Search Results (827)

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

As a pivotal carrier of emerging human–computer interaction technologies, artificial intelligence (AI) conversational agents (CAs) hold critical significance for research on the mechanisms of users’ continuance usage behaviour, which is essential for technological optimization and commercial transformation. However, the differential impact pathways of multidimensional perceived interactivity on continuance usage intention, particularly the synergistic mechanisms between technical and affective dual-path dimensions, remain unclear. This study investigates the personalized AI-based CAs project “Dialogue with Great Souls,” launched on a Chinese social platform, using survey data from 305 users. A hybrid approach combining partial least squares structural equation modelling (PLS-SEM) and artificial neural networks (ANN) was employed for empirical analysis. The results indicate that technical dimensions, such as control and responsiveness, are key factors influencing trust, while affective interactive dimensions, including communication, personalization, and playfulness, significantly affect social presence, thereby shaping users’ continuance usage intention. ANN results corroborated most PLS-SEM findings but revealed inconsistencies in the predictive importance of personalization and communication on social presence, highlighting the complementary nature of linear and nonlinear interaction mechanisms. By expanding the interactivity model and adopting a hybrid methodology, this study constructs a novel framework for AI CAs. The empirical findings suggest that developers should strengthen socio-emotional bonds in anthropomorphic interactions while ensuring technical credibility to enhance users’ continuance usage intention. This research not only advances theoretical perspectives on the integration of technical and affective dimensions in agent systems but also provides practical recommendations for optimizing the design and development of AI CAs. Full article

This paper presents a Neural Network-Based Symbolic Computation Algorithm (NNSCA) for solving the (2+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation. By combining neural networks with symbolic computation, NNSCA bypasses traditional method limitations, deriving and visualizing exact solutions. It designs neural network architectures, converts the PDE into algebraic constraints via Maple, and forms a closed-loop solution process. NNSCA provides a general paradigm for high-dimensional nonlinear PDEs, showing great application potential. Full article

It is well known that nonlinear partial differential equations (NLPDEs) can only be solved numerically and that fourth-order NLPDEs in their derivatives require unconventional methods. This paper explains spectral numerical methods for obtaining a numerical solution by Fast Fourier Transform (FFT), implemented under Python in tis version 3.1 and their libraries (NumPy, Tkinter). Examples of NLPDEs typical of Condensed Matter Physics to be solved numerically are the conserved Cahn–Hilliard, Swift–Hohenberg and conserved Swift–Hohenberg equations. The last two equations are solved by the first- and second-order exponential integrator method, while the first of these equations is solved by the conventional FFT method. The Cahn–Hilliard equation, a phase-field model with an extended Ginzburg–Landau-like functional, is solved in two-dimensional (2D) to reproduce the evolution of the microstructure of an amorphous alloy Ce₇₅Al_{25 − x}Gax, which is compared with the experimental micrography of the literature. Finally, three-dimensional (3D) simulations were performed using numerical solutions by FFT. The second-order exponential integrator method algorithm for the Swift–Hohenberg equation implementation is successfully obtained under Python by FFT to simulate different 3D patterns that cannot be obtained with the conventional FFT method. All these 2D/3D simulations have applications in Materials Science and Engineering. Full article

This study focuses on the nonlinear partial differential equation known as the Lonngren wave equation, which plays a significant role in plasma physics, nonlinear wave propagation, and astrophysical research. By applying a suitable wave transformation, the nonlinear model is reduced to an ordinary differential equation. Analytical wave solutions of the Lonngren wave equation are then derived using the extended direct algebraic method. The physical behavior of these solutions is illustrated through 2D, 3D, and contour plots generated in Mathematica. Finally, the stability analysis of the Lonngren wave equation is discussed. Full article

This study conducts an in-depth analysis of the dynamical characteristics and solitary wave solutions of the integrable Zhanbota-IIA equation through the lens of planar dynamic system theory. This research applies Lie symmetry to convert nonlinear partial differential equations into ordinary differential equations, enabling the investigation of bifurcation, phase portraits, and dynamic behaviors within the framework of chaos theory. A variety of analytical instruments, such as chaotic attractors, return maps, recurrence plots, Lyapunov exponents, Poincaré maps, three-dimensional phase portraits, time analysis, and two-dimensional phase portraits, are utilized to scrutinize both perturbed and unperturbed systems. Furthermore, the study examines the power frequency response and the system’s sensitivity to temporal delays. A novel classification framework, predicated on Lyapunov exponents, systematically categorizes the system’s behavior across a spectrum of parameters and initial conditions, thereby elucidating aspects of multistability and sensitivity. The perturbed system exhibits chaotic and quasi-periodic dynamics. The research employs the maximum Lyapunov exponent portrait as a tool for assessing system stability and derives solitary wave solutions accompanied by illustrative visualization diagrams. The methodology presented herein possesses significant implications for applications in optical fibers and various other engineering disciplines. Full article

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant. Full article

In a 1987 article of the last author dedicated to the memory of a pioneer of classical plasticity Aris Philips of Yale, the last author outlined three examples of self-organization during plastic deformation in metals: persistent slip bands (PSBs), shear bands (SBs) and Portevin Le Chatelier (PLC) bands. All three have been observed and analyzed experimentally for a long time, but there was no theory to capture their spatial characteristics and evolution in the process of deformation. By introducing the Laplacian of dislocation density and strain in the standard constitutive equations used for these phenomena, corresponding mathematical models and nonlinear partial differential equations (PDEs) for the governing variable were generated, the solution of which provided for the first time estimates for the wavelengths of the ladder structure of PSBs in Cu single crystals, the thickness of stationary SBs in metals and the spacing of traveling PLC bands in Al-Mg alloys. The present article builds upon the 1987 results of the aforementioned three examples of self-organization in plasticity within a unifying internal length gradient (ILG) framework and expands them in 2 major ways by: (i) introducing the effect of stochasticity and (ii) capturing statistical characteristics when PDEs are absent for the description of experimental observations. The discussion focuses on metallic systems, but the modeling approaches can be used for interpreting experimental observations in a variety of materials. Full article

