An optimal iteration method with application to the Thomas-Fermi equation

The aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.     

Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation

The linear stability of IMEX (IMplicit–EXplicit) methods and exponential integrators for stiff systems of ODEs arising in the discrete solution of PDEs is examined for nonlinear PDEs with both linear dispersion and dissipation, and a clear method of visualization of the linear stability regions is proposed. Predictions are made based on these visualizations and are supported by a series of experiments on five PDEs including quasigeostrophic equations and stratified Boussinesq equations. The experiments, involving 24 IMEX and exponential methods of third and fourth order, confirm the predictions of the linear stability analysis, that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis. The experiments also demonstrate that IMEX methods achieve comparable stability to exponential methods, and that exponential methods are significantly more accurate only when the problem is nearly linear. Novel IMEX predictor–corrector methods are also derived.     

New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection–reaction equation. By using a Fourier–Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.     

Approximate solutions of nonlinear PDEs by the invariant expansion

<正>It is difficult to obtain exact solutions of the nonlinear partial differential equations(PDEs) due to their complexity and nonlinearity,especially for non-integrable systems.In this paper,some reasonable approximations of real physics are considered,and the invariant expansion is proposed to solve real nonlinear systems.A simple invariant expansion with quite a universal pseudopotential is used for some nonlinear PDEs such as the Korteweg-de Vries(KdV) equation with a fifth-order dispersion term,the perturbed fourth-order KdV equation,the KdV-Burgers equation,and a Boussinesq-type equation.     

High-Frequency Multiconductor Transmission-Line Theory

This work presents a thorough derivation of the full-wave transmission-line equations on the basis of Maxwell’s theory. The multiconductor system is assumed to be composed of nonuniform thin wires. It is shown that the mixed potential integral equations are equivalent to generalized telegrapher equations. Novel, exact, and compact expressions for the multiconductor transmission-line parameters are derived, and their connection to radiation effects is shown. Iteration and perturbation procedures are proposed for the solution of the generalized transmission-line equations.     

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein－Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.     

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.     

A spectral FC solver for the compressible Navier–Stokes equations in general domains I: Explicit time-stepping

We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier–Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods—since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.     

限制加性许瓦兹预条件的变形及其在二维三温能量方程中的应用

给出标准限制加性许瓦兹预条件的变形,并应用当前流行的Newton-Krylov-Schwarz方法,结合该预条件子,求解由二维三温能量方程离散得到的非线性代数方程组,减少收敛所需要的迭代次数和所需的CPU时间.数值实验表明,该方法比标准限制加性许瓦兹预条件方法收敛所需要的迭代次数和CPU时间要少.     

A Multidimensional Superposition Principle: Classical Solitons IV

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition Formulae for nonlinear wave interactions. In the present research the technique of multiexpansions with constraints is considered for finding the above SFs and investigation of the related solitons. (The simplest case of such expansions technically is analogous to the so-called invariant truncated singular expansions.) As the applications, the soliton SFs of the MKdV_±, Kaup-Kupershmidt and new A_± equation are obtained for the bell-shape exponential solitons of the various families, algebraic solitons, and the configuration of the two noninteracting kinks. The linearized, parameterized versions of these SFs are investigated then, and the related analysis of the interactions is presented. The obtained results allow one to consider the one soliton solutions mentioned as the strong bound states of the simpler solitons. Concerning the results for the above concrete nonlinear PDEs, the approach being developed made it possible both to obtain the new results and to reveal new moments for the already known ones.     

Numerical Study of Unsteady MHD Flow and Entropy Generation in a Rotating Permeable Channel with Slip and Hall Effects

This article investigates an unbiased analysis for the unsteady two-dimensional laminar flow of an incompressible, electrically and thermally conducting fluid across the space separated by two infinite rotating permeable walls.The influence of entropy generation, Hall and slip effects are considered within the flow analysis. The problem is modeled based on valid physical arguments and the unsteady system of dimensionless PDEs (partial differential equations) are solved with the help of Finite Difference Scheme. In the presence of pertinent parameters, the precise movement of the flow in terms of velocity, temperature, entropy generation rate, and Bejan numbers are presented graphically, which are parabolic in nature. Streamline profiles are also presented, which exemplify the accurate movement of the flow. The current study is one of the infrequent contributions to the existing literature as previous studies have not attempted to solve the system of high order non-linear PDEs for the unsteady flow with entropy generation and Hall effects in a permeable rotating channel. It is expected that the current analysis would provide a platform for solving the system of nonlinear PDEs of the other unexplored models that are associated to the two-dimensional unsteady flow in a rotating channel.     

