分数阶偏微分方程数值算法及其在力学中的应用

发布时间：2018-08-01 08:15

【摘要】：近年来,分数阶微积分理论和方法被广泛应用于科学和工程中的各个领域。分数阶微积分提供了强有力的工具来描述各种各样的材料和过程中的记忆和继承性质。本文主要研究几类时间分数阶偏微分方程的数值计算方法及分数阶微积分理论与数值方法在力学中的一些应用。首先,对二维非线性分数阶反应次扩散方程,我们提出两种紧致有限差分格式,并用傅里叶分析方法给出了这两种格式的稳定性和收敛性的理论分析。其次,对加热下广义二阶流体分数阶Stokes第一问题,我们提出一种数值参数估计方法来估计Riemann-Liouville分数阶导数的阶。第三,对二维分数阶Cable方程,我们提出一种空间四阶的紧致有限差分格式,用傅里叶分析方法给出了稳定性和收敛性的理论证明。对于反问题,我们提出一种数值参数估计方法,给出了两个分数阶导数的阶的最优估计。第四,在肿瘤热疗实验中,我们构建了双层球形组织的一个时间分数阶热波模型,采用隐式差分方法,给出了模型的数值解。对于反问题,借助于热疗实验数据,我们提出一种非线性参数估计方法,给出了未知分数阶导数和松弛时间参数的最优估计。最后,对钠离子跨肠壁的输运过程,我们建立了一个在浓度梯度和电势梯度共同作用下的空间分数阶反常扩散模型,用有限差分方法得到了该问题的数值解,并对钠离子跨肠壁输运过程中的浓度变化进行了分析。具体来说：第一章,我们首先给出了关于分数阶微积分历史发展状况的一些简单介绍；其次,我们介绍了求解时间分数阶偏微分方程的几种数值方法以及本文中涉及的几类分数阶算子的定义；最后,我们简单介绍了本文的主要研究工作。第二章,我们研究了二维非线性分数阶反应次扩散方程：其中,非线性源项g(u,x,y,t)有二阶连续偏导数(?)g(u,x,y,t)/(?)t2且关于u满足Lips-chitz条件,即首先,我们构建了一种时间一阶、空间四阶的紧致有限差分格式：对方程两边关于时间积分,利用Riemann-Liouville分数阶积分的定义、四阶紧致差分格式近似空间二阶导数以及梯形公式近似非线性源项.我们用傅里叶分析的方法证明了该紧致有限差分格式的稳定性和收敛性.数值算例验证了我们的理论分析.其次,我们利用线性插值技术,构建了一种时间二阶、空间四阶的紧致有限差分格式,并用傅里叶分析的方法给出了该紧致有限差分格式稳定和收敛的条件.数值算例验证了该算法的精度和有效性.第三章,我们研究了加热下广义二阶流体分数阶Stokes第一问题我们提出一种数值方法来估计Riemann-Liouville分数阶导数的阶.首先,对于正问题,我们采用隐式数值方法来求解.对于反问题,我们借助于digamma函数,先求得分数阶敏感方程；其中进而引进Levenberg-Marquardt迭代算法来估计未知的Riemann-Liouville分数阶导数的阶。为了验证算法的有效性,我们给出了在测量值是否包含随机测量误差两种情形下的估计问题的解,并讨论了各个初始参数值的选取对估计结果的影响.数值算例表明,我们提出的数值算法对于估计Riemann-Liouville分数阶导数的阶是有效的.第四章,我们研究二维分数阶Cable方程：我们研究了二维分数阶Cable方程中两个分数阶导数的阶的估计问题。对于正问题,我们提出一种空间四阶的紧致有限差分格式,并用傅里叶分析的方法给出了该紧致有限差分格式稳定性和收敛性的理论证明。对于反问题,我们首先求得了分数阶敏感矩阵,进而引入Levenberg-Marquardt迭代方法,给出了两个分数阶导数的阶的最优估计,并讨论了各个初始参数值的选取对估计结果的影响。数值算例验证了我们的算法的有效性。第五章,针对双层球形组织,我们构建了一个时间分数阶热波模型,它包含肿瘤内(0≤r≤R)和健康组织(Rr≤a)的热传导,方程如下：采用隐式差分方法,我们给出了上述时间分数阶热波模型的数值解。对于反问题,借助于热疗实验数据,我们采用非线性参数估计方法,给出了未知分数阶导数和松弛时间参数的最优估计,并讨论了各个初始参数值的选取对估计结果的影响。数值实验结果表明：我们提出的时间分数阶热波模型对于模拟热疗试验中的热传导行为是比较适合的,并且我们提出的数值参数估计方法对于估计复合介质中分数阶热波模型中的参数是有效的。第六章,我们基于分数阶微积分理论,利用分数阶Fick定律,研究了在浓度梯度和电势梯度的共同作用下,钠离子跨肠壁的反常输运问题,建立了带有Riemann-Liouville空间分数阶导数的反常输运模型,并用有限差分方法得到了该问题的数值解。我们研究了在跨肠壁输运过程中,细胞内外两层钠离子的浓度以及毛细血管处钠离子的平均浓度随时间变化的过程,并根据各个不同参数的不同取值,描绘出了钠离子浓度变化的不同趋势；也讨论了空间分数阶导数的阶在取不同数值的变化过程中,钠离子浓度曲线的变化过程以及钠离子吸收速度的快慢问题。研究结果表明,分数阶反常扩散模型对于描述钠离子跨肠壁输运过程是适合的。第七章,我们给出本文的总结和未来可能的研究方向。
[Abstract]:In recent years, fractional calculus theory and method have been widely used in various fields of science and engineering. Fractional calculus provides a powerful tool to describe the memory and inheritance of various materials and processes. This paper mainly deals with the numerical calculation method of several class of time fractional partial differential equations and the fractional order micro Some applications of integral theory and numerical method in mechanics. First, we propose two compact finite difference schemes for the two-dimensional nonlinear fractional order rediffusion equation, and give the theoretical analysis of the stability and convergence of the two schemes by Fourier analysis. The second, the fractional order Stokes of the generalized two order fluid under heating. First, we propose a numerical parameter estimation method to estimate the order of the Riemann-Liouville fractional derivative. Third, for the two-dimensional fractional Cable equation, we propose a space four order compact finite difference scheme, and give the theoretical proof of the stability and convergence by the Fu Liye analysis method. In the method of estimating the numerical parameter, the optimal estimation of the order of two fractional derivatives is given. Fourth. In