Burgers Equation Research Articles

Solving viscid Burgers equation by Adomian decomposition Method (ADM), Regular Pertubation Method (RPM) and Homotopy Perturbation Method (HPM)

In this paper the Adomian decomposition Method (ADM), Regular Pertubation Method (RPM) and the Homotopy Perturbation Method (HPM) are used to study Burgers equation. Then we compare the solutions obtained by these three methods.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Some asymptotic profiles for the viscous Burgers equation with infinite mass

Some asymptotic profiles for the viscous Burgers equation with infinite mass

Neuroscience inspired neural operator for partial differential equations

We propose, in this paper, a Variable Spiking Wavelet Neural Operator (VS-WNO), which aims to bridge the gap between theoretical and practical implementation of Artificial Intelligence (AI) algorithms for mechanics applications. With recent developments like the introduction of neural operators, AI's potential for being used in mechanics applications has increased significantly. However, AI's immense energy and resource requirements are a hurdle in its practical field use case. The proposed VS-WNO is based on the principles of spiking neural networks, which have shown promise in reducing the energy requirements of the neural networks. This makes possible the use of such algorithms in edge computing. The proposed VS-WNO utilizes variable spiking neurons, which promote sparse communication, thus conserving energy, and its use is further supported by its ability to tackle regression tasks, often faced in the field of mechanics. Various examples dealing with partial differential equations, like Burger's equation, Allen Cahn's equation, and Darcy's equation, have been shown. Comparisons have been shown against wavelet neural operator utilizing leaky integrate and fire neurons (direct and encoded inputs) and vanilla wavelet neural operator utilizing artificial neurons. The results produced illustrate the ability of the proposed VS-WNO to converge to ground truth while promoting sparse communication.

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Non-local time evolution equation with singular integral and its application to traffic flow model

We consider an integro-differential equation model for traffic flow, which extends the Burgers equation model. To derive exact solutions to the equation, we first examine the features of integrable integro-differential equations and find that they are obtained through the residue formula from integrable equations in complex domains. Using this approach, we construct several new integrable equations with double singular integrals and elliptic singular integrals. Then, we discuss the traffic model and show that it exhibits interactions between free-flow and congested regions, as well as the deadlock phenomenon, depending on the non-locality parameter.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Analysis of nonlinear Burgers equation with time fractional Atangana-Baleanu-Caputo derivative

This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo (ABC) derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional (ABC) derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.

Non-intrusive reduced-order model for time-dependent stochastic partial differential equations utilizing dynamic mode decomposition and polynomial chaos expansion

In this study, we present a novel non-intrusive reduced-order model (ROM) for solving time-dependent stochastic partial differential equations (SPDEs). Utilizing proper orthogonal decomposition (POD), we extract spatial modes from high-fidelity solutions. A dynamic mode decomposition (DMD) method is then applied to vertically stacked matrices of projection coefficients for future prediction of coefficient fields. Polynomial chaos expansion (PCE) is employed to construct a mapping from random parameter inputs to the DMD-predicted coefficient field. These lead to the POD–DMD–PCE method. The innovation lies in vertically stacking projection coefficients, ensuring time-dimensional consistency in the coefficient matrix for DMD and facilitating parameter integration for PCE analysis. This method combines the model reduction of POD with the time extrapolation strengths of DMD, effectively recovering field solutions both within and beyond the training time interval. The efficiency and time extrapolation capabilities of the proposed method are validated through various nonlinear SPDEs. These include a reaction–diffusion equation with 19 parameters, a two-dimensional heat equation with two parameters, and a one-dimensional Burgers equation with three parameters.

Complex shock structure in multi-species dusty plasma with nonextensive electrons and two-temperature nonthermal ions

A theoretical investigation has been made to explore the behavior of nonplanar (cylindrical and spherical) electrostatic shock waves in a dusty plasma system that consists of arbitarily charged dust particles, negatively charged heavy ions following Cairn’s distribution, positively charged ions with two different temperatures, and nonextensive electrons. The reductive perturbation technique is used to derive a modified Burgers equation analytically. Analytical analysis shows that the characteristics of the nonplanar shock waves, including their polarity, amplitude, width, and phase speed, undergo significant alterations due to factors such as the charges on dust particles, the number density of particles, the temperatures of heavy and light ions, nonextensive electron behavior, and dust kinematic viscosity. Furthermore, this study found shock structures with either a positive or negative potential, depending on the critical value of the proportion of ions to the density of dust particles. The findings maybe useful in understanding the characteristics of shock waves both in space plasma environments and laboratory plasmas.