一种图像非线性形变的恢复方法

几何成像系统中非线性形变图像的恢复仍是一个未能很好解决的课题。针对常见的双二次、双三次形变图像,提出了以不完全双二次、双三次插值法拟合形变图像,根据形变的特点采用牛顿迭代法和双线性灰度插值法对图像进行恢复。用所提出的方法对失真图像进行恢复,结果表明,仅选用少量参考点便可得到良好的效果。     

Algorithm for solving the inverse problem of ballistics in the case of an asymmetric object

An algorithm to identify the aerodynamic characteristics of an asymmetric object from its trajectory data obtained in a ballistic experiment is developed based on the technique for estimating the nonlinear system’s parameters. Using the method of successive approximations, the coefficients of the aerodynamic function polynomial representation are found that best describe measuring data. The essence of the algorithm is the solution of the direct problem of the symmetric object’s dynamics using the complete set of Euler dynamic equations. The variation of the desired parameters is statistically estimated during calculations. The algorithm allows for jointly processing data of a series of experiments with similar models. Thereby, the volume of processed data is augmented and the final result becomes more accurate.     

Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods

Integration factor methods are a class of “exactly linear part” time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521–537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. In this paper, we solve this problem by applying the Krylov subspace approximations to the matrix exponential operator. Then we apply this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction–diffusion equations. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method in resolving the stiffness of the DG spatial operator for reaction–diffusion PDEs. Application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown.     

基于同伦技术的Burgers方程的小波精细积分算法

以Burgers方程为例,结合区间小波精细积分方法,将同伦摄动方法的应用范围推广到多维非线性问题,给出一种求解非线性偏微分方程的新的小波精细积分方法,得到一种近似解析解的数值结果,对时间步长不敏感,更适合于求解非线性问题.     

First passage time of nonlinear ship rolling in narrow band non-stationary random seas

The generalized extended stochastic central difference (GESCD) method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non-stationary random seas. The GESCD method is based on a combination of the extended stochastic central difference method with a statistical linearization technique, modified adaptive time scheme, and time coordinate transformation. The extended stochastic central difference method is, however, an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non-stationary random disturbances. Approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined. It is concluded that the GESCD method is very accurate, simple and efficient to apply compared with Monte Carlo simulation. The proposed method is applicable to cases with large nonlinearities and intensive random excitations. The approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the Monte Carlo simulation. It is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature.     

A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers’ equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers’ equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.     

Non-similar mixed convection analysis for magnetic flow of second-grade nanofluid over a vertically stretching sheet

The aspiration of this research is to explore the impact of non-similar modeling for mixed convection in magnetized second-grade nanofluid flow. The flow is initiated by the stretching of a sheet at an exponential rate in the upward vertical direction. The buoyancy effects in terms of temperature and concentration differences are inserted in the $x$-momentum equation. The aspects of heat and mass transfer are studied using dimensionless thermophoresis, Schmidt and Brownian motion parameters. The governing coupled partial differential system (PDEs) is remodeled into coupled non-similar nonlinear PDEs by introducing non-similar transformations. The numerical analysis for the dimensionless non-similar partial differential system is performed using a local non-similarity method via bvp4c. Finally, the quantitative effects of emerging dimensionless quantities on the non-dimensional velocity, temperature and mass concentration in the boundary layer are conferred graphically, and inferences are drawn that important quantities of interest are substantially affected by these parameters. It is concluded that non-similar modeling, in contrast to similar models, is more general and more accurate in convection studies in the presence of buoyancy effects for second-grade non-Newtonian fluids.     

基于改进布谷鸟算法识别瞬态热传导问题的导热系数

基于改进布谷鸟算法反演瞬态热传导问题随温度变化的导热系数.采用Kirchhoff变换将非线性热传导问题转换为线性热传导问题,使用边界元法求解瞬态热传导正问题.将导热系数的反演转化为函数表达式中未知参数的反演,使用改进布谷鸟算法求解未知参数.与共轭梯度法相比,改进布谷鸟算法对迭代初值不敏感;与布谷鸟算法相比,改进布谷鸟算法迭代次数大大减少.数值算例表明对改进布谷鸟算法,增加测点数量迭代次数增加;增加鸟巢数量迭代次数减少;减小测量误差计算结果更精确,同时迭代次数更少.数值算例验证了改进布谷鸟算法反演导热系数的准确性和有效性.     

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.