Numerical solution of one‐dimensional Burgers' equation

In this article, an iterative method for the approximate solution of a class of Burgers' equation is obtained in reproducing kernel space . It is proved the approximation converges uniformly to the exact solution u(x, t) for any initial function under trivial conditions, the derivatives of are also convergent to the derivatives of u(x, t), and the approximate solution is the best approximation under the system © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1251–1264, 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     

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     

Stochastic Burgers' equation on the real line: regularity and moment estimates

In this project, we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman–Kac representation which solves a stochastic partial differential equation such that its Hopf–Cole transformation solves Burgers' equation. Finally, we obtain Hölder regularity and moment estimates for the solution to Burgers' equation.     

Existence of positive periodic solution for variable–coefficient third‐order differential equation with singularity

In this paper, we investigate a class of singular third‐order differential equations with variable coefficients. By application of Green's functions and Schauder's fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established. Copyright © 2013 John Wiley & Sons, Ltd.     

The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

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.     

Constructive approximation of solution for fourth‐order nonlinear boundary value problems

In this paper, we obtain a sequence of approximate solution converging uniformly to the exact solution of a class of fourth‐order nonlinear boundary value problems. Its exact solution is represented in the form of series in the reproducing kernel space. The n‐term approximation u_n(x) is proved to converge to the exact solution u(x). Moreover, the derivatives of u_n(x) are also convergent to the derivatives of u(x). Copyright © 2008 John Wiley & Sons, Ltd.     

High order compact solution of the one‐space‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving one‐space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high‐order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one‐space‐dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of (Mohanty, 3 .© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

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     

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.     

A numerical method for the one‐dimensional sine‐Gordon equation

A numerical method based on a predictor–corrector (P‐C) scheme arising from the use of rational approximants of order 3 to the matrix‐exponential term in a three‐time level recurrence relation is applied successfully to the one‐dimensional sine‐Gordon equation, already known from the bibliography. In this P‐C scheme a modification in the corrector (MPC) has been proposed according to which the already evaluated corrected values are considered. The method, which uses as predictor an explicit finite‐difference scheme arising from the second order rational approximant and as corrector an implicit one, has been tested numerically on the single and the soliton doublets. Both the predictor and the corrector schemes are analyzed for local truncation error and stability. From the investigation of the numerical results and the comparison of them with other ones known from the bibliography it has been derived that the proposed P‐C/MPC schemes at least coincide in terms of accuracy with them. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

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     

A meshless approach for solution of Burgers’ equation

A new meshless method called gradient reproducing kernel particle method (GRKPM) is proposed for numerical solutions of one-dimensional Burgers’ equation with various values of viscosity and different initial and boundary conditions. Discretization is first done in the space via GRKPM, and subsequently, the reduced system of nonlinear ordinary differential equations is discretized in time by the Gear's method. Comparison with the exact solutions, which are only available for restricted initial conditions and values of viscosity, approves the efficacy of the proposed method. For challenging cases involving small viscosities, comparison with the results obtained using other numerical schemes in the literature further attests the desirable features of the presented methodology.     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

The use of Chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation

A numerical technique is presented for the solution of the second order one‐dimensional linear hyperbolic equation. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem is reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical solution of the one‐dimensional wave equation with an integral condition

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this research a numerical technique is developed for the one‐dimensional hyperbolic equation that combine classical and integral boundary conditions. The proposed method is based on shifted Legendre tau technique. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 282–292, 2007     

Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation _yt−_yxx+y_yx=u(t)$y_t-y_{xx}+y{y}_x=u(t)$ on the line segment [0,1]$[0,1]$. The right-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1)=0$y(t,1)=0$ and restrict ourselves to using only two controls (namely the interior one u(t)$u(t)$ and the boundary one y(t,0)$y(t,0)$). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole–Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.