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

Inviscid Burgers as a degenerate elliptic problem

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Data-driven identification of stable sparse differential operators using constrained regression

Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics forecasting problems. In this article, we propose a novel methodology for learning differential operators that are theoretically linearly stable and have sparsity patterns of common discretization schemes. These differential operators are obtained by solving a constrained regression problem, involving local constraints to ensure the linear stability of the global dynamical system. We further extend this approach for learning nonlinear differential operators by determining linear stability constraints for linearized equations around an equilibrium point. The applicability of the proposed method is demonstrated for both linear and nonlinear partial differential equations such as 1-D scalar advection-diffusion equation, 1-D Burgers equation and 2-D advection equation. The results indicated that solutions to constrained regression problems with linear stability constraints provide accurate and linearly stable sparse differential operators.

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

A third (fourth)-order computational scheme for 2D Burgers-type nonlinear parabolic PDEs on a nonuniformly spaced grid network

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

Comparative Study of Solutions of Fractional Order Mixed KdV Burger’s Equation

Objective: The aim of this study is to obtain numerical solutions of fractional order mixed KdV Burger’s equation using iterative methods. Methodology: The study is centered on a time fractional mixed KdV Burger’s equation in the Caputo sense. The approximate solutions of this equation are obtained using Adomian decomposition method and Homotopy Perturbation method. Also, a Python code is developed to obtained numerical solutions and simulate graphically. Finding: The obtained solutions are compared with the exact solution and presented graphically using Python program for better analysis. Novelty: These methods shall be used to obtain approximate solution of such type of non-linear fractional order differential equations. Keywords: Fractional Differential Equation, Caputo Derivative, ADM, HPM, KdV Burger equation, Python

Exact solutions of the generalized Huxley–Burgers’ equations

This study employs the Extended Kudryashov method to address the fractional-order generalized Burgers–Huxley equation, the fractional-order generalized Huxley equation, the fractional-order generalized Burgers equation, and the fractional-order generalized Fisher equation. Through this approach, exact solutions for these equations are derived. Furthermore, we visualize the obtained results in 3D using Maple [Formula: see text]. The outcomes underscore the effectiveness and reliability of the Extended Kudryashov method as a clear and efficient tool for solving the specified equations.

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

The advancement of scientific machine learning (ML) techniques has led to the development of methods for approximating solutions to nonlinear partial differential equations (PDE) with increased efficiency and accuracy. Automatic differentiation has played a pivotal role in this progress, enabling the creation of physics-informed neural networks (PINN) that integrate relevant physics into machine learning models. PINN have shown promise in approximating the solutions to the Navier–Stokes equations, overcoming the limitations of traditional numerical discretization methods. However, challenges such as local minima and long training times persist, motivating the exploration of domain decomposition techniques to improve it. Previous domain decomposition models have introduced spatial and temporal domain decompositions but have yet to fully address issues of smoothness and regularity of global solutions. In this study, we present a novel domain decomposition approach for PINN, termed domain-discretized PINN (DD-PINN), which incorporates complementary loss functions, subdomain-specific transformer networks (TRF), and independent optimization within each subdomain. By enforcing continuity and differentiability through interface constraints and leveraging the Sobolev (H 1) norm of the mean squared error (MSE), rather than the Euclidean norm (L 2), DD-PINN enhances solution regularity and accuracy. The inclusion of TRF in each subdomain facilitates feature extraction and improves convergence rates, as demonstrated through simulations of threetest problems: steady-state flow in a two-dimensional lid-driven cavity, the time-dependent cylinder wake, and the viscous Burgers equation. Numerical comparisons highlight the effectiveness of DD-PINN in preserving global solution regularity and accurately approximating complex phenomena, marking a significant advancement over previous domain decomposition methods within the PINN framework.

Stability analysis, lump and exact solutions to Sharma–Tasso–Olver–Burgers equation

Stability analysis, lump and exact solutions to Sharma–Tasso–Olver–Burgers equation