Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.

Analysis and Numerical Computations of the Multi-Dimensional,Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation

The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.

Exact solutions of a time-fractional modified KdV equation via bifurcation analysis

The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.

煤张开型裂隙三维宏细观演化特征及扰动因素探究

裂隙演化方式受控于诸如矿物特征及围压条件等内外环境,为探究含裂隙煤体裂隙宏细观演化特征及影响因素的围压效应,基于工业CT扫描系统及其搭载的三轴加载系统对含裂隙煤体开展三轴静载试验,以多角度联合表征,对原生裂隙、矿物及围压的内外条件相互作用机制做出合理解释。结果表明:①围压会改变煤体初始损伤显著区位置,使其随围压升高由裂隙尖端过渡至煤体上、下端,且微孔隙和大尺寸裂隙之间比微孔隙和微孔隙之间更易相互贯通,并产生新的宏观裂纹。②围压升高使得三维动态分形维数由缓慢增加、快速增加和平稳增加转变为平稳增加、快速增加和缓慢增加的发展阶段,可表征裂隙的时间演化规律。③含裂隙煤体在单轴或低围压下呈纵向拉伸破坏,高围压会使其破坏方式趋于剪切,并通过2种途径提升煤体强度。④起裂角理论值偏离试验值程度随围压增加而增加,与煤体由矿物分布引起的离散度数值关系一致。⑤根据裂隙的受力成分及矿物分布特征将裂隙扩展行为分为直驱、绕核和错核3种类型,该扩展行为受围压对裂隙的作用力成分影响,由相对纯粹拉应力、拉伸–剪切复合应力和相对纯粹剪应力作用的裂纹分别对应以上3种扩展行为,即对裂隙的扩展影响形式表现为以围压为主,矿物赋存形态为辅。

超临界CO_(2)作用下无烟煤孔隙结构演化时间效应规律

CO_(2)地质封存技术是实现减排和废弃矿井资源再利用的有效技术之一,但煤层孔隙结构在CO_(2)注入后的变化具有时间效应,其演化规律将直接影响CO_(2)地质封存潜力。为了探索作用时间增加后煤层孔隙结构变化的真实情况,提高CO_(2)地质封存量计算精度,通过对重庆市南桐矿区无烟煤进行超临界CO_(2)吸附实验、低温氮气吸附实验、X射线衍射测试、核磁共振测试、扫描电镜测试等实验,研究了不同超临界CO_(2)作用时间对该区域内煤样孔隙结构的演化规律。研究结果表明:①超临界CO_(2)作用后煤样孔隙度的变化主要发生在7 d内,处理7 d后煤样孔隙度增加了3.20%,7~14 d内煤样孔隙度仅增加了0.05%;②随作用时间增加,煤样孔隙的比表面积增加但变化量逐渐减小,7 d内日均增长19.74%~24.50%,7~14 d内日均增长2.37%~4.60%,其变化规律近似呈对数函数关系;③超临界CO_(2)与煤产生的化学反应引起矿物质的溶解,增大了煤体的表面粗糙度,表征煤样粗糙度的分形维数随着作用时间的增加持续增大;④超临界CO_(2)作用后能有效改变煤样的孔隙度及表面形貌,为CO_(2)吸附提供了更多吸附点位,超临界CO_(2)对孔隙结构的改造效果在7 d内比较明显。结论认为,超临界CO_(2)作用下无烟煤孔隙结构的变化随作用时间增加最终会趋于稳定,研究结果可为无烟煤储层CO_(2)地质封存潜力评价提供理论指导。

不同放牧时间对荒漠草原土壤颗粒组成及分形维数的影响

研究放牧时间对草原土壤颗粒组成和分形维数的影响,有助于了解放牧干扰下草地土壤质量和草地生态状况,为草地适时放牧提供理论依据。以宁夏荒漠草原为对象,设置禁牧封育(FY)、传统时间轮牧(FG)、延迟开始轮牧(YG)、提前结束轮牧(TG)、延迟开始并提前结束轮牧(YT)5种不同时间放牧处理,研究了不同放牧时间下荒漠草原土壤颗粒组成及分形维数的变化及其影响因素。结果表明:各处理下荒漠草原土壤质地以粉粒、极细砂粒和细砂粒为主,0~40 cm土层处理间土壤颗粒组成差异显著(P<0.05),土壤颗粒分形维数变化范围为2.11~2.75,且随土层加深逐渐增大。总体上土壤YG粉粒含量、分形维数最高;0~10 cm土层除FY最低外,其余土层粉粒含量与分形维数均以FG处理最低。土壤分形维数与粉粒含量呈极显著正相关(P<0.01),与极细砂粒和细砂粒含量呈显著或极显著负相关(P<0.01,P<0.05),土壤颗粒分形维数与有机碳、总孔隙度呈正相关,与容重、速效钾、毛管孔隙度呈负相关。研究认为,分形维数可表征荒漠草原土壤质地变化,延迟放牧改善了土壤颗粒组成及分形维数,有利于土壤质量发展,可作为荒漠草原适时放牧的参考依据。

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme.This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space.A stability analysis of the scheme is presented to confirm that the errors do not amplify.A convergence analysis is also presented.Computational experiments are carried out in addition to verify the theoretical analysis.Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

In this paper,a new type of the discrete fractional Gronwall inequality is developed,which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdifiFusion equation.Based on the temporal-spatial error splitting argument technique,the discrete fractional Gronwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdififusion equation.

Fractional Difference Approximations for Time-Fractional Telegraph Equation

In this paper, we approximate the solution to time-fractional telegraph equation by two kinds of difference methods: the Grünwald formula and Caputo fractional difference.

New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.