In this research paper, the negative order Korteweg–de Vries equation expressed as nonlinear partial differential equation, firstly introduced by Wazwaz, is solved for the exact Jacobi elliptic function solution. For this purpose, the Jacobi elliptic function scheme, one of the direct algebraic methods, was used. The obtained exact solutions of the negative-order Korteweg–de Vries equation, a symmetry evolution equation, contains the combination of Jacobi elliptic functions and incomplete elliptic integral of second function. The three unique families of exact solutions are classified and presented. The degeneration of the obtained Jacobi elliptic function solutions into various solitons, periodic and rational solutions, is reported using the modulus transformation of Jacobi elliptic function solutions. The necessary condition existence of certain Jacobi elliptic function solutions is presented. The two-dimensional graphs for certain Jacobi elliptic function solutions are drawn to show the variation in wave propogation with respect to velocity and modulus. The non-existence of certain Jacobi elliptic function solutions for negative-order Korteweg–de Vries equations is reported. Finally, the obtained solutions were compared with the previously obtained solutions of negative-order Korteweg–de Vries equation. Full article

This study presents the complete analysis of a (2 + 1)-dimensional nonlinear wave-type partial differential equation with anisotropic power-law nonlinearities and a general power-law source term, which arises in physical domains such as fluid dynamics, nonlinear acoustics, and wave propagation in elastic media, yet their symmetry properties and exact solution structures remain largely unexplored for arbitrary nonlinearity exponents. To fill this gap, a complete Lie symmetry classification of the equation is performed for arbitrary values of m and n, providing all admissible symmetry generators. These generators are then employed to systematically reduce the PDE to ordinary differential equations, enabling the construction of exact analytical solutions. Traveling wave and soliton solutions are derived using Jacobi elliptic function and sine-cosine methods, revealing rich nonlinear dynamics and wave patterns under anisotropic conditions. Additionally, conservation laws associated with variational symmetries are obtained via Noether’s theorem, yielding invariant physical quantities such as energy-like integrals. The results extend the existing literature by providing, for the first time, a full symmetry classification for arbitrary m and n, new families of soliton and traveling wave solutions in multidimensional settings, and associated conserved quantities. The findings contribute both computationally and theoretically to the study of nonlinear wave phenomena in multidimensional cases, extending the catalog of exact solutions and conserved dynamics of a broad class of nonlinear partial differential equations. Full article

Partial differential equations are an integral part of modern scientific development. Hyperbolic partial differential equations are encountered in many fields and have many applications—both linear and nonlinear types, with some being semilinear and quasilinear. In this paper, a family of implicit numerical schemes for solving hyperbolic partial differential equations is derived, utilizing finite differences and tridiagonal sweep. Through the discrete Fourier transform, a necessary and sufficient condition for convergence is proven for the linear version of the family of difference schemes, expanding the known results on boundary conditions that ensure convergence. Numerical verification confirms the found condition. A series of experiments on different boundary conditions and semilinear hyperbolic PDEs show that the same condition seems to also hold in those cases. In view of the results, an optimal subset of the family is found. A comparison between the implicit schemes and an explicit analogue is conducted, demonstrating the gained efficiency of the implicit schemes. Full article

This paper presents a computational framework for resolving a nonlinear extension of the Black–Scholes partial differential equation that accounts for transaction costs through a volatility function dependent on the Gamma of the option price. A meshfree radial basis function-generated finite difference procedure is developed using a modified multiquadric kernel. Analytical weight formulas for first- and second-order differentiations are discussed on 3-node stencils for both uniform and non-uniform point distributions. The proposed method offers an efficient scheme suitable for accurately pricing European scenarios when nonlinear transaction cost effects. 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

In cyber-physical systems governed by nonlinear partial differential equations (PDEs), real-time control is often limited by sparse sensor data and high-dimensional system dynamics. Deep reinforcement learning (DRL) has shown promise for controlling such systems, but training DRL agents directly on full-order simulations is computationally intensive. This paper presents a sensor-driven, non-intrusive reduced-order modeling (NIROM) framework called FAE-CAE-LSTM, which combines convolutional and fully connected autoencoders with a long short-term memory (LSTM) network. The model compresses high-dimensional states into a latent space and captures their temporal evolution. A DRL agent is trained entirely in this reduced space, interacting with the surrogate built from sensor-like spatiotemporal measurements, such as pressure and velocity fields. A CNN-MLP reward estimator provides data-driven feedback without requiring access to governing equations. The method is tested on benchmark systems including Burgers’ equation, the Kuramoto–Sivashinsky equation, and flow past a circular cylinder; accuracy is further validated on flow past a square cylinder. Experimental results show that the proposed approach achieves accurate reconstruction, robust control, and significant computational speedup over traditional simulation-based training. These findings confirm the effectiveness of the FAE-CAE-LSTM surrogate in enabling real-time, sensor-informed, scalable DRL-based control of nonlinear dynamical systems. 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