Search Results (244)

Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we show that the QTT approach can achieve logarithmic scaling in memory and computational cost for linear PDEs when the relevant QTT ranks remain moderate. We also develop QTT space-time formulations that treat time as an additional dimension, allowing the full temporal evolution to be represented and solved globally rather than through sequential time stepping. For the nonlinear Burgers equation, we study both time-stepping and a frozen-coefficient space-time Picard scheme in QTT form, and report empirical convergence behavior on smooth one-dimensional viscous test problems. Additionally, we present a proof-of-concept data-driven workflow within the quantum-inspired framework, in which sampled source data are interpolated into QTT form and then incorporated directly into the structured PDE solver. Full article

We present and analyze a weighted family of iterative methods for solving systems of nonlinear equations. The proposed schemes are constructed as a generalization of the Singh–Sharma fifth-order method by incorporating suitable weight functions into the correction step, thereby generating a flexible class of methods that includes the original scheme as a special case. Sufficient conditions on the weight functions are established to guarantee fifth-order local convergence, and the resulting error equation shows how the weights influence the leading error term. Several admissible choices are presented to illustrate the versatility of the family. The practical performance of the proposed variants is investigated on a collection of large-scale nonlinear systems. Furthermore, the family is applied to the nonlinear algebraic system obtained from the finite-difference discretization of a stationary one-dimensional viscous Burgers problem. Numerical experiments indicate that the proposed methods provide a competitive and accurate alternative for solving nonlinear systems of this type. 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

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena. Full article

Modelling plays a critical role in many engineering applications. Partial differential equations (PDEs) are ubiquitous, describing various physical phenomena such as fluid flow, electromagnetism, and quantum mechanics. Although some of these equations have analytical solutions, many require high-fidelity simulations of parametric PDEs. In general, high-fidelity simulations are computationally expensive and often infeasible for real-time or multi-query applications. This challenge has led to the development of reduced-order models (ROMs). Over the past few decades, ROMs have emerged as a practical solution for simulating, controlling, and optimizing large-scale and complex dynamical systems. This paper introduces a novel Calibrated Intrusive Reduced-Order Modelling (CIROM) approach for the efficient and accurate simulation of the one-dimensional Burgers’ equation, employed as a canonical benchmark because it is a simplified fundamental partial differential equation that captures the behaviour of many real-world phenomena. The proposed method, combining the strengths of proper orthogonal decomposition (POD) and long short-term memory (LSTM) networks, effectively reduces computational complexity while addressing inherent instabilities in classical reduced-order models. Unlike traditional POD-ROMs, which often suffer from error accumulation and instability at high Reynolds numbers, the CIROM employs an iterative LSTM-based error correction mechanism to learn and compensate for truncation and projection errors. This study is benchmark-oriented and does not aim to provide a general PDE solver. The performance of the proposed method is rigorously evaluated across a broad range of Reynolds numbers, including interpolation and extrapolation scenarios, demonstrating robust extrapolation within moderate ranges. Comprehensive numerical experiments confirm that the CIROM outperforms both pure intrusive ROMs and purely data-driven LSTM models in terms of prediction accuracy, stability, and computational cost. Full article

In this paper, we study the dynamical analysis and solutions of the fractional Benjamin–Bona–Mahony–Burger equation. We demonstrate various derived solutions using different definitions of fractional derivatives, namely the $\beta$-derivative, conformable derivative, and M-truncated derivative, to examine their kinetic characteristics. Firstly, we find the solution of the fractional Benjamin–Bona–Mahony–Burger equation using two different approaches. We then discuss the effects of the fractional derivative on the solutions using 3D graphical discussion. Finally, we discuss the dynamical analysis using sensitivity and chaos analysis. We also discuss the chaos analysis using permutation entropy, 2D and 3D phase portrait, fractal dimension, time analysis, return map, Lyapunov exponent, and multistability through Poincare map and basins of attraction. To explore a diverse range of phenomena across the fields of physical science and engineering, this study highlights the computational strength and flexibility of the proposed method. Full article

