Search Results (496)

In this paper, we study a class of time–space fractional partial differential equations involving Caputo time-fractional derivatives and Riesz space-fractional derivatives. A computational scheme is developed by combining a discrete approximation for the Caputo derivative in time with a modified trapezoidal method (MTM) for the Riesz derivative in space. We establish the stability and convergence of the scheme and provide detailed error analysis. The novelty of this work lies in the construction of an MTM-based spatial discretization that achieves ${\displaystyle \beta }$-order convergence in space and a ${\displaystyle (3-\alpha )}$-order convergence in time, while improving accuracy and efficiency compared to existing methods. Numerical experiments are carried out to validate the theoretical findings, confirm the stability of the proposed algorithm under perturbations, and demonstrate its superiority over a recent scheme from the literature. Full article

Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into the learning process. However, standard PINNs often suffer from training instabilities and unbalanced optimization when handling multi-term loss functions, especially in problems involving singular perturbations, fractional operators, or multi-scale behaviors. To address these limitations, we propose a novel gradient variance weighting physics-informed neural network (GVW-PINN), which adaptively adjusts the loss weights based on the variance of gradient magnitudes during training. This mechanism balances the optimization dynamics across different loss terms, thereby enhancing both convergence stability and solution accuracy. We evaluate GVW-PINN on three representative PDE models and numerical experiments demonstrate that GVW-PINN consistently outperforms the conventional PINN in terms of training efficiency, loss convergence, and predictive accuracy. In particular, GVW-PINN achieves smoother and faster loss reduction, reduces relative errors by one to two orders of magnitude, and exhibits superior generalization to unseen domains. The proposed framework provides a robust and flexible strategy for applying PINNs to a wide range of integer- and fractional-order PDEs, highlighting its potential for advancing data-driven scientific computing in complex physical 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

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

Many types of exact solutions to the truncated M-fractional classical Lonngren wave model are explored in this paper. The classical Lonngren wave model is a significant electronics equation. This model is used to explain the electronic signals within semiconductor materials, especially tunnel diodes. Through the application of a modified $(G^{\prime }/G^2)$-expansion technique and an extended sinh-Gordon equation expansion (EShGEE) method, we obtained various wave solutions, including periodic, kink, singular, dark, bright, and dark–bright types, among others. To ensure that the solutions in question are stable, linear stability analysis is also carried out. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. The obtained results are useful in various areas, including electronic physics, soliton physics, plasma physics, nonlinear optics, acoustics, etc. Both techniques are useful for solving nonlinear partial fractional differential equations. Both techniques provide exact solutions, which can be important for understanding complex phenomena. Both techniques are reliable and yield distinct types of exact soliton solutions. Full article

In this paper, we develop a systematic review of the existing literature on the mathematical modeling of several aspects of pollinators. We selected the MathSciNet and Wos databases and performed a search for the words “pollinator” and “mathematical model”. This search yielded a total of 236 records. After a detailed screening process, we retained 107 publications deemed most relevant to the topic of mathematical modeling in pollinator systems. We conducted a bibliometric analysis and categorized the studies based on the mathematical approaches used as the central tool in the mathematical modeling and analysis. The mathematical theories used to obtain the mathematical models were ordinary differential equations, partial differential equations, graph theory, difference equations, delay differential equations, stochastic equations, numerical methods, and other types of theories, like fractional order differential equations. Meanwhile, the topics were positive bounded solutions, equilibrium and stability analysis, bifurcation analysis, optimal control, and numerical analysis. We summarized the research findings and identified some challenges that could inform the direction of future research, highlighting areas that will aid in the development of future research. Full article

We present a hybrid parallel scheme for efficiently solving Caputo time-fractional partial differential equations (CTFPDEs) with integer-order spatial derivatives on multicore CPU and GPU platforms. The approach combines a second-order spatial discretization with the $L1$ time-stepping scheme and employs MATLAB parfor parallelization to achieve significant reductions in runtime and memory usage. A theoretical third-order convergence rate is established under smooth-solution assumptions, and the analysis also accounts for the loss of accuracy near the initial time $t=t_0$ caused by weak singularities inherent in time-fractional models. Unlike many existing approaches that rely on locally convergent strategies, the proposed method ensures global convergence even for distant or randomly chosen initial guesses. Benchmark problems from fractional biological models—including glucose–insulin regulation, tumor growth under chemotherapy, and drug diffusion in tissue—are used to validate the robustness and reliability of the scheme. Numerical experiments confirm near-linear speedup on up to four CPU cores and show that the method outperforms conventional techniques in terms of convergence rate, residual error, iteration count, and efficiency. These results demonstrate the method’s suitability for large-scale CTFPDE simulations in scientific and engineering applications. Full article

