theta (t,x)$ and $\phi (t,x)$. Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, $f_1$, $f_2$, $w_1$, $w_2$, $\alpha$, $\xi$, ${\theta }_0$, ${\alpha }_0$, ${\phi }_0$, and ${\xi }_0$, we prove the well-posedness of a solution $\theta ,\alpha ,\phi ,\xi$, which is $\left(\theta (t,x),\alpha (t,x)\right)\in W_p^{1,2}\left(Q\right)\times W_p^{1,2}(\Sigma )$, $\left(\phi (t,x),\xi (t,x)\right)\in W_{\nu }^{1,2}\left(Q\right)\times W_p^{1,2}(\Sigma )$, $\nu =min\{q,\mu \}$. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future. Full article