A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities

A new aftertreatment technique is developed in this paper to deal with the truncated series derived by the differential transformation method (DTM) to obtain approximate periodic solutions. The proposed aftertreatment technique splits into two types, named as (Sine-AT technique, SAT) and (Cosine-AT technique, CAT). It is shown that the differential transformation method with the proposed aftertreatment technique is very effective and convenient for a class of nonlinear oscillatory problems with fractional nonlinearities without any need for Padé approximants or Laplace transform.     

改进的微分变换法对不连续冲击波的求解

提出了一种改进的微分变换法,将Padé逼近法与标准微分变换法结合,这种改进的微分变换法主要应用于对冲击波的分析处理方面,能够改善级数的收敛性,并且给出收敛的渐进级数解,甚至精确解,从而为求解强非线性间断问题提供了一种有效的解析方法．     

Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM

In this study, generalized Hirota-Satsuma coupled KdV equation is solved using by two recent semi-analytic methods, differential transform method (DTM) and reduced form of differential transformation method (so called RDTM). The concepts of DTM and RDTM is briefly introduced, and their application for generalized Hirota-Satsuma coupled KdV equation is studied. The results obtained employing DTM and RDTM are compared with together and exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by classic DTM. The numerical results reveal that the RDTM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the RDTM can be found widely applicable in engineering.     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method

In this paper, by introducing the fractional derivative in the sense of Caputo, the generalized two-dimensional differential transform method (DTM) is directly applied to solve the coupled Burgers equations with space- and time-fractional derivatives. The presented method is a numerical method based on the generalized Taylor series formula which constructs an analytical solution in the form of a polynomial. Several illustrative examples are given to demonstrate the effectiveness of the generalized two-dimensional DTM for the equations.     

Solution of free vibration equations of beam on elastic soil by using differential transform method

Free vibration differential equations of motion of one end fixed, the other simply supported and axial loaded beams on elastic soil is solved using differential transform method (DTM), analytical solution and frequency factors are obtained.     

微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials

The free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams are studied through solving the governing differential equations of motion. Observing the fact that the conventional differential transform method (DTM) does not necessarily converge to satisfactory results, a new approach based on DTM called differential transform element method (DTEM) is introduced which considerably improves the convergence rate of the method. In addition to DTEM, differential quadrature element method of lowest-order (DQEL) is used to solve the governing differential equation, as well. Carrying out several numerical examples, the competency of DQEL and DTEM in determination of free longitudinal and free transverse frequencies and critical buckling load of tapered Euler–Bernoulli beams made of axially functionally graded materials is verified.     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

Numerical solution of special 12th-order boundary value problems using differential transform method

In this paper, a differential transform method (DTM) is used to find the numerical solution of a special 12th-order boundary value problems with two point boundary conditions. The analysis is accompanied by testing differential transform method both on linear and nonlinear problems from the literature [Wazwaz AM. Approximate solutions to boundary value problems of higher-order by the modified decomposition method. Comput Math Appl 2000:40;679–91; Siddiqi SS, Ghazala Akram. Solutions of 12th order boundary value problems using non-polynomial spline technique. Appl Math Comput 2007. doi:10.1016/j.amc.2007.10.015; Siddiqi SS, Twizell EH. Spline solutions of linear 12th-order boundary value problems. J Comput Appl Math 1997;78:371–90]. Numerical experiments and comparison with existing methods are performed to demonstrate reliability and efficiency of the proposed method.     

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

Optimal shape analysis of a column structure under various loading conditions by using differential transform method (DTM)

The effects of loading on the optimal shape of an Euler-Bernoulli column is investigated by considering four loading conditions which are mainly classified as eccentric compressive and follower type. The governing equations obtained from the structural stability condition of the column are used as a constraint to determine the minimum value of the volume by applying Hamilton principle. In the preceding task, the analysis is presented in step-by-step manner. The calculations are carried out by using differential transform method (DTM) which is a seminumerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the non-linear constrained governing equations are reduced to recurrence relations and related boundary conditions are transformed into a set of algebraic equations. The optimal distribution of cross-sectional area along column-length is obtained. Then, the volume of such column is calculated and compared to that of the uniform column which is also stable under given loading. The results obtained revealed out that DTM is a quite powerful solution technique for optimal shape analysis of a column structure.     

A new method for a generalized Hirota-Satsuma coupled KdV equation

In this paper, differential transform method (DTM), which is one of the approximate methods is implemented for solving the nonlinear Hirota-Satsuma coupled KdV partial differential equation. A variety of initial value system is considered, and the convergence of the method as applied to the Hirota-Satsuma coupled KdV equation is illustrated numerically. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient. Numerical examples are illustrated the pertinent features of the proposed algorithm.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Application of the differential transformation method for the solution of the hyperchaotic Rössler system

The aim of this paper is to investigate the accuracy of the differential transformation method (DTM) for solving the hyperchaotic Rössler system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the DTM solutions and the fourth-order Runge–Kutta (RK4) solutions are made. The DTM scheme obtained from the DTM yields an analytical solution in the form of a rapidly convergent series. The direct symbolic-numeric scheme is shown to be efficient and accurate.     

三重介质油藏模型及其Laplace空间解

本文建立了裂缝、洞与井筒连通的三重介质油藏的试井解释模型,利用Laplace变换和构造通解法,研究了这一类线性偏微分方程的边值问题,得到了其Laplace空间解,为裂缝、洞与井筒连通油藏的试井分析提供了坚实的理论基础。     

A Laplace variational iteration strategy for the solution of differential equations

The aim of this article is to introduce a novel Laplace variational numerical scheme, based on the variational iteration method (VIM) and Laplace transform, for the solution of certain classes of linear and nonlinear differential equations. The strategy is outlined and then illustrated through a number of test examples. The results assert that this alternative approach yields accurate results, converges rapidly and handles impulse functions and the ones with discontinuities.     

A random differential transform method: Theory and applications

In this article, a Differential Transform Method (DTM) based on the mean fourth calculus is developed to solve random differential equations. An analytical mean fourth convergent series solution is found for a nonlinear random Riccati differential equation by using the random DTM. Besides obtaining the series solution of the Riccati equation, we provide approximations of the main statistical functions of the stochastic solution process such as the mean and variance. These approximations are compared to those obtained by the Euler and Monte Carlo methods. It is shown that this method applied to the random Riccati differential equation is more efficient than the two above mentioned methods.     

The Laplace transform and polynomial Trefftz method for a class of time dependent PDEs

In this paper, we present a new method for solving 1D time dependent partial differential equations based on the Laplace transform (LT). As a result, the problem is converted into a stationary boundary value problem (BVP) which depends on the parameter of LT. The resulting BVP is solved by the polynomial Trefftz method (PTM), which can be regarded as a meshless method. In PTM, the source term is approximated by a truncated series of Chebyshev polynomials and the particular solution is obtained from a recursive procedure. Talbot’s method is employed for the numerical inversion of LT. The method is tested with the help of some numerical examples.     

凝析油-气两相流偏微分方程组问题的求解及应用

根据凝析气藏开发的实际需要，引入凝析油－气两相拟压力、拟时间函数，建立了凝析气在双重渗透率介质中的不稳定渗流新模型，该模型为一类偏微分方程组的混合问题，本文研究了这类偏微分方程组问题的求解方法。