We develop a unified Koopman–von Neumann (KvN) operator and Weyl–Wigner phase-space framework for inviscid ideal (barotropic) Euler flows. Our approach reformulates the nonlinear fluid dynamics as a linear KvN evolution on an enlarged field phase space, thereby enabling us to apply tools developed for quantum mechanics (Weyl quantization, Moyal ⋆-products, and Wigner functionals) to a classical fluid. We construct the appropriate KvN generator (including the required Jacobian term for unitarity) and derive the evolution equation for the corresponding Wigner functional. This framework clarifies when the classical Liouville (Vlasov) description is exact—namely, in quadratic or linear regimes where the Moyal bracket reduces to the Poisson bracket—and when higher-order quantum-like corrections become significant in fully nonlinear regimes. As an analytic example, we obtain a closed-form Wigner solution for a one-dimensional Burgers flow (pressureless Euler) and verify, term by term, that it reproduces the expected Liouville transport (with distributional contributions at the shock). We also compare the phase-space approach with a kinetic (Vlasov–monokinetic) formulation and outline the extension of the framework to three-dimensional flows using a Clebsch variable representation. Full article

The current manuscript presents an application of the Sumudu decomposition method (SDM) in efficiently tackling the systems of coupled nonlinear partial fractional differential equations. The technique combines the strengths of the Adomian decomposition method and the Sumudu transform, enabling the transformation of complex systems into rapidly converging series solutions. The efficacy of the technique is then portrayed on various nonlinear coupled fractional models, where approximate solutions are successfully obtained. Furthermore, the computational results indicate efficient numerical performance of the proposed approach for the cases considered. Certainly, the study’s results demonstrate that SDM is an effective and reliable technique for solving the examined class of fractional-order systems. Full article

In the present study, we propose a new nonlinear finite difference method for solving the generalized nonlinear time-fractional Burgers-type equation, which can achieve fourth-order accuracy in the spatial direction. The L1 formula is employed to discretize the time-fractional derivative on a graded mesh. Spatial discretization is accomplished by introducing a nonlinear fourth-order difference operator and a linear compact difference operator, and ultimately a nonlinear difference scheme with a temporal accuracy of order $2-\alpha$ and a spatial accuracy of the fourth order is deduced. For the proposed difference scheme, the existence and boundedness of its solution have been theoretically verified; meanwhile, combined with the cut-off function method, the uniqueness and convergence of the solution to this scheme are further proved. The optimal convergence result is attained under the $L_2$ norm. Eventually, two numerical examples are provided, both of which match the theoretical analysis well. Full article

This study presents a systematic comparative benchmark between two distinct paradigms for solving nonlinear partial differential equations (PDEs): the data-driven Physics-Informed Neural Networks (PINNs) and the analytical Homotopy Analysis Method (HAM). We apply both methods to a unified family of canonical PDEs, the Burgers, Huxley, Fisher, Burgers–Huxley, and Burgers–Fisher equations, under identical problem setups, domain discretization, and validation metrics. PINNs incorporate physical laws directly into neural network training by minimizing a loss function that enforces PDE residuals, yielding physically consistent solutions even for strongly nonlinear problems. HAM provides approximate analytical solutions using a unified framework, and the same initial guess, auxiliary linear operator, and auxiliary function across all equations despite their distinct nonlinearities. The controlled, consistent application of both methods enables a fair, reproducible comparison across this equation family. The results provide a quantitative performance map under identical conditions, delineating when PINNs (high accuracy, long-term stability, and generalization capability) are preferable, versus when HAM (computational speed, short-term analytic approximation, and lower memory footprint) offers advantages. While the finite radius of convergence of the truncated HAM series is theoretically expected, our controlled comparison quantifies for the first time how this degradation varies across equation types, revealing that the choice between methods depends on specific problem requirements including error tolerance, available computational resources, and temporal horizon. The novelty lies not in solving each equation individually, but in deriving a performance taxonomy that systematically connects equation features (shocks, stiffness, and reaction–diffusion coupling) to optimal solver choice—providing previously unavailable, evidence-based guidance for the scientific computing community. This study establishes the first rigorous, controlled comparative benchmark between analytic and data-driven PDE solvers across a spectrum of nonlinearities, providing a reproducible baseline for future hybrid scientific machine learning solvers. 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

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). 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 each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation. Full article

Two exact solutions are constructed for viscous compressible gas dynamics in two and three dimensions. The first is a steady vortex, with explicit solutions for the full Navier–Stokes system of velocity, density, temperature and pressure. In contrast, the second is a time-dependent radial solution to the 3D vector Burgers’ equation, with a constant injection rate from a spherical interior surface. That solution is shock-like at low Reynolds numbers. In both cases, expressions are given for the local density of entropy production. 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