基于概率穿越可视图的时间序列网络多重分形研究与应用

对有限穿越可视图进行改进,提出了概率穿越可视图.首先,将间接可视的节点之间的关联处理为穿越距离的函数,而且节点之间的关联随穿越距离的增加而减小,从而将无权图形式的时间序列网络推广到带权图;其次,采用复杂网络中的网络维数计算方法处理所得到的时间序列网络的分形维数,从而对其自相似特性进行分析;最后,通过划分不同的时间粒度得到多个不同的时间序列网络,对应有多个不同的分形维数,进而分析了所得到的时间序列网络的多重分形特性.在与经典的可视图法、水平可视图法及有限穿越可视图法的对比中论证了所提出的概率穿越可视图的优势.

HIGH-ORDER NUMERICAL METHOD FOR SOLVING A SPACE DISTRIBUTED-ORDER TIME-FRACTIONAL DIFFUSION EQUATION

This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation.First,we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives.Second,based on the piecewise-quadratic polynomials,we construct the nodal basis functions,and then discretize the multi-term fractional equation by the finite volume method.For the time-fractional derivative,the finite difference method is used.Finally,the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ^(2)+τ^(2-β)+h^(3)),whereτand h are the time step size and the space step size,respectively.A numerical example is presented to verify the effectiveness of the proposed method.

Variational iteration method for solving time-fractional diffusion equations in porous the medium

The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models with fractional derivatives are investigated analytically, and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.

Numerical Computation for Time-fractional Gas Dynamics Equations by Fractional Reduced Differential Transforms Method

The present article is concerned with the implementation of a recent semi-analytical method referred to as fractional reduced differential transform method （FRDTM） for computation of approximate solution of time-fractional gas dynamics equation （TFGDE） arising in shock fronts. In this approach, the fractional derivative is described in the Caputo sense. Four numeric experiments have been carried out to confirm the validity and the efficiency of the method. It is found that the exact or a closed approximate analytical solution of a fractional nonlinear differential equations arising in allied science and engineering can be obtained easily. Moreover, due to its small size of calculation contrary to the other analytical approaches while dealing with a complex and tedious physical problems arising in various branches of natural sciences and engineering, it is very easy to implement.

基于复杂网络的空中交通流分形特征分析

为准确把握空中交通流量变化规律,掌握空中交通系统内在特性,需要对空中交通流时间序列进行基于复杂网络的分形特征分析。收集空中交通流量数据,利用可视图方法构建复杂网络模型,分析网络拓扑结构,验证了该网络度分布服从幂律分布,拟合直线斜率为-2.086,证明了网络是无标度网络,具有单分形特征。验证了覆盖整个网络所有节点所需要的最少盒子数目与盒子直径成幂律关系,拟合直线斜率为-0.2121,相关系数为-0.8722,再次证明了网络具有单分形特征。通过验证网络广义分形维数关于参数的图像为非线性,拟合直线斜率分别为-1.942、-1.936、-1.78,相关系数均在0.8以上,拟合效果较好,证明了网络具有多重分形特性。通过计算重整化前后网络的幂指数相似,证明了网络具有自相似性。结果表明,应用复杂网络的理论分析空中交通流时间序列是可行有效的,为进一步深入应用研究奠定了基础。

正极性操作冲击电压下流注放电路径随机发展特性

流注放电具有分散性和随机性的典型特征,研究流注放电过程中起始时延分散性、发展路径随机性等因素对于加深放电过程机理认识、提升间隙绝缘性能具有重要意义。本文结合流注起始判据及分形理论建立了考虑空间电荷屏蔽效应的空气间隙动态流注随机发展计算模型,仿真模拟了棒-板、锥-板等间隙结构放电路径发展过程,得到了流注起始时延、发展形态、发展速度及空间电荷量等特征参数,并搭建光电联合检测平台获取试验值进行对比分析,验证了模型的有效性。仿真结果表明:在标准操作冲击电压作用下,电极曲率半径增大,起始时延分散性增加;随着外施电压上升率从0.45 kV/μs上升至11.15 kV/μs,棒-板间隙流注轴向发展速度从5.70×10^(5) m/s增加至8.32×10^(6) m/s,放电通道分形维数自1.19增加至1.28,放电树枝分叉增多,放电路径随机性增加。

不同应力比下砂岩时滞变形破坏特征研究

高地应力环境下地下洞室中的硬质围岩在开挖过程中极易发生时滞型岩爆.针对地下洞室开挖后围压的实际应力状态,以细砂岩为研究对象开展不同应力水平下的时滞变形破坏试验,并基于分形理论定量描述了破坏后宏观碎屑和破坏断面微观结构的分形特征.研究结果表明:在高应力水平下砂岩表现出显著的时滞变形特征,时滞变形段持续时间随应力比增加呈下降趋势,两者间呈指数函数关系;基于分形理论,通过粒度-数量、质量-粒度、几何参数-数量、盒数维数法等方法计算宏微观分形维数可知,时滞变形破坏程度与分形特征相关性显著,宏微观分形维数随破坏程度增大呈线性增加.研究结果可为阐明岩体时滞变形破坏机理提供一定参考价值.

基于时间切片序列图像的数据流量检测技术

为实现I/O设备数据发送状态的智能快速检测,采用开源Wireshark软件对设备的时间序列数据流量进行采集,提出对I/O设备时间序列流量图像进行切片处理的方法。针对切片图像应用分形理论表征设备通信数据流量的发送状态,基于通信网络数据流量折线图利用Matlab编写计算程序计算数据流的分形维数,将I/O设备正常传输数据状态和停止传输数据状态两种情况下的分形维数进行对比分析。计算数据表明分形维数可以用来表征设备通信数据流量的变化,根据分形维数数值能够区分数据正常传输与异常传输两种状态,证实该方法可用于设备的网络数据流量检测。