A systematic literature review of Burgers’ equation with recent advances

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.     

Painlevé Analysis and Some Solutions of(2+1)-Dimensional Generalized Burgers Equations

Burgers equation ut = 2uux + uxx describes a lot of phenomena in physics fields, and it has attracted much attention.In this paper,the Burgers equation is generalized to (2+1) dimensions.By means of the Painlev(e') analysis,the most generalized Painlev(e') integrable(2+1)-dimensional integrable Burgers systems are obtained.Some exact solutions of the generalized Burgers system are obtained via variable separation approach.     

Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

Incompressible turbulence as non-local field theory

It is well-known that incompressible turbulence is non-local in real space because sound speed is infinite in incompressible fluids. The equation in Fourier space indicates that it is non-local in Fourier space as well. However, the shell-to-shell energy transfer is local. Contrast this with Burgers equation which is local in real space. Note that the sound speed in Burgers equation is zero. In our presentation we will contrast these two equations using non-local field theory. Energy spectrum and renormalized parameters will be discussed.     

Effect of time profile of laser pulse on the excitation of sound signals in a liquid with a series of laser pulses

We have studied the sound generation with high repetition rate pulsed laser. We have solved the inhomogeneous wave equation for acoustic pressure in a liquid generated by a laser, using Green’s function formalism and convolution technique. To obtain the maximum pressure of the sound waves, we found the conditions on repetition rate and on period of laser pulse of various shapes. Our analysis shows that the sound generated in a liquid with a series of laser pulses is highly affected by the time profile of the pulses besides other parameters, namely laser beam diameter, laser beam optical wavelength, repetition rate and period of laser pulse. This effect is pronounced particularly in frequency domain. We found that the noise of higher harmonics in the generated sound can be greatly removed with the proper choice of the time profile of the laser pulses. It is found that the pressure is generated around the fundamental frequency for the half-sine and rectangular pulses, with the proper choice of repetition rate and period of pulse. The application of the present analysis for underwater communication is pointed out.     

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

In this article, we study the (2+1)-extension of Burgers equation and the KP equation. At first, based on a known Bäcklund transformation and corresponding Lax pair, an invariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation is worked out. Given a known solution and using the invariance, we can find solutions of the (2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgers equation which cannot be directly obtained by constraining the invariance of the (2+1)-extension of Burgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgers equation can help us find the solutions of KP equation. At last, based on the invariance of Burgers equation, the corresponding recursion formulae for finding solutions of KP equation are digged out. As the application of our theory, some examples have been put forward in this article and some solutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods

This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.     

Time fractional effect on ion acoustic shock waves in ion-pair plasma

The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal’s method. Using the Adomian decomposition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth’s ionosphere.     

高精度球面阵聚焦声源定位方法研究

提出了一种高精度高空间分辨率球面阵聚焦声源定位方法——虚拟源法。该方法通过球面阵波束扫描获得实际声源的空间聚焦谱,并假定各扫描点为虚拟声源,将实际声源聚焦谱看作是全体虚拟源共同作用的结果,由此得到各虚拟源对声场的贡献量,从而可实现声源精确定位。仿真研究分析了频率,阵列孔径,声场模态阶数,信噪比等参数对声源定位性能的影响,并与常规算法进行对比。结果显示,该方法不受频率和阵列孔径的限制,避免了空间“混淆”,能够进行高精度高分辨率声源定位,并具有良好的背景噪声抑制能力。      

Numerical simulation of parametric sound generation and its application to length-limited sound beam

In this study we propose a simulation model for predicting the nonlinear sound propagation of ultrasound beams over a distance of a few hundred wavelengths, and we estimate the beam profile of a parametric array. Using the finite-difference time-domain method based on the Yee algorithm with operator splitting, axisymmetric nonlinear propagation was simulated on the basis of equations for a compressible viscous fluid. The simulation of harmonic generation agreed with the solutions of the Khokhlov–Zabolotskaya–Kuznetsov equation around the sound axis except near the sound source. As an application of the model, we estimated the profiles of length-limited parametric sound beams, which are generated by a pair of parametric sound sources with controlled amplitudes and phases. The simulation indicated a sound beam with a narrow truncated array length and a width of about one-quarter to half that of regular a parametric beam. This result confirms that the control of sound source conditions changes the shape of a parametric beam and can be used to form a torch like low-frequency sound beam.     

New Multiple Soliton-like Solutions to (3+1)-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1 )-dimensional Burgers equation with variable coefficients.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications

A governing differential equation for predicting the effective thermal conductivity of composites with spherical inclusions is shown to be simply derived by using the result of the generalized self-consistent model. By applying the equation to composites including spherical inclusions such as graded spherical inclusions, microballoons, mutiply-coated spheres, and spherical inclusions with an interphase, their effective thermal conductivities are easily predicted. The results are compared with those in the literatures to be consistent. It can be stated from the investigations that the effective thermal conductivity of composites with spherical inclusions can be estimated as long as their conductivities are expressed as a function of their radius.     

Generalized space-time paraxial acoustic ray tracing

Paraxial ray tracing has gained popularity in seismology and underwater acoustics for modelling the propagation of sound when the medium is stationary and time independent. In this article differential geometry is used to derive a generalized paraxial ray-tracing procedure valid for any fluid media described by a local sound speed and velocity depending arbitrarily on position and time. Geodesic deviation is used to model acoustic beam deformation, and the sectional curvature along a ray to determine convergence and divergence zones in space. The resulting paraxial equations presented here are the most general that can be derived for the acoustic field and apply to any environment including those with time dependence and fluid motion. Applied to layered media the geodesic deviation equation is solved exactly. Some illustrative examples are included.     

Modified Burgers equation by local discontinuous Galerkin method

In this paper, we present the local discontinuous Galerkin method for solving Burgers’ equation and the modified Burgers’ equation. We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail. The method is applied to the solution of the one-dimensional viscous Burgers’ equation and two forms of the modified Burgers’ equation. The numerical results indicate that the method is very accurate and efficient.     

Complex ray representation of the astigmatic Gaussian beam propagation

The method of Gaussian beam ray-equivalent modelling, first proposed by Arnaud, is generalized to the case of general astigmatism. It has been shown that a generally astigmatic Gaussian beam can be properly represented by two complex rays, or equally by four real rays, which are treated by the well-known propagation equation and ray tracing method in geometric optics, and from which the beam parameters are easily obtained. Illustrative numerical examples are given. The equivalence between the complex-ray treatment and the generalized ABCD law is also shown.     

Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang–Mills Equations

With the help of some reductions of the self-dual Yang Mills （briefly written as sdYM） equations, we introduce a Lax pair whose compatibility condition leads to a set of （2 ＋ 1）-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new （2 ＋ 1）-dimensional nonlinear integrable systems, in particular, obtains a kind of （2 ＋ 1）-dimensional integrable couplings of a new （2 ＋ 1）- dimensional integrable nonlinear equation.