New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Numerical Solutions of a New Type of Fractional Coupled Nonlinear Equations

In this paper, we investigate a new type of fractional coupled nonlinear equations. By introducing the fractional derivative that satisfies the Caputo＇s definition, we directly extend the applications of the Adomian decomposition method to the new system. As a result, with the aid of Maple, the realistic and convergent rapidly series solutions are obtained with easily computable components. Two famous fractional coupled examples： KdV and mKdV equations, are used to illustrate the efficiency and accuracy of the proposed method.

The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid

In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.

Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.

Approximate solutions to fractional differential equations

In this paper, the time-fractional coupled viscous Burgers' equation(CVBE)and Drinfeld-Sokolov-Wilson equation(DSWE) are solved by the Sawi transform coupled homotopy perturbation method(HPM). The approximate series solutions to these two equations are obtained. Meanwhile, the absolute error between the approximate solution given in this paper and the exact solution given in the literature is analyzed. By comparison of the graphs of the function when the fractional order α takes different values, the properties of the equations are given as a conclusion.

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.

基于CLSVOF-IB方法海洋管道流固耦合特性研究

本文利用CLSVOF-IB方法研究海洋管道所受的流体作用力及其涡激振动特性,其中CLSVOF(Coupled Level-Set and VOF)方法用来模拟海洋自由波面,浸入边界(immersed boundary,IB)方法用于模拟海洋管道与流体之间的相互作用力。计算结果表明,CLSVOF-IB方法能够准确分析海洋立管的流固耦合特性,可通过选择合适的管道直径和来流速度来解决管道的流致振动问题。此外通过改变流体傅汝德数(Fr数),结果还表明该数值方法能较好捕捉海洋管道与海洋自由波面的相互作用过程。

酸分馏效应对GasBench-IRMS测定碳酸盐碳、氧同位素的影响

本文利用在线制样装置(GasBench)和稳定同位素比质谱仪(IRMS)测试一系列碳酸盐标准物质和现代海洋浮游有孔虫样品,在不同测试条件下,分析不同性状碳酸盐矿物的氧同位素分馏程度,计算氧同位素酸分馏系数,探讨酸分馏效应对不同碳酸盐矿物氧同位素测定的影响机制和控制因素,以及对氧同位素测试值(δ^(18)O_(m))的校正方法。结果表明,酸分馏效应对氧同位素值的影响主要是由于生成的CO_(2)气体与顶空瓶中气态水(高温时主要影响)或酸中游离态自由水(低温时主要影响)之间发生氧同位素交换引起的。不同性状碳酸盐标准物质的氧同位素酸分馏系数具有显著差异,且与氧同位素组成相关。δ^(18)O_(m)值发生偏移的程度与反应温度、样品量、磷酸含水量以及酸中游离态和气态水的氧同位素组成与样品氧同位素组成之间的差异等因素有关。氧同位素酸分馏效应与信号强度具有显著相关性。建议精确称量相同质量的标准物质和样品,遵循信号强度匹配的原则,并采用多种标准物质线性校正的方法有效校正氧同位素分馏现象。在较低的反应温度(25℃)下,氧同位素分馏程度较轻微,δ^(18)O分析精度更高,优于0.05‰。本方法为精确测定碳酸盐中碳、氧同位素组成提供了参考,可有效提高相关的测试精度。

Exact Solution of Space-Time Fractional Coupled EW and Coupled MEW Equations Using Modified Kudryashov Method

In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation(CEWE) and the space-time fractional coupled modified equal width wave equation(CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.

CFD Simulation of a container ship in random waves using a coupled level-set and volume of fluid method

The coupled level-set and volume of fluid(CLSVOF)method is an advanced interface-capturing method that has been extended to handle overset grid systems.However,artificial uneven interface may be observed across block boundaries of different sizes and geometries.We present an improved inter-grid VOF interpolation and mass correction scheme to address the issue.To demonstrate the capability of the improved CLSVOF method,it is applied to the simulation of a container ship in pitch and heave motions under both head sea and following sea irregular wave conditions.Our simulation proves that the improved CLSVOF method is capable of revealing detailed physics difficult to see with other methods.Those phenomena simulated in our work include the extensive greenwater propagation on the ship deck,the breakup of overtopping waves into small droplets,and the formation and collapse of air pockets in sudden bow and stern slamming which cause strong and highly localized impacts on the ship bow,stern,and rudder.

THE UPWIND FINITE DIFFERENCE FRACTIONAL STEPS METHOD FOR NONLINEAR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, trod two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.

多收集器等离子体质谱快速精确测定钕同位素比值

应用多收集器等离子体质谱,采用双分馏系数内部校正法对国际标样Shin-EtsuJN di-1和实验室标样Nd-GIG的143Nd/144Nd比值进行了为期3个月的测量统计,结果分别为0.512121±0.000011(2σ,SD)、0.511531±0.000010(2σ,SD)。采用单分馏系数外部校正法对Nd-GIG标样的测量统计结果为0.511522±0.000008(2σ,SD)。测量结果在分析误差范围内与其推荐值或表面热电离质谱测量值完全一致。对m(Ce)/m(Nd)约为0.10的两份Nd-GIG混合样进行了测量,结果表明,利用多收集器等离子体质谱可直接分析残留铈的样品,其测量结果在误差范围内。

应用Riccati展开法求非线性分数阶偏微分方程的新精确解(英文)

应用Riccati展开法和复变换获得非线性分数阶Sharma-Tasso-Olever方程和时空分数阶耦合Burgers方程的精确解,这些解包括三角函数解和双曲函数解.因此,我们介绍这种方法对于研究非线性分数阶偏微分方程具有十分重要的意义.

分数阶对偶Burger方程的精确解

将分数阶复变换方法和tanh函数方法相结合,得到了一种用来求解时-空分数阶非线性微分方程精确解的复变换-tanh函数方法。借助于软件Mathematica的符号计算功能,使用该方法求解了分数阶对偶Burger方程,得到了分数阶对偶Burger方程的新的精确解。

基于分数阶黏弹性地基模型的群桩时效行为研究

基于有限单元法,将群桩划分为有限个2节点单元,从而构建了竖向受荷群桩的有限元求解方程;为了模拟饱和软黏土的流变特性,通过Laplace变换,推导了分数阶导数Merchant模型中的应力–应变关系;引入弹性-黏弹性对应原理,得到横观各向同性分数阶黏弹性饱和软土地基的边界元解答;考虑桩-土界面处的位移协调,将群桩有限元求解方程和地基边界元解答进行耦合,推导出群桩-土相互作用方程;再根据群桩承台的位移协调条件,求解得到群桩基础状态量的时变解答;基于以上理论,设计了数值算例,以验证方法的正确性,并分析了分数阶次对群桩时效行为的影响。

模件式脉冲转换系统三维流动的计算

阐述了用分形耦合贴体网格坐标和流体有限控制容积法，求解具有多分支、变曲率和变截面的复杂几何形状和边界条件的柴油机模件式脉冲转换系统中的非定常粘性可压三维流动计算问题，编制了计算程序进行计算．计算结果表明，这种计算方法对柴油机模件式脉冲转换系统的流动计算是相当有效的．其计算程序不仅可用于柴油机模件式脉冲转换系统，亦可用于脉冲系统和定压系统的流动计算，具有一定的通用性．