Solution Of Burgers Research Articles

Fractional viscoelastic constitutive modelling of real-time strain response for asphalt pavement composites subjected to simulating wheel loadings

ABSTRACT This paper proposed a replacement of the Kelvin–Voigt model with fractional basic elements, and the model constructed with this method was used to describe the viscoelastic strain response of asphalt pavement materials under the simulated wheel loading. Similarities between Abel element and Kelvin–Voigt model were explored, and strain response analytical solutions of Burgers and 1S1A1D models under intermittent have sine loading were derived. Then, different intermittent loading tests of asphalt binder and mixture were carried out. The results showed that both the Abel element and the Kelvin–Voigt model could represent the delayed viscoelastic behaviours accurately, and the strain response represented by them could be fully recovered during the unloading stage. Furthermore, parameters of the Abel element and Kelvin–Voigt model showed a high degree of correlation, and the 1S1A1D model exhibited the best effects for asphalt binders, particularly during the unloading period, with correlation coefficients mostly above 0.999. Moreover, when expressing the strain response of asphalt mixture under simulated wheel loading, the 1S1A1D model outperformed the Burgers model in terms of peak strain value, residual strain and changing trends. This study demonstrated that the fractional constitutive model was advantageous in representing the strain response of asphalt pavement composites.

Streamlined numerical solutions of burgers' equations bridging the tailored finite point method and the Cole Hopf transformation

This paper explores the possibility of solving the one-dimensional Burgers' equation with a tailored finite point method (TFPM) based on an explicit stencil along with utilizing the Cole Hopf transformation. The proposed TFPM procedure operates on an explicit four-point stencil where the nodal solution at the advanced temporal level is written as a linear combination of the nodal solutions at the preceding temporal level. In order to bring the essence of the local exact solutions into the numerical approximations the scalar coefficients involved in the linear sum are determined by the application of the fundamental solutions into the stencil. The efficiency of the method is demonstrated through the comparisons of TFPM solutions for various examples with the exact solutions and solutions from well-established methodologies in the literature.

A comprehensive study of subdivision collocation method for Burgers’ equation

This study explores the use of subdivision schemes to efficiently solve Burgers’ equation. Burgers’ equation is a fundamental fluid dynamics equation that describes the nonlinear behavior of fluid flow. This type of nonlinear equation is difficult to solve analytically, which makes the numerical solution an important tool. The subdivision collocation method (SCM) converts Burgers’ equation into a system of algebraic linear equations using the quasilinearization technique. The results of this study demonstrate that the proposed approach yields accurate numerical solutions for Burgers’ equation. Additionally, the subdivision approach is computationally efficient and requires fewer computational resources than existing numerical methods, making it a promising tool for solving Burgers’ equation in practical applications. Overall, this study provides valuable insights into the approximate solution of Burgers’ equation by implementing subdivision schemes.

Numerical Solution of Burgers–Huxley Equation Using a Higher Order Collocation Method

In this paper, the collocation method with cubic B-spline as basis function has been successfully applied to numerically solve the Burgers–Huxley equation. This equation illustrates a model for describing the interaction between reaction mechanisms, convection effects, and diffusion transport. Quasi-linearization has been employed to deal with the nonlinearity of equations. The Crank–Nicolson implicit scheme is used for discretization of the equation and the resulting system turned out to be semi-implicit. The stability of the method is discussed using Fourier series analysis (von Neumann method), and it has been concluded that the method is unconditionally stable. Various numerical experiments have been performed to demonstrate the authenticity of the scheme. We have found that the computed numerical solutions are in good agreement with the exact solutions and are competent with those available in the literature. Accuracy and minimal computational efforts are the key features of the proposed method.

Nonlinear electrostatic kelvin-helmholtz shock waves in a viscous electron-positron-ion plasma with non-maxwellian distribution

A theoretical investigation is carried out for nonlinear electrostatic Kelvin-Helmholtz (K-H) shock waves in a magnetized electron-positron-ion viscous plasma in the presence of transport equations and non-Maxwellian particles by following the generalized (r, q) distribution function. The propagation of electrostatic K-H modes are studied both in the presence of trapped and free electrons. The nonlinear analysis with inclusion of plasma transport properties (magnetic viscosity and heat conduction) lead to nonlinear electrostatic K-H mode in the form of shock like waves by solving the modified Burgers’ equation. The electrostatic K-H shocks are investigated numerically with effect of different plasma parameters such as shear velocity and non-Maxwellian distributed particles. It is observed that the striking features (viz., amplitude and width of dissipative shock through the solution of Burgers’ equation) of the K-H mode are significantly modified by the effects of non-thermality of electrons and positrons both at shoulder and tails along with shear velocity due to viscosity. The relevancy of our work to the observations in space (viz., cometary comae and earth’s ionosphere), astrophysical (viz., pulsars) and laboratory (viz., solid-high intense laser plasma interaction experiments) plasmas is highlighted.

