A Study of Some Nonlinear Partial Differential Equations by Using Adomian Decomposition Method and Variational Iteration Method

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing nonlinear partial differential equations(PDEs)to a model of coupled systems of nonlinear ordinary differential equations using similarity variables(ODEs).In order to obtain the results of a coupled system of nonlinear ODEs,we discuss a method which is known as the differential transform method(DTM).The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs.To observe beast agreement between analytical method and numerical method,we compare our result with the Rung-Kutta method of order four(RK4).We also provide simulation plots to the obtained result by using Mathematica.Onthese plots,we discuss the effect of different parameters which arise during the calculation of the flow model equations.

Mean Square Numerical Methods for Initial Value Random Differential Equations

In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.

Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.

Differential transform method for solving Richards' equation

An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.

IMPROVEMENT ON STABILITY AND CONVERGENCE OF A. D. I. SCHEMES

Alternating direction implicit (A.D.I.) schemes have been proved valuable in the approximation of the solutions of parabolic partial differential equations in multi-dimensional space. Consider equations in the form partial derivative u/partial derivative t - partial derivative/partial derivative x(a(x,y,t) partial derivative u/partial derivative x) - partial derivative/partial derivative y(b(x,y,t) partial derivative u partial derivative y) = f Two A.D.I. schemes, Peaceman-Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence have been analysed with Fourier Method, which cannot be extended beyond the model problem with constant coefficients. Additionally, L-2 energy method has been introduced to analyse the case of non-constant coefficients, however, the conclusions are too weak and incomplete because of the so-called 'equivalence between L-2 norm and H-1 semi-norm'. In this paper, we try to improve these conclusions by H-1 energy estimating method. The principal results are that both of the two A.D.I. schemes are absolutely stable and converge to the exact solution with error estimations O(Delta t(2) + h(2)) in discrete H-1 norm. This implies essential improvement of existing conclusions.

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

EXACT ANALYTIC METHOD FG(?) SOLVING VARIABLE COEFFICIENT DIFFERENTIAL EQUATION

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation formal is obtained. Its convergence in proved. We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given, which indicate that satisfactory results can he obtanedby this method.

Using reproducing kernel for solving a class of partial differential equation with variable-coefficients

How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.

幂函数形式连续变截面梁振动的弯曲固有频率分析

为给变截面变幅杆弯曲振动提供设计依据,该文使用微分方程解析法求解变截面梁固有频率。首先,建立变截面梁模型,其中截面面积和惯性矩均按幂次函数变化。得到变截面梁自由振动时挠度的解析表达式,并获得不同边界条件下梁弯曲振动的固有频率方程。其中惯性矩所对应幂指数与截面面积的幂指数的差值为4时,可得自振频率方程的精确形式;而幂指数差值不等于4时,给出近似解法。其次,对4种具体的变截面梁求解不同边界下的自振频率,并与瑞利-里兹法所得的自振频率解比较。验证精确解法结果的正确性,并发现近似解法结果的相对偏差在5%以内。该解析方法较瑞利-里兹法具有能快速求解的特点,且易于分析截面参数对梁固有频率的影响。由算例可得,边界和其他参数不变时,梁的同阶次无量纲自振频率随着幂次指数的增加而增加。几何参数中仅截面形状参数改变时,随着形状参数的增加,梁的同阶次无量纲自振频率随之减小,但固定-自由梁的第1阶自振频率除外。

基于TRNSYS太阳能地源热泵供暖系统仿真研究

利用DEST建筑能耗模拟软件,对张家口市某建筑进行全年逐时热负荷模拟,利用太阳能和地热能两种可再生能源作为供暖热源,以TRNSYS建立太阳能地源热泵系统仿真模型,在供暖期采用定流量和变流量两种不同的系统运行方式进行模拟分析,分析结果表明,两种供暖系统COP分别为3.74和4.29。采用定流量方式运行时,循环水泵总能耗为9610.87kW/h,热泵机组总能耗为26122.69kW/h,系统总能耗为35733.56kW/h。负荷侧采取变流量方式运行时循环水泵总能耗为6229.89kW/h,热泵机组总能耗为24961.36kW/h,系统总能耗为31191.25kW/h。根据结果可看出,相对于定流量运行,用户侧变流量运行下系统COP提高14.7%,系统总能耗减少14.56%。负荷侧变流量运行下太阳能地源热泵系统整体性能更优。

差压法测定NEG吸氢性能

通过对定压法进行改进提出了差压法,用于测试非蒸散型吸气剂(NEG)的吸氢性能。该方法在进气室和吸气室之间安装固定流导元件,利用调节微调阀提供可控的进气流量,通过测量吸气室、进气室和参考室的压力计算NEG的吸气性能参数。利用两种NEG比较了差压法和改进的定压法所测吸气性能。结果表明:差压法测得的吸气容量比改进的定压法测得的吸气容量低0.07268 Pa·L;腔室壁面吸附能力使改进定压法的试验结果高于理论结果,而差压法的试验结果不受影响;对改进的定压法进行优化后,其计算结果与差压法的接近。差压法可以获得更准确、更有效的吸气剂性能参数,在差压法中,参考部分和试验部分同时进行测试,可缩短测试时间,减少控制进气带来的误差。

杨氏模量实验测量方法的改进

介绍了梁弯曲法测量杨氏模量的实验方法,并在该方法基础上采用LVDT电压位移传感器去测钢尺弯曲形变的同时,增加了利用光杠杆进行读数的方法,对两种测量方法测得的实验数据进行比对,得到利用LVDT电压位移传感器测量数据的方法有实验数据的波动小、精确度高,并且实验装置更加简易,操作更加简单,读取数据直接等优点.

基于改进移动极限常数的变密度拓扑优化方法

为提高基于变密度法的拓扑优化模型的迭代速度,将模型中的极限移动常数,改进为随单元密度变动的变量。结果表明:改进后,在目标函数变动微小的情况下,极大地提高了模型的迭代速率。可见此种方法,有效地改进了拓扑优化模型,具有可行性。

数理方程中分离变量法的思路探究

国内很多“数学物理方程”教材在讲述分离变量法时,解答的第一步设y(x,t)=X(x)T(t),让初学者产生疑问,为什么方程的解具有乘积形式?加法形式的解是否可以?这里给出一个比较浅显的解释,以期让初学者能够比较容易地理解.

余风量控制法在化学实验室压差控制中的应用

实验室作为科研人员核心工作场所,其室内有毒有害物质职业暴露健康风险直接关系着每一位科研人员的身心健康。余风量控制法作为实验室广泛应用的压差控制方法,是防止有毒有害物质外溢、避免交叉污染的有效工程控制手段,对降低科研人员职业暴露健康风险具有显著效果。文章深入探讨了余风量控制法的应用条件、适用性、设计流程及影响因素,并以某化学实验室为例进行系统阐述,旨在为实验室设计与工程实践提供参考依据。