Analytic solution of natural convection flow of a non‐newtonian fluid between two vertical flat plates by using decomposition method

An analysis has been performed to study the natural convection of a non‐Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non‐Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by Adomian decomposition method (ADM) and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non‐linear equations and so can be found widely applicable in engineering and science. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1384–1395, 2010     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems

The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm-Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler-Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k = 1, 2, 3, … , can be obtained by using minimally 2(k + 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

二阶和四阶椭圆型偏微分方程的镜像基本解方法及应用

将基本解方法推广到二阶和四阶椭圆型偏微分方程的对称问题,在边界上不需要处理奇异积分.通过坐标变换,将一般二阶和四阶椭圆型偏微分方程化为目前研究较为成熟的调和或双调和方程.再根据镜像法构造出适合对称条件的基本解函数,简化了计算,且不影响计算的精度.通过数值计算结果可以看出,利用镜像技术构造出的基本解,前期准备数据少,可保持精度,是一种有效的数值方法.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

不定最小二乘问题的改进的不完全双曲Gram-Schmidt预处理算法

应用改进的不完全双曲Gram-Schmidt(IHMGS)方法预处理不定最小二乘问题的共轭梯度法(CGILS)、正交分解法(ILSQR)与广义的最小剩余法(GMRES)等迭代算法来求解大型稀疏的不定最小二乘问题.数值实验表明,IHMGS预处理方法可有效提高相应算法的迭代速度,且当矩阵的条件数比较大时,效果更加显著.     

求解微分方程初值问题的一种弧长法

对于连续介质力学问题中导出的微分方程初值问题,常常具有解奇异性,如不连续、Ｓｔｉｆ性质或激波间断·本文通过在相应空间,引入一个或数个弧长参数变量,克服解的奇异性·对于常微分方程组引入弧长参数变量后,奇异性得以消除和削弱,应用一般的解常微分方程组的方法（如Ｒｕｎｇｅ＿Ｋｕｔａ法）求解·对于偏微分方程引入弧长参数变量后,在相应的空间离散成常微分方程组,用解奇异性常微分方程组相同的方法即可求解·本文给出了两个算例     

Analytical solutions to nonlinear equations arising in heat transfer by variational iteration,homotopy perturbation,and Adomian decomposition methods

As thermal conductivity plays an important role on fin efficiency, we tried to solve heat transfer equation with thermal conductivity as a function of temperature. In this research, some new analytical methods called homotopy perturbation method, variational iteration method, and Adomian decomposition method are introduced to be applied to solve the nonlinear heat transfer equations, and also the comparison of the applied methods (together) is shown graphically. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature.

In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them.

Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods.

In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).

     

On the Convergence of Decoupled Optimal Power Flow Methods

This paper investigates the convergence of decoupled optimal power flow (DOPF) methods used in power systems. In order to make the analysis tractable, a rigorous mathematical reformation of DOPF is presented first to capture the essence of conventional heuristic decompositions. By using a nonlinear complementary problem (NCP) function, the Karush–Kuhn–Tucker (KKT) systems of OPF and its subproblems of DOPF are reformulated as a set of semismooth equations, respectively. The equivalent systems show that the sequence generated by DOPF methods is identical to the sequence generated by Gauss–Seidel methods with respect to nonsmooth equations. This observation motivates us to extend the classical Gauss–Seidel method to semismooth equations. Consequently, a so-called semismooth Gauss–Seidel method is presented, and its related topics such as algorithm and convergence are studied. Based on the new theory, a sufficient convergence condition for DOPF methods is derived. Numerical examples of well-known IEEE test systems are also presented to test and verify the convergence theorem.     

A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L₂,L_∞, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems

A new analytical method called He’s variational iteration method (VIM) is introduced to be applied to solve nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equations and free vibration of a nonlinear system having combined linear and nonlinear springs in series in this article. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with the results of the homotopy analysis method and also with the exact solution. He’s Variational iteration method in this problem functions so better than the homotopy analysis method and exact solutions one of them in per section.     

The method of lines for nonlinear parabolic differential equations with mixed derivatives

Summary By the so-called longitudinal method of lines the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper, for a wide class of nonlinear parabolic differential equations the spatial derivatives occuring in the original problem are replaced by suitable differences such that monotonicity methods become applicable. A convergence theorem is proved. Special interest is devoted to the equationu _t=f(x,t,u,u _x,u _xx), if the matrix of first order derivatives off(x,t,z,p,r) with respect tor may be estimated by a suitable Minkowski matrix.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials

According to the linear theory of elasticity, there exists a combination of different orders of stress singularity at a V-notch tip of bonded dissimilar materials. The singularity reflects a strong stress concentration near the sharp V-notches. In this paper, a new way is proposed in order to determine the orders of singularity for two-dimensional V-notch problems. Firstly, on the basis of an asymptotic stress field in terms of radial coordinates at the V-notch tip, the governing equations of the elastic theory are transformed into an eigenvalue problem of ordinary differential equations (ODEs) with respect to the circumferential coordinate θ around the notch tip. Then the interpolating matrix method established by the first author is further developed to solve the general eigenvalue problem. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the amplitude of the stress field described as generalized stress intensity factors of the V-notches. The present method is also available to deal with the plane V-notch problems in bonded orthotropic multi-material. Finally, numerical examples are presented to illustrate the accuracy and the effectiveness of the method.     

Epsilon-ritz method for solving optimal control problems: Useful parallel solution method

Using Balakrishnan's epsilon problem formulation (Ref. 1) and the Rayleigh-Ritz method with an orthogonal polynomial function basis, optimal control problems are transformed from the standard two-point boundary-value problem to a nonlinear programming problem. The resulting matrix-vector equations describing the optimal solution have standard parallel solution methods for implementation on parallel processor arrays. The method is modified to handle inequality constraints, and some results are presented under which specialized nonlinear functions, such as sines and cosines, can be handled directly. Some computational results performed on an Intel Sugarcube are presented to illustrate that considerable computational savings can be realized by using the proposed solution method.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 John Wiley & Sons, Ltd.