One of the important techniques for solving several partial differential equations is the residual power series method, which provides the approximate solutions of differential equations in power series form.In this work, we use Aboodh transform in the analogical structure of the residual power series method to obtain a new method called the Aboodh residual power series method (ARPSM). By using this technique, we calculate the coefficients of some power series solutions of time Caputo fractional Kawahara equations. To obtain analytical and numerical solutions for the TCFKEs, we use ARPSM, first with the approximate initial condition and then with the exact initial condition. We present ARPSM’s reliability, efficiency, and capability by graphically describing the numerical results for analytical solutions and by comparing our solutions with other solutions for the TCFKEs obtained using two alternative methods, namely, the residual power series method and the natural transform decomposition method. Full article

In this work, we introduce the conformabletriple Laplace–Sumudu transform (CTLST), a novel integral transform designed to solve both linear and nonlinear conformable FPDEs. This new approach builds on the recent development of the triple Laplace–Sumudu transform and incorporates the conformable derivative to extend its applicability to fractional models. We first present the foundational definitions and key properties of the CTLST, followed by its application to a variety of two- and three-dimensional conformable FPDEs. The effectiveness of the proposed method is demonstrated through several examples, where exact and approximate solutions are derived, illustrative 3D plots are presented, and symmetry analysis is employed to verify the obtained results. The CTLST provides a promising analytical tool for tackling complex conformable FPDEs in mathematical physics and engineering. Full article

Modeling risk evolution in financial networks presents both practical and theoretical challenges, particularly during periods of heightened systemic stress. This issue has gained urgency recently in China as it faces unprecedented financial strain, largely driven by structural shifts in the real estate sector and broader economic vulnerabilities. In this study, we combine Fractional-order Partial Differential Equations (FoPDEs) with network-based analysis methods, proposing a hybrid framework for capturing and modeling systemic financial risk, which is quantified using the $\Delta CoVaR$ algorithm. The FoPDEs model is formulated based on reaction–diffusion equations and discretized using the Caputo fractional derivative. Parameter estimation is conducted through a composite optimization strategy, and numerical simulations are carried out to investigate the underlying mechanisms and dynamic behavior encoded in the equations. For empirical evaluation, we utilize data from China’s financial and real estate sectors. The results demonstrate that our model achieves a Mean Relative Accuracy (MRA) of 95.5% for daily-frequency data, outperforming LSTM and XGBoost under the same conditions. For weekly-frequency data, the model attains an MRA of 91.7%, exceeding XGBoost’s performance of 90.25%. Further analysis of parameter dynamics and event studies reveals that the fractional-order parameter $\alpha$, which controls the memory effect of the model, tends to remain low when $\Delta CoVaR$ exhibits sudden surges. This suggests that the model assigns greater importance to past data during periods of financial shocks, capturing the persistence of risk dynamics more effectively. Full article

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is developed for a broad class of time-fractional system equations. In particular, we check the validity and effectiveness of the approach by providing three illustrative examples, confirming its accuracy and applicability in solving both linear and nonlinear fractional problems. 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

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article

The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the time-varying coefficient $\theta \left(t\right)$ in the following evolution equation:$$\left\{\begin{array}{c}{\displaystyle du(t,x)=(A_0+\theta \left(t\right)A_1)u(t,x)dt+d{\xi }^H(t,x),x\in [0,1],t\in (0,T],}\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill \end{array}\right.$$ where ${\xi }^H(t,x)$ is a cylindrical sub-fractional Brownian motion in ${\mathbb{L}}^2\left([0,T]\times [0,1]\right)$, and $A_0+\theta \left(t\right)A_1$ is a strongly elliptic differential operator. We obtain the asymptotic mean square error and the limiting distribution of the proposed estimator. These results are proved under some standard conditions on the kernel and some mild conditions on the model. Finally, we give an application for the confidence interval construction. Full article

This work analyzes the aero-elastic oscillations of cantilever beams reinforced by carbon nanotubes (CNTs). Four different distributions of single-walled CNTs are assumed as the reinforcing phase, in the thickness direction of the polymeric matrix. A modified third-order piston theory is used as an accurate tool to model the supersonic air flow, rather than a first-order piston theory. The nonlinear dynamic equation governing the problem accounts for Von Kármán-type nonlinearities, and it is derived from Hamilton’s principle. Then, the Galerkin decomposition technique is adopted to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation. This is solved analytically according to a multiple time scale method. A comprehensive parametric analysis was conducted to assess the influence of CNT volume fraction, beam slenderness, Mach number, and thickness ratio on the fundamental frequency and lateral dynamic deflection. Results indicate that FG-X reinforcement yields the highest frequency response and lateral deflection, followed by UD and FG-A patterns, whereas FG-O consistently exhibits the lowest performance metrics. An increase in CNT volume fraction and a reduction in slenderness ratio enhance the system’s stiffness and frequency response up to a critical threshold, beyond which a damped beating phenomenon emerges. Moreover, higher Mach numbers and greater thickness ratios significantly amplify both frequency response and lateral deflections, although damping rates tend to decrease. These findings provide valuable insights into the optimization of CNTR composite structures for advanced aeroelastic applications under supersonic conditions, as useful for many engineering applications. Full article