Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

基于扰动波的HVAS液滴耦合振动特性分析

高速电弧喷涂过程中熔融金属液滴的振动特性对喷涂涂层性能具有决定性作用,但限于实验技术很难检测液滴的动力学行为,因而利用扰动波理论建立了气流和液滴的耦合振动方程组,并数值求解获得了喷涂气流最不稳定波数,与Bradley的数据曲线进行了比较,二者呈现出较好的吻合性,误差在士2％之间,验证了所建立的HVAS小扰动液滴模型是有效的;根据HVAS液滴耦合振动模型分析了不同喷涂气流及其速度对不同熔融金属的相应HVAS液滴振动特性。结果表明,HVAS过程中用N2具有一定的优势,且增加气流速度有利于提高喷涂涂层的结合强度,从而为HVAS过程中的材料选择与工艺控制提供了基本的理论依据。     

Wave-like deformations for oscillating carbon nanotubes

Carbon nanostructures such as nanotubes and fullerenes, represent future materials because of their remarkable mechanical, electrical and thermal properties. Double-walled carbon nanotubes are widely studied as possible gigahertz oscillators, where the inner tube oscillates within the outer tube. These oscillators are believed to generate frequencies in the gigahertz range and typically of the order of 1–74 GHz. They are also known to generate wave-like formations on the outer surface. In this paper, we study such induced deformations on the surface of the outer tube, as generated by the moving inner tube. Following previous authors we assume that double-walled carbon nanotubes can be modelled as transversely isotropic linearly elastic materials. Using a previously derived approximate force distribution for the resultant van der Waals forces arising from the interatomic interactions, we solve a dynamic linearly elastic problem, and show that the resulting solution exhibits wave-like behaviour.     

An analytical framework of 2D diffusion,wave-like,telegraph, and Burgers’ models with twofold Caputo derivatives ordering

The purpose of the current work is to provide an analytical solution framework based on extended fractional power series expansion to solve 2D temporal–spatial fractional differential equations. For this purpose, we first present a new trivariate expansion endowed with twofold Caputo-fractional derivatives ordering, namely \(\alpha ,\,\beta \in (0,1]\), to study the combined effect of fractional derivatives on both temporal and spatial coordinates. Then, by virtue of this expansion, a parallel scheme of the Taylor power series solution method is utilized to extract both closed-form and supportive approximate series solutions of 2D temporal–spatial fractional diffusion, wave-like, telegraph, and Burgers’ models. The obtained closed-form solutions are found to be in harmony with the exact solutions exist in the literature when \(\alpha =\beta =1\), which exhibits the legitimacy and the validity of the proposed method. Moreover, the accuracy of the approximate series solutions is validated using graphical and tabular tools. Finally, a version of Taylor’s Theorem that associated with our proposed expansion is derived in terms of mixed fractional derivatives.     

Wave-Like Solutions for a Class of Parabolic Models

A reduction aproach is developed for determining exact solutions of anonlinear second order parabolic PDE. The method in point makes acomplementary use of the leading ideas of the theory of quasilinearhyperbolic systems of first order endowed by differential constraintsand of the techniques providing multiple wave-like solutions ofnonlinear PDEs. The searched solutions exhibit a inherent wave featuresand they are obtained by solving a consistent overdetermined system ofPDEs. Remarkably, in the process it is possible to define nonlinearmodel equations which allow special classes of initial or boundary valueproblems to be solved in a closed form. Within the present reductionapproach exact solutions and model material response functions areobtained for an equation of widespread application in many fields ofinterest.     

New exact solutions to KdV equations with variable coefficients or forcing

Jacobi elliptic function expansion method is extended to construct the exact solutions to another kind of KdV equations, which have variable coefficients or forcing terms. And new periodic solutions obtained by this method can be reduced to the soliton-typed solutions under the limited condition.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

A generalized auxiliary equation method and its applications

In this paper, a generalized auxiliary equation method with the aid of the computer symbolic computation system Maple is proposed to construct more exact solutions of nonlinear evolution equations, namely, the higher-order nonlinear Schrödinger equation, the Whitham–Broer–Kaup system, and the generalized Zakharov equations. As a result, some new types of exact travelling wave solutions are obtained, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions. The method is straightforward and concise, and its applications are promising.     

