Search Results (78)

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

The KdV–Burgers equation $u_t+u{u}_x-v{u}_{xx}+\beta u_{xxx}=0$ models the interplay of nonlinearity, dispersion and dissipation. Through the potential $u=v_x$, a Lagrangian density is constructed for the resulting potential system. Application of Noether’s theorem yields an infinite-dimensional symmetry $V=f(x,t)\partial v$, where f satisfies the linearized equation $f_t-v{f}_{xx}+\beta f_{xxx}=0$. This symmetry generates an infinite family of continuity equations of the form $D_{t}T+D_{x}X=0$ with $T=f(x,t)$ and $X=-f_{x}u+v{f}_x-\beta f_{xx}$. For ν > 0, these relations constitute linear potential balance laws rather than classical conservation laws, as the density T depends explicitly on the auxiliary field f and on time. In the inviscid limit ν = 0, the family reduces to the classical KdV conservation hierarchy. High-accuracy spectral simulations confirm the validity of these identities up to discretization error. Full article

This study presents a fourth-order implicit compact finite difference scheme for the Benjamin–Bona–Mahony–Burgers (BBMB) equation, a nonlinear long-wave equation describing the dynamics of various wave phenomena. By employing an order-reduction framework via an auxiliary variable, we construct a compact difference scheme that yields a nonlinear algebraic system with a narrowly banded structure. Because the continuous BBMB model is governed by an underlying conservation law, the proposed numerical method is designed to preserve this structural property in the discrete sense. The discrete conservation, boundedness, and unique solvability of the scheme are firmly established, and an optimal discrete maximum norm error estimate is derived. Finally, comprehensive numerical experiments are conducted to validate both the conservative properties and the theoretical order of accuracy of the proposed scheme. Full article

The Korteweg–de Vries (KdV) equation is widely known as a canonical model for weakly nonlinear and weakly dispersive waves, notably and historically for water waves in shallow depths. Being integrable, it has a rich solution set of interacting solitary and periodic waves. Recently, we extended it with several forcing/friction terms to describe the evolution of wind-driven water wave packets in shallow water. The outcome is a modified KdV–Burgers equation, whose relevant solutions are principally solitary wave trains forming a soliton gas. In this article that is extended further by allowing the water depth to be slowly spatially varying, and introducing a basic horizontal current, also slowly spatially varying. The outcome is a modified KdV–Burgers equation with spatially slowly varying coefficients. We adapt the Whitham modulation theory for a slowly varying solitary wave train, allowing for the prediction of wave amplitude growth/decay due to a combination of the slowly varying background and the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to exhibit and validate the modulation theory. Full article

Structure-preserving numerical methods are well-established for purely conservative or purely dissipative systems but remain underdeveloped for mixed-type equations coupling dispersion, dissipation, and nonlinearity. We investigate the Korteweg–de Vries–Burgers equation as a canonical model of this class. We develop a geometric covering method based on nonlocal symmetries that lifts the equation to an extended manifold, enabling exact conservation law preservation. As a pedagogical counterexample, we also analyze a naive recursive approximation. Both methods are implemented using sixth-order compact finite differences and fourth-order Runge–Kutta (RK4) time integration. Numerical experiments on sinusoidal waves, two-soliton collisions, and perturbed traveling waves show that the covering method reduces numerical dissipation by 50% and phase error by 90% relative to a standard second-order scheme, achieving one to two orders of magnitude higher accuracy. Mass and momentum are conserved to machine precision (below ${10}^{-14}$), and soliton amplitudes are preserved to within 0.3% after collision, with only 15% computational overhead. The framework offers a generalizable template for embedding nonlocal symmetries into high-order numerical methods for nonlinear wave equations. Full article

