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Applied Mathematics 2018年第7期9卷 821-835页

作者： Masho Jima Alemayehu Shiferaw Ali Tsegaye Department of Mathematics Jimma University Jimma Ethiopia

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.

关键词： Differential Quadrature Method Fourier Expansion Coupled Viscous Coupled Viscous burgers’ Equation Initial and Boundary Conditions

Flow of Generalized burgers’ Fluid Induced by Sawtooth Pulses Stress with Quadratic Edges

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Journal of Applied Mathematics and Physics 2018年第6期6卷 1258-1277页

作者： Qamar Sultan Zaib Un Nisa Mudassar Nazar Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan Pakistan School of Mathematical Sciences University of Science and Technology of China Hefei China

This paper presents the unsteady magnetohydrodynamic (MHD) flow of a generalized burgers’ fluid between two side walls perpendicular to a plate. The flow is generated from rest at time ?induced by stress applied on the bottom plate. The stress is assumed to be like sawtooth pulses with quadratic edges. The solutions obtained by means of the Laplace, and Fourier cosine and sine transforms are presented as a sum between the corresponding Newtonian and non-Newtonian contributions. Graphs are sketched for various parameters of interest.

关键词： Generalized burgers’ Fluid Sawtooth Pulses Stress Fourier Cosine and Sine Transforms

A Note on Crank-Nicolson Scheme for burgers’ Equation

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Applied Mathematics 2011年第7期2卷 883-889页

作者： Kanti Pandey Lajja Verma 不详

In this work we generate the numerical solutions of the burgers’ equation by applying the Crank-Nicolson method directly to the burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-merical solutions as well as exact solutions for different values of viscosity are calculated and we find that the numerical solutions are very close to exact solution.

关键词： Hopf-Cole Transformation burgers’ Equation Crank-Nicolson Scheme Nonlinear Partial Differential Equations

Upwind Finite-Volume Solution of Stochastic burgers’ Equation

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Applied Mathematics 2012年第11期3卷 1818-1825页

作者： Mohamed A. El-Beltagy Mohamed I. Wafa Osama H. Galal Department of Engineering Mathematics & Physics Engineering Faculty Cairo University Giza Egypt Department of Engineering Mathematics & Physics Faculty of Engineering Fayoum University Fayoum Egypt

In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The burgers’ equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic burgers’ solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.

关键词： Polynomial Chaos Expansion Stochastic burgers’ Equation Upwind Finite-Volume Technique

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

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Journal of Applied Mathematics and Physics 2018年第9期6卷 1825-1839页

作者： Yuexing Bai Sudao Bilige Temuer Chaolu College of Sciences Inner Mongolia University of Technology Hohhot China College of Arts and Sciences Shanghai Maritime University Shanghai China

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.

关键词： Potential Symmetry One-Dimensional Optimal System Invariant Solution Coupled burgers’ Equations

A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations

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American Journal of Computational Mathematics 2011年第1期1卷 26-38页

作者： Katsuhiro Sakai Daishi Watabe 不详

According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

关键词： Numerical Scheme Numerical Analysis Numerical Stability Positivity Condition Advection-Diffusion Equation Advection Equation High-Order Scheme Godunov Theorem burgers’ Equation

An O(k2+kh2+h2) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

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American Journal of Computational Mathematics 2011年第1期1卷 11-17页

作者： Ranjan Kumar Mohanty Vijay Dahiya 不详

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

关键词： Quasi-Linear Parabolic Equation Implicit Method Cubic Spline Approximation Diffusion-Convection Equation Singular Equation burgers’ Equation Reynolds Number

High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

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Applied Mathematics 2011年第10期2卷 1243-1251页

作者： Ranjan K. Mohanty Deepika Dhall 不详

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.

关键词： Variable Mesh Arithmetic Average Discretization Non-Linear Integro-Differential Equation Diffusion Equation Simpson’s 1/3 Rd Rule Singular Coefficients burgers’ Equation Maximum Absolute Errors