The Solution Operator of the Korteweg-de Vries Equation is Computable

The initial value problem of the Korteweg-de Vries (KdV) equation posted on the real line R: defines a nonlinear map K from the space H^s( ) to the space C( , H^s( )) for any given real numbers s ≥ = 0. In this paper we prove that the map K_R is computable for any integer s ≥ = 3.     

Numerical integration of a coupled Korteweg-de Vries system

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for an arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence, which gives the conditions and the appropriate choice of the grid sizes. The method is applied to the Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to nonintegrable cases.     

Solitary wave simulations of Complex Modified Korteweg-de Vries Equation using differential quadrature method

Complex Modified Korteweg-deVries Equation is solved numerically using differential quadrature method based on cosine expansion. Three test problems, motion of single solitary wave, interaction of solitary waves and wave generation, are simulated. The accuracy of the method is measured via the discrete root mean square error norm L₂, maximum error norm L_∞ for the motion of single solitary wave since it has an analytical solution. A rate of convergency analysis for motion of single solitary wave containing both real and imaginary parts is also given. Lowest three conserved quantities are computed for all test problems. A comparison with some earlier works is given.     

Energy transfer in a dispersion-managed Korteweg-de Vries system

We consider the propagation of weakly nonlinear, weakly dispersive waves in an inhomogeneous media within the framework of the variable-coefficient Korteweg-de Vries equation. An analytical formula with which to compute the energy transfer between neighboring solitary waves is derived. The resulting expression shows that the energy change in a variable KdV system is essentially due the two-wave mixing, contrary to the energy change in a nonlinear Schrödinger system, which results from the intrachannel four-wave mixing. By considering the case of Gaussian solitary wave solutions, we have determined the transfer of energy in the system analytically and numerically.     

Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation

We propose spectral Chebyshev collocation algorithms for the approximation of the initial and boundary value problem for the Korteweg-de Vries equation. Both single and multidomain approaches are discussed. Different methods for the treatment of the boundary conditions are considered. The numerical analysis of the eigenvalues' behaviour of the spectral differentiation operators involved in the approximation suggests appropriate finite difference methods for time-marching. Several numerical experiments have been performed, which prove spectral convergence and stability of the proposed schemes.     

A split-step Fourier method for the complex modified Korteweg-de Vries equation

In this study, the complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem. Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation.     

The application of shape preserving splines for the modified Korteweg-de Vries equation

This paper investigates the use of shape-preserving splines for the numerical solution of the Korteweg-de Vries equation modified to include such effects as dissipation or time varying dispersion. Using a fractional time-step method, a simple algorithm is obtained, which is shown to be very accurate, even though the soliton becomes very steep.     

An adaptive method of lines solution of the Korteweg-de Vries equation

Following a method of lines formulation, the Korteweg-de Vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle, which allows the number of nodes to be significantly reduced as compared to a fixed-grid solution. Several finite difference schemes, including direct and stagewise procedures, are compared and the results of a large number of computational experiments are presented, which demonstrate that the selection of a spatial approximation scheme for the third-order derivative term is the primary determinant of solution accuracy.     

A lattice Boltzmann model for the Korteweg-de Vries equation with two conservation laws

A lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model to the KdV equation, our method has higher-order accuracy. Two key steps in the development of this model are the addition of a momentum conservation condition, and the construction of a correlation between the first conservation law and the second conservation law. The numerical example shows the higher-order moment method can be used to raise the truncation error of the lattice Boltzmann scheme.     

Asymptotic Expansions of Eigenvalues of the First Boundary-Value Problem for Singularly Perturbed Second-Order Differential Equation with Turning Points

For singularly perturbed second-order equations, the dependence of the eigenvalues of the first boundary-value problem on a small parameter at the highest derivative is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable. This leads to the emergence of so-called turning points. Asymptotic expansions with respect to a small parameter are obtained for all eigenvalues of the considered boundary-value problem. It turns out that the expansions are determined only by the behavior of the coefficients in the neighborhood of the turning points.     

On the calculation of the timing shifts in the variable-coefficient Korteweg-de Vries equation

The variable-coefficient Korteweg-de Vries equation that governs the dynamics of weakly nonlinear long waves in a periodically variable dispersion management media is considered. For general bit patterns, an analytic expression describing the evolution of the timing shift produced by nonlinear interactions between neighboring solitons is derived. The general result with a Gaussian like solitary wave profile for a special case of negligible average dispersion is tested. By considering a piecewise-constant map, we found that the timing shift is reduced substantially in the dispersion-managed Korteweg-de Vries system.     

Soliton fission and fusion of a new two-component Korteweg-de Vries (KdV) equation

In this paper, the Painlevé test is performed for a new two-component Korteweg-de Vries (KdV) equation proposed by Foursov. It is shown that this equation passes the integrability test and is P-integrable. By means of the truncated singular expansion, some explicit solutions from the trivial zero solution are derived. The phenomena of soliton fission and fusion are studied in detail.     

Indirect Boundary Integral Equation Method for the Cauchy Problem of the Laplace Equation

In this paper, we examine the Cauchy problem of the Laplace equation. Motivated by the incompleteness of the single-layer potential function method, we investigate the double-layer potential function method. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of the Cauchy problem, which are well known to be highly ill-posed in nature. The ill-posedness is dealt with Tikhonov regularization, whilst the optimal regularization parameter is chosen by Morozov discrepancy principle. Convergence and stability estimates of the proposed method are then given. Finally, some examples are given for the efficiency of the proposed method. Especially, when the single-layer potential function method does not give accurate results for some problems, it is shown that the proposed method is effective and stable.     

解一类Poisson方程反演问题的并行算法

本文讨论了一类特殊的Ｐｏｉｓｓｏｎ方程反演问题的数值解法和相应的并行算法。     

On a Boundary Controllability Problem for a System Governed by the Two-Dimensional Wave Equation

Journal of Computer and Systems Sciences International - The boundary controllability of oscillations of a plane membrane is studied. The magnitude of the control is bounded. The controllability...     

Wiener滤波问题的一种Diophantine方程解

用时域上的现代时间序列分析方法,基于ARMA新息模型和白噪声估值器,提出Wiener滤波问题的一种Diophantine方程解。它可统一处理平稳或非平稳ARMA信号的最优滤波、平滑和预报问题。仿真例子说明了其有效性。     

Closed form of the Solutions of some Boundary-Value Problems for Anomalous Diffusion Equation with Hilfer’s Generalized Derivative

The concept of double-order fractional derivative generalizing the well-known Hilfer’s derivative is introduced. The formula is given for the Laplace transform of double-order fractional derivative, which is used to solve the Cauchy-type problem for equations of fractional order with this derivative. The closed solutions to some boundary-value problems for the equation of anomalous diffusion with double-order fractional derivative in time are obtained.     

Solving the Cauchy Problem for a Two-Dimensional Difference Equation at a Point Using Computer Algebra Methods

Programming and Computer Software - An algorithm for finding the solution to the Cauchy problem for a two-dimensional difference equation with constant coefficients at a point using computer...     

Integral Equation Formulation of the Biharmonic Dirichlet Problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman–Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.