A novel approach based on minimization for numerical solutions of Burgers’ equation

This paper deals with the numerical solution of Burgers’ equation. By using radial basis functions (RBFs), an optimization approach is used to derive the approximated solution. Furthermore, we use the regularization term to avoid the growth of RBF coefficients. Also, the stability of the proposed method is verified, and finally we present some numerical examples to confirm our given approach.

An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation

Burgers’ equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers’ equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers’ equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications.

Forecasting Pollution Using Numerical Simulation Implementing Artificial Bee Colony Optimization

In this article, an optimization strategy is presented for the numerical solution of Burgers’ equations, which play an important role in estimating and forecasting pollution. The method involves the exponential B-spline basis function as the basis function in the differential quadrature method. Since exponential B-spline involves a parameter, the artificial bee colony optimization algorithm is implemented to find the unknown parameters that result in the minimum error. Among the metaheuristic optimization algorithms, the artificial bee colony (ABC) is one that has received the greatest attention from researchers and has been successfully implemented to solve various problems in engineering and sciences. The proposed work emphasizes the calculation of the parameter of exponential basis functions, a major factor that plays a role in the error calculation using the ABC optimization algorithm. The acquired findings are provided as tables, and the physical behaviour is showcased in the form of figures and tables. The results are in good conformity with the earlier studies.

Solution of burgers’ equation appears in fluid mechanics by multistage optimal homotopy asymptotic method

In this article, we approximate analytical solution of Burgers? equations using the multistage homotopy asymptotic method which are utilized in modern physics and fluid mechanics. The suggested algorithm is an accurate and simple to-utilize semi-analytic tool for non-linear problems. In the current research we investigation the efficiency and accuracy of the method for the solution of non-linear PDE for large time span. Numerical comparison with the variational iteration method shows the efficacy and accuracy of the proposed method.

Retraction: Solution of Burgers' equation appears in fluid mechanics by multistage optimal homotopy asymptotic method, doi https://doi.org/10.2298/TSCI23S1087W

Editor-in-Chief of the journal Thermal Science request that it is necessary to retract paper SOLUTION OF BURGERS' EQUATION APPEARS IN FLUID MECHANICS BY MULTISTAGE OPTIMAL HOMOTOPY ASYMPTOTIC METHOD DOI: https://doi.org/10.2298/TSCI23S1087W by Fuzhang WANG, Niaz ALI SHAH, Imtiaz AHMAD Hijaz AHMAD, Muhammad Kamran ALAM, and Phatiphat THOUNTHONG published in the journal Thermal Science, Vol. 27, Year 2023, Special Issue 1, S87-S92 since by technical error of the Editorial staff, this paper has already been published in the journal Thermal Science Vol.26, Part 1B, pp. 815-821, 2022 https://doi.org/10.2298/TSCI210302343W <br><br><font color="red"><b> Link to the retracted article <u><a href="http://dx.doi.org/10.2298/TSCI23S1087W">10.2298/TSCI23S1087W</a></b></u>

A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers

This computational study is concerned with the numerical solutions of Burgers’-type equations at high Reynolds numbers. The high Reynolds numbers drive the nonlinearity to play an essential role and the equations to become more convection-dominated, which causes the solutions obtained with the standard numerical methods to involve spurious oscillations. To overcome this challenge, the Galerkin finite element formulation is stabilized by using the streamline-upwind/Petrov–Galerkin method. The stabilized formulation is further supplemented with YZβ shock-capturing to achieve better solution profiles around strong gradients. The nonlinear equation systems arising from the space and time discretizations are solved by using the Newton–Raphson (N–R) method at each time step. The resulting linearized equation systems are solved with the BiCGStab technique combined with ILU preconditioning at each N–R iteration. A comprehensive set of test examples is provided to demonstrate the robustness of the proposed formulation and the techniques used.

An example of intrinsic randomness in deterministic PDES

A new mechanism leading to a random version of Burgers’ equation is introduced: it is shown that the Totally Asymmetric Exclusion Process in discrete time (TASEP) can be understood as an intrinsically stochastic, non-entropic weak solution of Burgers’ equation on [Formula: see text]. In this interpretation, the appearance of randomness in the Burgers’ dynamics is caused by random additions of jumps to the solution, corresponding to the random effects in TASEP.