A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to ana-lyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With themethod of complex variable function, a series of conformal mapping functions are obtained from dif-ferent cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And,the general expressions for the equations of a cylindrical shell, including the solutions of stress concen-trations meeting the boundary conditions of the large openings' edges, are given in the mapping plane.Furthermore, by applying the orthogonal function expansion technique, the problem can be summa-rized into the solution of infinite algebraic equation series. Finally, numerical results are obtained forstress concentration factors at the cutout's edge with various opening's ratios and different loadingconditions. This new method, at the same time, gives a possibility to the research of cylindrical shellswith large non-circular openings or with nozzles.     

Study on Exact Analytical Solutions for Two Systems of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｃｏｕｒｓｅｏｆｓｔｕｄｙｉｎｇｔｈｅｗａｔｅｒｗａｖｅ,ｍａｎｙｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓｗｅｒｅｏｂｔａｉｎｅｄ ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ ,ｍＫｄＶｅｑｕａｔｉｏｎ ,(2 1 )＿ｄｉｍｅｎｓｉｏｎａｌＫＰｅｑｕａｔｉｏｎ ,ｃｏｕｐｌｅｄＫｄＶｅｑｕａｔｉｏｎｓ,ｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓ ,ＷＫＢｅｑｕａｔｉｏｎｓｅｔｃ .[1- 13 ].Ｉｎｏｒｄｅｒｔｏｆｉｎｄｅｘｐｌｉｔｉｃｅｘａｃｔｓｏｌｕｔｉｏ…     

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates

A novel superposition method based on the symplectic geometry approach is presented for exact bending analysis of rectangular cantilever thin plates. The basic equations for rectangular thin plate are first transferred into Hamilton canonical equations. By the symplectic geometry method, the analytic solutions to some problems for plates with slidingly supported edges are derived. Then the exact bending solutions of rectangular cantilever thin plates are obtained using the method of superposition. The symplectic superposition method developed in this paper is completely rational compared with the conventional analytical ones because the predetermination of deflection functions, which is indispensable in existing methods, is dispelled.     

New truncated expansion method and soliton-like solution of variable coefficient KdV-MKdV equation with three arbitrary functions

The truncated expansion method for finding explicit and exact soliton-like solution of variable coefficient nonlinear evolution equation was described. The crucial idea of the method was first the assumption that coefficients of the truncated expansion formal solution are functions of time satisfying a set of algebraic equations, and then a set of ordinary different equations of undetermined functions that can be easily integrated were obtained. The simplicity and effectiveness of the method by application to a general variable coefficient KdV-MKdV equation with three arbitrary functions of time is illustrated.     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｒｕｓｓｅｌａｔｏｒｒｅａｃｔｉｏｎｍｏｄｅｌｐｌａｙｓａｎｉｍｐｏｒｔａｎｔｒｏｌｅｂｏｔｈｉｎｂｉｏｌｏｇｙａｎｄｉｎｃｈｅｍｉｓｔｒｙ .ＳｉｎｃｅｔｈｅｍｏｄｅｌｗａｓｐｕｔｆｏｒｗａｒｄｂｙＰｒｉｇｏｇｉｎｅａｎｄＬｅｆｅｖｅｒｉｎ 1 968,ｍｕｃｈａｔｔｅｎｔｉｏｎｈａｄｂｅｅｎｐａｉｄｔｏｔｈｅｍｏｄｅｌａｎｄｍａｎｙｐｒｏｐｅｒｔｉｅｓｏｆｉｔｈａｄｂｅｅｎｒｅｓｅａｒｃｈｅｄｂｙｍａｎｙｐｅｏｐｌｅｖｉａｕｓｉｎｇｄｉｆｆｅｒｅｎｔｍｅｔｈｏｄｓ[1- 5…     

Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

Differentiator series solution of linear differential ordinary equation

１Ｄｉｆｅｒｅｎｔｉａｔｏｒ,ＩｎｖｅｒｓＯｐｅｒａｔｏｒａｎｄＴｈｅｉｒＰｒｏｐｅｒｔｉｅｓ１．１ＤｉｆｅｒｅｎｔｉａｔｏｒａｎｄｉｎｖｅｒｓｏｐｅｒａｔｏｒＳｕｐｐｏｓｅｔｈｅｌｉｎｅａｒｏｒｄｉｎａｒｙｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆ...     

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators,in which the perturbation procedure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived.The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method.Several strongly nonlinear oscillators with quadratic,cubic,and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.