Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

The rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

A multidomain integrated‐radial‐basis‐function collocation method for elliptic problems

This article is concerned with the use of integrated radial‐basis‐function networks (IRBFNs) and nonoverlapping domain decompositions (DDs) for numerically solving one‐ and two‐dimensional elliptic problems. A substructuring technique is adopted, where subproblems are discretized by means of one‐dimensional IRBFNs. A distinguishing feature of the present DD technique is that the continuity of the RBF solution across the interfaces is enforced with one order higher than with conventional DD techniques. Several test problems governed by second‐ and fourth‐order differential equations are considered to investigate the accuracy of the proposed technique. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

This paper presents an MLP‐type neural network with some fixed connections and a backpropagation‐type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with n equations and n unknowns. The proposed structure is based on a backpropagation‐type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

一组十二参高精度矩形板元

Abstract. Using the method of undetermined function, a set of 12 parameter rectangular p|atedement with doub[e set parameter and geometry symmetry is constructed. Their consistencyerror are O(h2) , one order higher than the usua[ 12 parameter rectangu|ar p[ate elements.     

The Runge‐Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second‐order scheme for solving the system of Euler equations, which combines the Runge‐Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method‐of‐lines approach will be discretized using a Runge‐Kutta method. Finally, a second‐order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one‐ and two‐dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A continuum theory for first‐order phase transitions based on the balance of structure order

First‐order phase transitions are modelled by a non‐homogeneous, time‐dependent scalar‐valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius–Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg‐de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh‐order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite‐order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite‐difference discretization, in which a lattice generalized solitary wave solution is found.     

Local existence and uniqueness for a geometrically exact membrane‐plate with viscoelastic transverse shear resistance

We prove the local existence and uniqueness to a geometrically exact, observer‐invariant membrane‐plate model introduced by the author. The model consists of an elliptic partial differential system of equations describing the equilibrium response of the membrane which is non‐linearly coupled with a viscoelastic evolution equation for exact rotations, taking on the role of an orthonormal triad of directors. This coupling introduces a viscoelastic transverse shear resistance. Refined elliptic regularity results together with a new extended Korn's first inequality for plates and shells allow to proceed by a fixed point argument in appropriately chosen Sobolev‐spaces in order to prove existence and uniqueness. Copyright © 2004 John Wiley & Sons, Ltd.     

Generalization of the Maxwell equation and relation to the elastic equation

In this paper, we propose a hyperbolic system of first‐order pseudo‐differential equations as generalization of the Maxwell equation. We state basic properties of this system corresponding to the ones of the (usual) Maxwell equation and explain that several known generalized Maxwell equations presented by some researchers can be integrated into the system. Namely, their equations can be regarded as our equation in special cases. Their generalized equations admit not only transversal but also longitudinal waves and are examined from the physical viewpoint. Using the present system, from the mathematical viewpoint, we interpret the meaning for presence of the longitudinal wave (with the transversal one) in their generalized equations. This presence means existence of more than one non‐zero characteristic root for the system (ie, non‐zero eigenvalue of the symbol). We prove also that our system becomes a first‐order expression of (generalized) elastic equations. Furthermore, it is shown that introducing the elastic equations implies expressing the generalized Maxwell equations by the potentials.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.     

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

Based on Givens‐like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher‐order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher‐order tensor with orthogonal factors. Furthermore, based on cross‐high‐order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem. The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

关于圆簿板大挠度问题的正交条件解法

本文重新考察了钱伟长教授求解圆薄板大挠度问题的系统近似法,发现此法实质上可视为奇异摄动理论中的变形参数法.以无量纲中心挠度为小参数,将挠度、中面薄膜力和载荷参数作渐近展开,我们对所得的递推方程给出了正交条件(可解性条件),据此可确定圆薄板的刚度特性.本文指出,利用圆薄板小挠度解和正交条件,可以不经求解方程而导得载荷参数与中心挠度关系的三阶近似以及中心点、边缘处的薄膜力的首项近似.文中对若干特例(均布载荷、复合载荷、各种边界条件)进行了具体计算,所得的结果与钱伟长、叶开沅、黄黔等人在文[1～4]中给出的结果完全相符.     

Numerical bifurcation analysis of a ‘car‐following’ model

We study a system of ordinary differential equations describing a car‐following model for the motion of N car around a circular highway. All cars behave in the same way. The acceleration of each car is determined as a function of the headway (optimal velocity function). This model is known to have a solution with constant velocities and headways which, in a certain parameter regime, is stable and, varying the density of the cars, the loss of stability is generally due to a super‐ or subcritical Hopf bifurcation. Guided by analytical results, we numerically investigate the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of the model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Finite element solution of double-diffusive boundary layer flow of viscoelastic nanofluids over a stretching sheet

This paper deals with the double-diffusive boundary layer flow of non-Newtonian nanofluid over a stretching sheet. In this model, where binary nanofluid is used, the Brownian motion and thermophoresis are classified as the main mechanisms which are responsible for the enhancement of the convection features of the nanofluid. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of group theory transformations. The variational finite element method (FEM) is used to solve these ordinary differential equations. We have examined the effects of different controlling parameters, namely, the Brownian motion parameter, the thermophoresis parameter, modified Dufour number, viscoelastic parameter, Prandtl number, regular Lewis number, Dufour Lewis number, and nanofluid Lewis number on the flow field and heat transfer characteristics. Graphical display of the numerical examine are performed to illustrate the influence of various flow parameters on the velocity, temperature, concentration, reduced Nusselt, reduced Sherwood and reduced nanofluid Sherwood number distributions. The present study has many applications in coating and suspensions, movement of biological fluids, cooling of metallic plate, melt-spinning, heat exchangers technology, and oceanography.