Linear and nonlinear modelling of far-field propagation of broadband shock-associated noise

A triple-scale computational model is implemented to simulate noise generated by a supersonic under-expanded screeching jet corresponding to the LTRAC (Laboratory for Turbulence Research in Aerospace and Combustion) experiment at different distances from the source. The investigation is focused on the broadband-associated noise, which is a prominent feature of the acoustic field of the LTRAC jet. In the jet near-field, the compressible Navier–Stokes equations are solved using the high-resolution CABARET Large Eddy Simulation (LES) method accelerated on Graphics Processing Units. The LES solution is substituted in the Ffowcs Williams–Hawkings (FW–H) model to obtain the noise solution in the acoustic mid field at 20 initial jet diameters from the jet nozzle exit. The mid-field acoustic solution is used as the input for the spherical generalised Burgers’ equation. The general form of Burgers’ equation is solved numerically in the frequency domain for a wide range of observer distances up to 18 million initial jet diameters, where viscous dissipation fully dominates for most frequencies. To answer the question if the nonlinear acoustic wave propagation effects for the LTRAC jet are important, the nonlinear and linear solutions of Burgers’ equation are compared.

Generalized characteristics for finite entropy solutions of Burgers’ equation

We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers’ equation ∂tu+∂xu22=0,whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution’s hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.

Solution of Burgers’ equation appears in fluid mechanics by multistage optimal homotopy asymptotic method

In this article, we approximate analytical solution of Burgers? equations using the Multistage homotopy asymptotic method which are utilized in modern physics and fluid mechanics. The suggested algorithm is an accurate and simple to-utilize semi-analytic tool for non-linear problems. In the current research we investigation the efficiency and accuracy of the method for the solution of non-linear PDE for large time span. Numerical comparison with the variational iteration method shows the efficacy and accuracy of the proposed method.

New Exact Solutions of Burgers’ Equation Using Power Index Method

In this article, we focus on the new exact solutions of Burger’s equation by using a new technique which is known as the power index method (PIM). In this method, we choose suitable indexes of independent variables and similarity transformation so that the partial differential equation may be converted into ODE. We have obtained analytic solution of the ODE by using symbolic package Maple. We have got exact solution of Burgers’ equation by using analytic solution of ODE and similarity transformation. The proposed method has been effectively employed to find new exact solutions for the nonlinear Burgers’ equation. Finally, the proposed resulting answers are compared with the homotopy perturbation, decomposition, and variational iteration solutions.

On the L2 stability of shock waves for finite-entropy solutions of Burgers

We prove L2 stability estimates for entropic shocks among weak, possibly non-entropic, solutions of scalar conservation laws ∂tu+∂xf(u)=0 with strictly convex flux function f. This generalizes previous results by Leger and Vasseur, who proved L2 stability among entropy solutions. Our main result, the estimate∫R|u(t,⋅)−u0shock(⋅−x(t))|2dx≤∫R|u0−u0shock|2+Cμ+([0,t]×R), for some Lipschitz shift x(t), includes an error term accounting for the positive part of the entropy production measure μ=∂t(u2/2)+∂xq(u), where q′(u)=uf′(u). Stability estimates in this general non-entropic setting are of interest in connection with large deviation principles for the hydrodynamic limit of asymmetric interacting particle systems. Our proof adapts the scheme devised by Leger and Vasseur, where one constructs a shift x(t) which allows to bound from above the time-derivative of the left-hand side. The main difference lies in the fact that our solution u(t,⋅) may present a non-entropic shock at x=x(t) and new bounds are needed in that situation. We also generalize this stability estimate to initial data with bounded variation.

Group Classification Invariant Solutions of Burgers' Equation

The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.

The analysis of conservation laws, symmetries and solitary wave solutions of Burgers–Fisher equation

In this paper, the conservation laws, significant symmetries’ application, and traveling wave solutions are obtained for Burger–Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained.

Dynamical structures of exact soliton solutions to Burgers’ equation via the bilinear approach

The Cole–Hopf transformation is used to search for exact multiple wave solutions of Burgers’ equation. Particularly, solitary wave solutions and their interaction solutions are explicitly given, and a fission phenomenon is found, which is tough to be found by other traditional methods. Different solutions are presented with different combinations of elementary functions and a necessary condition for fission is proposed. Based on the bilinear formalism and with the aid of symbolic computation, we also determine two classes of rational solutions, rogue wave solutions and rogue–kink wave solutions, using various ansätze. To exhibit dynamics, we make diverse graphical analyses for the presented solutions, which show that the solutions are reliable in both mathematical physics and engineering.