Generalized differential transform method to differential-difference equation

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Padé technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.     

Maintaining the stability of nonlinear differential equations by the enhancement of HPM

Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained. In this Letter, the need for stability verification is shown through some examples. Consequently, HPM is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic, even the selected linear part is not.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities

This Letter applies He's parameter-expansion method to oscillators with smooth nonlinearities. The method does not depend upon small parameter assumption, hence it is very better than the perturbation method. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain. The method can be easily extended to other nonlinear oscillations.     

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.     

Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation

In this Letter we propose a pseudospectral method for solving Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.     

Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations

The multiple soliton solutions of the approximate equations for long water waves and soliton-like solutions for the dispersive long-wave equations in 2+1 dimensions are constructed by using an extended homogeneous balance method. Solitary wave solutions are shown to be a special case of the present results. This method is simple and has a wide-ranging practicability, and can solve a lot of nonlinear partial differential equations.     

Response of non-linear non-conservative systems of two degrees of freedom to transient excitations

This paper deals with the approximate analysis of non-linear non-conservative systems of two degrees of freedom subjected to transient excitations. By using a transformation of the co-ordinates, the governing differential equations of the system are brought into a form to which the method of averaging of Krylov and Bogoliubov can be applied. The response of a representative spring-mass-damper system to typical pulses like a blast pulse and a half sinusoidal pulse is determined. The validity of the approach is demonstrated by comparison of the approximate solutions with numerical results obtained on a digital computer.     

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

Moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data. In particular, they are often employed when solving partial differential equations on unstructured meshes, which are typically needed when the geometry defining the domain is complex. It is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements, however little research has specifically addressed this issue. In this paper, a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions, and the precise conditions under which singularities occur are identified. The theory is used to develop a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner, while attempting to maintain stencil symmetry as closely as possible. Finally, the scheme is applied in a convection–diffusion equation solver and an incompressible Navier–Stokes solver, and the results are shown to compare favourably with known analytical solutions and previously published results.     

Application of ultraspherical polynomials to non-linear autonomous systems

This paper deals with the approximate solutions of non-linear autonomous systems by the application of ultraspherical polynomials. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by a generalized averaging technique based on the ultraspherical polynomial expansions. The method is illustrated with examples and the results are compared with the digital and analog computer solutions. There is a close agreement between the analytical and exact results.     

A modified stroboscopic method for non-linear vibration equations

A technique is developed for determining approximate periodic solutions to a class of non-linear differential equations which often arise in vibration theory. Although it is akin to the stroboscopic method of Minorsky, no topological arguments are utilized. The method is illustrated by application to a well known example and it is shown to give correct results.     

Exact solutions of the Fokker-Planck equations with moving boundaries

By means of time-dependent similarity transformations, we derive exact solutions of the Fokker-Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker-Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures.     

Piecewise optimal linearization method for nonlinear stochastic differential equations

A method for estimating the dynamical statistical properties of the solutions of nonlinear Langevin-type stochastic differential equations is presented. The non-linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval. The linearization procedure is optimal in the sense of the Chebyshev inequality. Long-term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals. The method is applied to a model nonlinear equation for which the exact time-dependent moments can be obtained by numerical methods. The calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium.Supported in part by the National Science Foundation under Grants CHE77-16307 and PHY76-04761.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

远场模型方程的同伦近似对称约化

以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性. 关键词： 同伦近似对称法 远场模型方程 同伦级数解     

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

In this research, two powerful analytical methods are introduced to handle nonlinear good Boussinesq, heat transfer and coupled Burgers' equations. One is the homotopy-perturbation method (HPM) and the other is the variational iteration method (VIM). VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. HPM converts a difficult problem into a simple one, which can be easily handled. The results attained in this paper confirm the idea that HPM and VIM are powerful mathematical tools and that they can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.     

APPROXIMATE ANALYTICAL SOLUTIONS FOR PRIMARY CHATTER IN THE NON-LINEAR METAL CUTTING MODEL

This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates.     

On non-linear oscillations with slowly varying system parameters

The paper deals with an approximate analysis of non-linear oscillation problems with slowly varying system parameters. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by using the generalized averaging method of Sinha and Srinivasan based on ultraspherical polynomial expansions. The Bogoliubov-Mitropolsky results are given by a particular set of these polynomials. Problems of a single degree of freedom system as well as monofrequency oscillations in systems with multiple degrees of freedom are considered. The approach has been illustrated by an example and the results are compared with the numerical solutions. A close agreement is found.     

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

In this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.     

Optical beams in lossy non-local Kerr media

It is discussed that optical beams propagate in non-local Kerr medium waveguides with losses. A variational principle is carried out for the 1 + 1-D non-local non-linear Schrödinger equation in the presence of the losses. In the strongly non-local case, the approximate analytical solutions are obtained. The lossy soliton solution shows that, Unlike its local counterpart, such lossy strongly non-local soliton does not possess the adiabatic property anymore. In addition, the general approximate results for non-soliton cases are gained. The comparisons between our approximate analytic solutions and numerical simulations confirm our variational approximate solutions.