Due to excellent performance, polytetrafluoroethylene (PTFE), being sealing material, is widely used in chemical engineering, aerospace engineering, mechanical engineering, civil engineering, energy engineering and other sectors. However, due to obvious temperature drops in supplying or storing fluids, the mechanical behavior of PTFE under cryogenic conditions is still unclear. In this study, the creep mechanical performance of PTFE gaskets after cryogenic aging in liquid oxygen is experimentally investigated. The circular PTFE gasket samples are immersed into liquid oxygen for cryogenic aging treatment. The universal testing machine is utilized for material mechanic tests. Three different load levels, including 10 MPa, 15 MPa and 20 MPa, are designed and accounted for. It is found that the creep strain of PTFE exhibits three typical stages, namely the initial rapid increase phase, transition phase with a reducing growth rate, and stable linear growth phase. Moderate cryogenic immersion aging can effectively improve the creep resistance of PTFE, but excessive aging treatments will lead to mechanical property degradation of PTFE. The Burgers life prediction model is improved by introducing a nonlinear correction term, which can accurately predict the creep behavior of PTFE under different aging states. The present study can provide experimental evidence and a theoretical basis for a deep understanding of the mechanical response of PTFE materials under extreme cryogenic intermittent service conditions. Full article

Utilizing waste tire crumb rubber to modify asphalt enhances the damping and noise reduction performance of pavements. This study systematically evaluated the damping performance of crumb rubber–modified asphalt over a wide temperature range. A high-temperature damping index based on the loss factor and a low-temperature energy dissipation ratio derived from the Burgers model were proposed for quantitative characterization. The results show that damping performance is primarily controlled by temperature and crumb rubber content, while particle size plays a secondary role. Increasing crumb rubber content markedly improves damping performance. When the crumb rubber content exceeds 20%, the damping temperature stability, peak loss factor, and its retention tend to level off, whereas the low-temperature enhancement diminishes when the content exceeds 25%. Accordingly, the robust combinations are 80-mesh (≈180 μm) with 20% content for high-temperature conditions and 80-mesh with 25% content for low-temperature conditions. Multivariate nonlinear regression models achieved high predictive accuracy (R² = 0.927 and 0.985). Microscopic analyses indicate that crumb rubber increases constrained interfacial phases and system viscosity, and partial particle exposure at 20–25% further enhances interfacial friction and energy dissipation, consistent with the observed macroscopic damping behavior. These findings provide a theoretical basis for robust, noise-reducing pavements. Full article

The coupled nonlinear system of fractional Boussinesq–Burger equations that may be utilized to model the propagation of shallow water waves is solved in this study using a novel numerical approach. The fractional derivatives in Caputo–Fabrizio and Atangana–Baleanu manner are executed in the system under consideration. The exact solutions of the proposed nonlinear fractional system are shown in the classical scenario of fractional order at $\mathrm{ß}=1$, whereas the approximate solutions are derived using the natural decomposition method. The series solution is generated such that it is simple to compute. Our results are compared with the exact results which clearly show that the suggested approach solutions quickly converge to the known accurate results. We acquire some analysis of the absolute error by comparing the approximate values with their corresponding precise solutions throughout the provided computations. Numerical and graphical simulations are used to confirm the usefulness of the suggested approach, and the outcomes are compared with well-known methods like the fractional decomposition method (FDM) and Laplace residual power series method (LRPSM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of the proposed system. The obtained outcomes ensure that the suggested approach is more effective and examines the highly nonlinear problems arising in engineering and science. Full article

The Burgers–Huxley equation is a nonlinear partial differential equation that incorporates convective, diffusive and reactive effects and arises in various reaction–diffusion and fluid flow models. In this paper, a numerical method based on the method of lines is proposed for its solution. The spatial derivatives are approximated using a third-order finite difference scheme, which converts the governing partial differential equation into a system of ordinary differential equations. The resulting semi-discrete system is solved in time using the classical fourth-order Runge–Kutta method. The stability and convergence properties of the proposed scheme are analyzed to establish its numerical reliability. Several numerical experiments are presented to illustrate the accuracy and efficiency of the method. The computed results confirm that the proposed approach provides accurate and stable solutions for the Burgers–Huxley equation. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article

