Coupled Burgers Research Articles

一类耦合Burgers方程组的数值计算

一类耦合Burgers方程组的数值计算

Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers Equation

This paper deals with the numerical solutions of one dimensional time dependent coupled Burgers' equation with suitable initial and boundary conditions by using Chebyshev wavelets in collaboration with a collocation method. The proposed method converts coupled Burgers' equations into system of algebraic equations by aid of the Chebyshev wavelets and their integrals which can be solved easily with a solver. Benchmarking of the proposed method with exact solution and other known methods already exist in the literature is made by three test problems. The feasibility of the proposed method is demonstrated by test problems and indicates that the proposed method gives accurate results in short cpu times. Computer simulations show that the proposed method is computationally cheap, fast and quite good even in the case of less number of collocation points.

Finite Genus Solutions to the Coupled Burgers Hierarchy

The coupled Burgers hierarchy is derived with the aid of Lenard recursion sequences. Based on the characteristic polynomial of Lax matrix, a trigonal curve of arithmetic genus $$m-2$$ is introduced, from which the meromorphic functions $$\phi _2,\phi _3$$ and the Baker–Akhiezer $$\psi $$ function are defined. The finite genus solutions for the coupled Burgers hierarchy are achieved by using asymptotic expansion of $$\phi _2,\phi _3$$ and their Riemann theta function representation.

Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

One of the most challenging PDE forms in fluid dynamics namely Burgers equations is solved numerically in this work. Its transient, nonlinear, and coupling structure are carefully treated. The Hermite type of collocation mesh-free method is applied to the spatial terms and the 4th-order Runge Kutta is adopted to discretize the governing equations in time. The method is applied in conjunction with the Gaussian radial basis function. The effect of viscous force at high Reynolds number up to 1,300 is investigated using the method. For the purpose of validation, a conventional global collocation scheme (also known as “Kansa” method) is applied parallelly. Solutions obtained are validated against the exact solution and also with some other numerical works available in literature when possible.

A Linearized High-Order Combined Compact Difference Scheme for Multi-Dimensional Coupled Burgers’ Equations

A Linearized High-Order Combined Compact Difference Scheme for Multi-Dimensional Coupled Burgers’ Equations

Approximate Solutions of Two-Dimensional Burgers' and Coupled Burgers' Equations by Residual Power Series Method

In this study, two-dimensional Burgers' and coupled Burgers' equations are examined by the residual power series method. This method provides series solutions which are rapidly convergent and their components are easily calculable by Mathematica. When the solution is polynomial, the method gives the exact solution using Taylor series expansion. The results display that the method is more efficient, applicable and accuracy and the graphical consequences clearly present the reliability of the method.

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers’ equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher’s result and even generates a value at the nodes or mesh points that the results have not seen yet.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations

This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.

Explicit Solutions and Conservation Laws of a Coupled Burgers’ Equation

Abstract Based on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers’ equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers’ equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.

Explicit solutions for a modified 2+1-dimensional coupled Burgers equation by using Darboux transformation

A Modified 2+1-Dimensional Coupled Burgers equation and its Lax pairs are proposed. A Darboux transformation for the Modified 2+1-Dimensional Coupled Burgers equation and its Lax pairs is established with the help of the gauge transformation between spectral problems. As an application of the Darboux transformation, we give some explicit solutions of the Modified 2+1-Dimensional Coupled Burgers equation.

HPM for Solving the Time-fractional Coupled Burgers Equations

In this paper, the homotopy perturbation method is implemented to derive the explicit approximate solutions for the time-fractional coupled Burger's equations. The including fractional derivative is in the Caputo sense. Special attention is given to prove the convergence of the method. The results are compared with those obtained by the exact at special cases of the fractional derivatives. The results reveal that the proposed method is very effective and simple.

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers’ Equations

We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the time variable. As an application, we developed the proposed numerical scheme for solving a numerical solution of the two-dimensional coupled Burgers’ equations. The main advantages of our scheme are higher accurate accuracy and facility to implement. The good accuracy of the proposed numerical scheme is tested by comparing the approximate numerical and the exact solutions for several two-dimensional coupled Burgers’ equations.

Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers’ equations

In this paper, we propose a new two-level implicit compact operator method of order two in time (t) and four in space (x) for the solution of time dependent coupled viscous Burgers’ equations. In this method, we did not use any transformation or linearization technique to handle nonlinearity. We use only 3-spatial grid points and the obtained tridiagonal nonlinear system has been solved by Newton’s iterative method. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed method. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. We show that the proposed method enables us to obtain high accurate solution for high Reynolds number.

Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations

Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.

An Efficient Numerical Scheme for Coupled Nonlinear Burgers’ Equations

In this paper, coupled nonlinear Burgers' equations are sol ved through a variety of meshless methods known as multiquadric quasi-interpolation scheme. In this scheme, the extension of univariate quasi-interpolation method is used to approximate the unknown functions and their spatial derivatives and the Taylor seri es expansion is used to discretize the temporal derivatives . The multiquadric quasi-interpolation scheme is a one-dimensional method that can be extend to the two-dimensional by converting to a compact form and using of a tensor product scheme. The method is tested on three experiments to show the efficiency and accuracy of it. Al so, we demonstrate the validity and applicability of our method by error analysis technique based on residual function.

An Efficient Numerical Scheme for Coupled Nonlinear Burgers’ Equations

An Efficient Numerical Scheme for Coupled Nonlinear Burgers’ Equations

A novel lattice Boltzmann model for the coupled viscous Burgers’ equations

In this work, a new lattice Boltzmann model for coupled Burgers’ equations is proposed through selecting proper distribution functions. Unlike the previous models, the present model can exactly recover the coupled equations without any assumptions. A detailed numerical study on several coupled Burgers’ equations is performed to validate the present model, and the results show that the present model not only has a second-order convergence rate in space, but also is more accurate than the previous model.

A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations

In this paper, we propose a collocation–based numerical scheme to obtain approximate solutions of coupled Burgers’ equations. The scheme employs collocation of modified cubic B-spline functions. We have used modified cubic B-spline functions for unknown dependent variables u, v, and their derivatives w.r.t. space variable x. Collocation forms of the partial differential equations result in systems of first–order ordinary differential equations (ODEs). In this scheme, we did not use any transformation or linearization method to handle nonlinearity. The obtained system of ODEs has been solved by strong stability preserving the Runge-Kutta method. The proposed scheme needs less storage space and execution time. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. The scheme is simple as well as easy to implement. The scheme provides approximate solutions not only at the grid points, but also at any point in the solution range.