Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

DIFFERENTIATOR SERIES SOLUTION OF LINEAR DIFFERENTIAL ORDINARY EQUATION

In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential difference between this method and the others is that by this method special and general solutions can be obtained directly with the operations of the differentor in the differential equation and without the enlightenment of other scientific knowledge.

Solution of Delay Differential Equations Using a Modified Power Series Method

This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this new approach is applicable to both linear and nonlinear problems. The method produces a system of algebraic equations which is solved to determine the coefficients in the trial solution. The method provides the solution in form of a rapid convergent series. The obtained results for numerical examples demonstrate the reliability and efficiency of the method.

Adomian Modification Methods for the Solution of Chebyshev’s Differential Equations

The current study examines the important class of Chebyshev’s differential equations via the application of the efficient Adomian Decomposition Method (ADM) and its modifications. We have proved the effectiveness of the employed methods by acquiring exact analytical solutions for the governing equations in most cases;while minimal noisy error terms have been observed in a particular method modification. Above all, the presented approaches have rightly affirmed the exactitude of the available literature. More to the point, the application of this methodology could be extended to examine various forms of high-order differential equations, as approximate exact solutions are rapidly attained with less computation stress.

爆破型Volterra积分微分方程的高精度计算

本文求解爆破型非线性Volterra积分微分方程,使用Picard迭代求出解在初始点的有限项分数阶级数展开式.使用级数解分离出方程的初始奇点,提出一种高效的Chebyshev配置法.对于爆破问题,使用待定系数法得到方程解在爆破时刻的渐近展开式主项,并把级数解Padé逼近分母的最小正根作为爆破时间的初步估计,进一步通过积分或微分变换提高爆破时间的预测精度.最后,数值算例验证级数解、Chebyshev配置解和爆破时间预测方法的正确性和有效性.

第二类Fredholm模糊积分方程的模糊数值解

研究了一类模糊集意义下的第二类Fredholm模糊积分方程的数值解。采用残余幂级数法得到第二类Fredholm模糊积分方程的k级截断级数解,将第二类Fredholm模糊积分方程的数值解用泰勒光滑公式展开,并通过代数方程组求解出相关系数。最后,结合数值算例证明了残余幂级数方法的稳定性和收敛性。

A New Technique for Solving Fractional Order Systems: Hermite Collocation Method

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

Some of Semi Analytical Methods for Blasius Problem

In this paper, the Adomian methods, differential transform methods, and Taylor series methods are applied to non-linear differential equations which is called Blasius problem in fluid mechanics. The solutions of the Blasius problem for two cases are obtained by using these methods and their results are shown in table.

A Semi-Analytical Method for Solutions of a Certain Class of Second Order Ordinary Differential Equations

This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.

A New Algorithm Based on Differential Transform Method for Solving Partial Differential Equation System with Initial and Boundary Conditions

In this paper, a new Fourier-differential transform method (FDTM) based on differential transformation method (DTM) is proposed. The method can effectively and quickly solve linear and nonlinear partial differential equations with initial boundary value (IBVP). According to boundary condition, the initial condition is expanded into a Fourier series. After that, the IBVP is transformed to an iterative relation in K-domain. The series solution or exact solution can be obtained. The rationality and practicability of the algorithm FDTM are verified by comparisons of the results obtained by FDTM and the existing analytical solutions.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

综合利用核磁谱差分与谱位移测井提高油层解释精度

以辽河油田应用实例分析了核磁谱差分测井在油气层解释识别应用中的多解性 ,提出综合利用核磁谱差分及谱位移测井减少核磁测井烃检测多解性 ,提高测井解释精度的观点以及解释分析方法 。

一种基于微分方程法的串补线路精确故障测距算法

串联补偿电容的接入使得从线路两端测得的稳态时的电压电流关系不再一一对应,用代数法无法区分故障发生在串补电容的哪一侧,这给以前常用的代数方程定位算法增加了新的困难。文中利用在线路两侧同步采集(用全球定位系统(GPS)进行同步)的电压、电流信号,采用微分方程数学模型并结合串补线路的特点推导了一种新的故障定位算法。该算法分别假定故障点在串补电容的两侧,通过计算得到2个故障定位解,其中一个为真根,另一个为伪根。针对定位过程中出现的真伪根,由暂态过程中电容两侧发生故障时的电压波形不同这一事实,根据线路两侧获取的数据分别计算出的故障点电压应相等这一原理,提出了一种简单、可靠地找出真根去除伪根的方法,可正确判定事故地点。仿真研究表明,该算法具有较高的精度和较强的适应性,能可靠区分真伪根,精确确定事故地点。

串联隔震体系的大变形力学行为分析与试验

研究了由叠层橡胶隔震支座和地下室悬臂柱组成的串联隔震体系的大变形行为。考虑轴向伸长、一阶剪切变形,建立了串联隔震体系大变形分析的几何非线性控制方程,并利用微分求积单元法进行了数值求解。结果表明,串联隔震体系大变形主要出现在叠层橡胶隔震支座处,地下室悬臂柱各处内力值呈近似均匀分布,计算结果与试验结果规律吻合。

线性微分方程的微分算子级数解法

介绍了微分算子级数法及其求解线性常微分方程通解、特解的原理、方法和实例·这个方法和其它解法的差别，在于不借助其它学科知识的启示。

考虑底面摩阻效应的弹性地基梁微分算子级数法

假定地基为具有切向和法向反力的弹性支承体,考虑弹性地基梁的纵横向耦合变形,根据弹性力学基本原理,从梁体微段分析出发,结合小变形条件下梁体几何方程和线弹性本构关系,导得考虑摩阻效应时关于法向位移和切向位移的微分方程组:在此基础上利用微分算子级数法对微分方程组进行求解,得到考虑纵横向耦合变形和底面摩阻效应时地基梁法向挠度和切向相对位移计算公式。最后对一个地基梁实例进行了计算分析,并对摩阻效应进行了讨论。结果表明,微分算子级数法解答能有效地考虑地基梁纵横向耦合变形特征和底面摩阻效应。

基于时序InSAR的常州市2015-2018年地面沉降监测

地面沉降不仅会对城市公共基础设施造成安全隐患,也会对当地的经济、环境和可持续发展造成影响.常州市位于我国著名的苏锡常地面沉降带,也是长三角地区的重要城市之一.地面沉降监测一直是该地区掌握地面沉降状态、预防地面沉降灾害的重要举措.利用2015年5月至2018年5月期间的36景Sentinel-1A的IW(Interferometric Wide Swath)模式的SLC(Single Look Complex) SAR(Synthetic Aperture Radar)数据,基于时序InSAR(Interferometric Synthetic Aperture Radar)方法对常州市地面沉降进行了监测,获得了常州地区该时间段年均沉降速率和时序地表累计形变图,揭示了观测期间常州市各地区详细的地面沉降时空分布和变化特征.结果表明,在2015-2018年整个观测期间,常州地区地表形变呈现出"整体相对稳定,武进区局部沉降和抬升较为突出"的特征,与该地区历史沉降监测结果具有良好的一致性和延续性.这一监测结果有助于全面掌握常州地区最近的地面沉降状况,建立并完善长期、连续的地面沉降时空演化监测记录,为城市基础设施选址、规划以及建立预警预报机制等提供决策依据.