A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

A new domain decomposition algorithm for generalized Burger’s–Huxley equation based on Chebyshev polynomials and preconditioning

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Huxley equation (GBHE). To reduce round-off error in spectral collocation (pseudospectral) method we use preconditioning. Firstly, theory of application of Chebyshev spectral collocation method with preconditioning (CSCMP) and domain decomposition on the generalized Burger’s–Huxley equation presented. This method yields a system of ordinary differential algebric equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

This paper proposes operational matrix of rth integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of rth integration is taken as rth power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain ith natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.     

非线性发展方程行波解的符号计算

Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation.     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

Numerical solutions for convection–diffusion equation using El-Gendi method

The method of El-Gendi [El-Gendi SE. Chebyshev solution of differential integral and integro-differential equations. J Comput 1969;12:282–7; Mihaila B, Mihaila I. Numerical approximation using Chebyshev polynomial expansions: El-gendi’s method revisited. J Phys A Math Gen 2002;35:731–46] is presented with interface points to deal with linear and non-linear convection–diffusion equations.The linear problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using three-level time scheme.The non-linear problem is reduced to three systems of ordinary differential. Each one of these systems is, then, solved using three-level time scheme. Numerical results for Burgers’ equation and modified Burgers’ equation are shown and compared with other methods. The numerical results are found to be in good agreement with the exact solutions.     

On the new solutions of the generalized Burgers–Huxley equation

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

A spectral domain decomposition approach for the generalized Burger’s–Fisher equation

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Fisher equation (B–F). Firstly, theory of application of Chebyshev spectral collocation method (CSCM) and domain decomposition on the generalized Burger’s–Fisher equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equationa*

We present a nonlinear method to approximate solutions of a Burgers–Huxley equation with generalized advection factor and logistic reaction. The equation under investigation possesses travelling-wave solutions that are temporally and spatially monotone functions; the travelling-wave fronts considered are bounded and connect asymptotically the stationary solutions of the model. For the linear regime, the method is consistent of first order in time and second order in space. In the nonlinear scenario, we investigate conditions under which bounded initial profiles evolve into bounded new approximations. The main results report on parametric conditions that guarantee the boundedness, the positivity and the monotonicity preservation of the method. As a consequence, our recursive method is capable of preserving the temporal and the spatial monotonicity of the solutions. We provide simulations that show that, indeed, our technique preserves the positivity, the boundedness and the temporal and spatial monotonicity of solutions.     

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

Journal of Theoretical Probability - In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability...     

Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers–Huxley equation

In this paper, a numerical solution of the generalized Burgers–Huxley equation is presented. This is the application of spectral collocation method. To reduce roundoff error in this method we use Darvishi’s preconditionings. The numerical results obtained by this method have been compared with the exact solution. It can be seen that they are in a good agreement with each other, because errors are very small and figures of exact and numerical solutions are very similar.     

Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness and the spatial and the temporal monotonicity. The main results provide conditions that guarantee the existence and the uniqueness of monotone and bounded solutions of our scheme. The technique was implemented and tested computationally, and the results confirm both a good agreement with respect to the travelling-wave solutions reported in the literature and the preservation of the mathematical features of interest.     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical solution of non‐stationary aero‐autoelasticity integro‐differential operator equations

The spectral method of G. N. Elnagar, which yields spectral convergence rate for the approximate solutions of Fredholm and Volterra–Hammerstein integral equations, is generalized in order to solve the larger class of integro‐differential functional operator equations with spectral accuracy. In order to obtain spectrally accurate solutions, the grids on which the above class of problems is to be solved must also be obtained by spectrally accurate techniques. The proposed method is based on the idea of relating, spectrally constructed, grid points to the structure of projection operators which will be used to approximate the control vector and the associated state vector. These projection operators are spectrally constructed using Chebyshev–Gauss–Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. Simulation studies demonstrate computational advantages relative to other methods in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

Chebyshev‐Legendre spectral method for solving the two‐dimensional vorticity equations with homogeneous Dirichlet conditions

The Chebyshev‐Legendre spectral method for the two‐dimensional vorticity equations is considered. The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev‐Gauss collocation points. The numerical analysis results under the L²‐norm for the Chebyshev‐Legendre method of one‐dimensional case are generalized into that of the two‐dimensional case. The stability and optimal order convergence of the method are proved. Numerical results are given. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Analytic approximate solutions to Burgers,Fisher, Huxley equations and two combined forms of these equations

In this paper, we are giving analytic approximate solutions to a class of nonlinear PDEs using the homotopy analysis method (HAM). The Burgers, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations are considered. We aim two goals: one is to highlight the efficiency of HAM in solving this class of PDEs and the other is that, although the considered equations have different combinations of nonlinear terms, when applying HAM, we use the same initial guess, the same auxiliary linear operator and the same auxiliary function for all of them.     

On the convergence of a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

In this note, we establish the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The numerical method was previously investigated in the literature and, amongst other features of interest, it is a fast and nonlinear technique that is capable of preserving positivity, boundedness and monotonicity. In the present work, we establish that the method is convergent with linear order of convergence in time and quadratic order in space. Some numerical experiments are provided in order to support the analytical results.     

Numerical solution of the coupled viscous Burgers’ equation

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers’ equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points. The scheme is based on Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The method is shown to be unconditionally stable using von-Neumann method. The accuracy of the proposed method is demonstrated by applying it on three test problems. Computed results are depicted graphically and are compared with those already available in the literature. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions.     

一维Burgers方程和KdV方程的广义有限谱方法

给出了高精度的广义有限谱方法．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth 预报格式和Adams-Moulton校正格式,为了避免由Korteweg-de Vries(KdV)方程的弥散项引起的数值振荡, 给出了两种数值稳定器．以Legendre多项式、Chebyshev多项式和Hermite多项式为基函数作为例子,给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

Exact solutions for two nonlinear wave equations with nonlinear terms of any order

In this paper, based on a variable-coefficient balancing-act method, by means of an appropriate transformation and with the help of Mathematica, we obtain some new types of solitary-wave solutions to the generalized Benjamin–Bona–Mahony (BBM) equation and the generalized Burgers–Fisher (BF) equation with nonlinear terms of any order. These solutions fully cover the various solitary waves of BBM equation and BF equation previously reported.