An efficient numerical method for solving coupled Burgers' equation by combining homotopy perturbation and pade techniques

In this study, we combined homotopy perturbation and Pade techniques for solving homogeneous and inhomogeneous two‐dimensional parabolic equation. Also, we apply our combined method for coupled Burgers' equations. The numerical results demonstrate that our combined method gives the approximate solution with faster convergence rate and higher accuracy than using the classic homotopy perturbation method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 982–995, 2011     

Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential decay rate.     

Operator splitting for numerical solution of the modified Burgers' equation using finite element method

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L₂ and L_∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method.     

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

In this article, we introduce a new, simple, and accurate computational technique for one‐dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017     

A second‐order accurate difference scheme for the two‐dimensional Burgers' system

In this article, a Crank‐Nicolson‐type finite difference scheme for the two‐dimensional Burgers' system is presented. The existence of the difference solution is shown by Brouwer fixed‐point theorem. The uniqueness of the difference solution and the stability and L₂ convergence of the difference scheme are proved by energy method. An iterative algorithm for the difference scheme is given in detail. Furthermore, a linear predictor–corrector method is presented. The numerical results show that the predictor–corrector method is also convergent with the convergence order of two in both time and space. At last, some comments are provided for the backward Euler scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L² norm is established. Numerical experiments are presented to validate the theoretical analysis.     

The pointwise estimates of a conservative difference scheme for Burgers' equation

In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L^∞ -norm when the initial value is bounded in H¹-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L^∞ -norm. Finally, numerical examples are carried out to verify our theoretical results.     

Multilevel augmentation methods for solving the Burgers' equation

In this article, multilevel augmentation method (MAM) for solving the Burgers' equation is developed. The Crank–Nicolson–Galerkin scheme of the Burgers' equation results in nonlinear algebraic systems at each time step, the computational cost for solving these nonlinear systems is huge. The MAM allows us to solve the nonlinear system at a fixed initial lower level and then compensate the error by solving a linear system at the higher level. We prove that the method has the same optimal convergence order as the projection method, while reducing the computational complexity greatly. Finally, numerical experiments are presented to confirm the theoretical analysis and illustrate the efficiency of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1665–1691, 2015     

A marker method for the solution of the damped Burgers' equation

A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h⁴ + k²). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L₂ and L_∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.     

A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph

Water wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and efficiency is shown using CPU time.     

Exact solutions for nonlinear Burgers' equation by homotopy perturbation method

The aim of this article is to construct a new efficient recurrent relation to solve nonlinear Burgers' equation. The homotopy perturbation method is used to solve this equation. Because Burgers' equation arises in many applications, it is worth trying new solution methods. Comparison of the results with those of Adomian's decomposition method leads to significant consequences. Four standard problems are used to illustrate the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Mathematical studies of the solution of Burgers' equations by Adomian decomposition method

In this paper, a novel Adomian decomposition method (ADM) is developed for the solution of Burgers' equation. While high level of this method for differential equations are found in the literature, this work covers most of the necessary details required to apply ADM for partial differential equations. The present ADM has the capability to produce three different types of solutions, namely, explicit exact solution, analytic solution, and semi-analytic solution. In the best cases, when a closed-form solution exists, ADM is able to capture this exact solution, while most of the numerical methods can only provide an approximation solution. The proposed ADM is validated using different test cases dealing with inviscid and viscous Burgers' equations. Satisfactory results are obtained for all test cases, and, particularly, results reported in this paper agree well with those reported by other researchers.     

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.     

Conservative numerical method for solving the averaged Boltzmann equation

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step.     

On Newton's method for solving generalized equations

In this paper, we study the convergence properties of a Newton-type method for solving generalized equations under a majorant condition. To this end, we use a contraction mapping principle. More precisely, we present semi-local convergence analysis of the method for generalized equations involving a set-valued map, the inverse of which satisfying the Aubin property. Our analysis enables us to obtain convergence results under Lipschitz, Smale and Nesterov-Nemirovski's self-concordant conditions.     

Compact ADI method for solving parabolic differential equations

A compact alternating direction implicit (ADI) method has been developed for solving two‐dimensional parabolic differential equations. In this study, the second‐order derivatives with respect to space are discretized using the high‐order compact finite differences. The Peaceman‐Rachford ADI method is then used for developing a new ADI scheme. It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable. The method can be generalized to the three‐dimensional case and an unconditionally stable compact Douglas ADI scheme is obtained. The method is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 129–142, 2002; DOI 10.1002/num.1037