Physics–Informed Neural Networks (PINNs) have demonstrated efficacy in solving both forward and inverse problems for nonlinear partial differential equations (PDEs). However, they frequently struggle to accurately capture multiscale physical features, particularly in regions exhibiting sharp local variations such as shock waves and discontinuities, and often suffer from optimization difficulties in complex loss landscapes. To address these issues, we propose LocRes–PINN, a physics–informed neural network framework that integrates local awareness mechanisms with residual learning. This framework integrates a radial basis function (RBF) encoder to enhance the perception of local variations and embeds it within a residual backbone to facilitate stable gradient propagation. Furthermore, we incorporate a residual–based adaptive refinement strategy and an adaptive weighted loss scheme to dynamically focus training on high–error regions and balance multi–objective constraints. Numerical experiments on the Extended Korteweg–de Vries, Navier–Stokes, and Burgers equations demonstrate that LocRes–PINN reduces relative prediction errors by approximately 12% to 67% compared to standard benchmarks. The results also verify the model’s robustness in parameter identification and noise resilience. Full article

Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent Scientific Foundation Models (SFMs) aim to alleviate this cost by learning universal surrogates from large collections of simulated systems, yet they typically rely on monolithic architectures with limited interpretability and high pretraining expense. In this work, we introduce Compositional Neural Operators (CompNO), a compositional neural operator framework for parametric PDEs. Instead of pretraining a single large model on heterogeneous data, CompNO first learns a library of Foundation Blocks, where each block is a parametric Fourier neural operator specialized to a fundamental differential operator (e.g., convection, diffusion, nonlinear convection). These blocks are then assembled, via lightweight Adaptation Blocks, into task-specific solvers that approximate the temporal evolution operator for target PDEs. A dedicated boundary-condition operator further enforces Dirichlet constraints exactly at inference time. We validate CompNO on one-dimensional convection, diffusion, convection–diffusion and Burgers’ equations from the PDEBench suite. The proposed framework achieves lower relative $L^2$ error than strong baselines (PFNO, PDEFormer and in-context learning-based models) on linear parametric systems, while remaining competitive on nonlinear Burgers’ flows. The model maintains exact boundary satisfaction with zero loss at domain boundaries, and exhibits robust generalization across a broad range of Péclet and Reynolds numbers. These results demonstrate that Compositional Neural Operators provide a scalable and physically interpretable pathway towards foundation models for PDEs. Full article

Considering the complexity and diversity of water-rich soft soil strata, indoor triaxial shear tests and creep tests were conducted on soft soil to explore its deformation law and creep characteristics. To address the nonlinear characteristics of soft soil creep, a nonlinear pot element was proposed and substituted for the two linear pot elements in the Burgers model, thus establishing an unsteady parametric Burgers model. The one-dimensional creep equation of the unsteady Burgers model was derived, theoretically determining that the unsteady model can describe three stages of creep. Based on this, the creep equation of the unsteady Burgers model was extended to a three-dimensional stress state, and the triaxial compression creep test curves of Ningbo soft soil were fitted and parameters identified. The above model was derived from a three-dimensional finite difference scheme suitable for numerical solution in FLAC3D. A custom constitutive creep model was developed in FLAC3D, and the non-accelerated creep stage and accelerated creep stage of the improved model were analyzed to verify the accuracy and reliability of the constitutive model. The results show that the numerical simulation results and the indoor creep test results are in good agreement in terms of strain increment and the creep change curve, which confirms the effectiveness and applicability of the proposed unsteady Burgers creep constitutive model and its secondary development application. Full article

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the $L1-2$ formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the $L_2$ norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of $O({\tau }^{1+\alpha }+{\tau }^2{t}_n^{\alpha -2})$ for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations. Full article

This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than $0.5\%$ in the Burgers limit and $3.2\%$ in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article