the tumor thermotherapy experiment, we construct a time fractional thermal wave model of the double spherical tissue, and use the implicit difference method to give the numerical solution of the model. For the inverse problem, we propose a kind of heat therapy experiment data. In the nonlinear parameter estimation method, the optimal estimation of the unknown fractional derivative and relaxation time parameters is given. Finally, we establish a space fractional order anomalous diffusion model under the interaction of the concentration gradient and the potential gradient for the transport process of the sodium ion cross wall, and the numerical solution of the problem is obtained by the finite difference method. In the first chapter, we give a brief introduction to the historical development of fractional calculus. Secondly, we introduce several numerical methods for solving the partial differential equation of time fractional order and several fractional order involved in this paper. In the second chapter, we study the two-dimensional nonlinear fractional order reaction sub diffusion equation, in which the nonlinear source term g (U, x, y, t) has a two order continuous partial derivative (?) g (U, x, y, t) / (?) T2 and the U satisfies the condition, that is, first, we construct a kind of time first order, The compact finite difference scheme of space four order: on both sides of the equation, we use the definition of the Riemann-Liouville fractional integral, the four order compact difference scheme approximating the two order derivative and the trapezoid formula to approximate the nonlinear source term. Convergence. Numerical examples verify our theoretical analysis. Secondly, we use linear interpolation technique to construct a compact finite difference scheme of time two order, space four order, and use Fourier analysis to give the condition of stability and convergence of the compact finite difference scheme. Numerical examples verify the accuracy and effectiveness of the algorithm. In the third chapter, we study the first problem of the fractional order Stokes of the generalized two order fluid under heating. We propose a numerical method to estimate the order of the fractional derivative of the Riemann-Liouville. First, we use the implicit numerical method to solve the positive problem. For the inverse problem, we help the Digamma function to obtain the fractional sensitive equation. In addition, the Levenberg-Marquardt iterative algorithm is introduced to estimate the order of the unknown Riemann-Liouville fractional derivative. In order to verify the effectiveness of the algorithm, we give the solution of the estimation problem in two cases where the measured value contains random measurement error, and the influence of the selection of the initial parameter values on the estimated results is discussed. The numerical example shows that the numerical algorithm proposed by us is effective for estimating the order of the fractional derivative of Riemann-Liouville. In the fourth chapter, we study the two-dimensional fractional Cable equation: we study the estimation of the order of the two fractional derivatives in the two-dimensional fractional Cable equation. For the Yu Zheng problem, we propose a space four order compact. The finite difference scheme is presented and the theoretical proof of the stability and convergence of the compact finite difference scheme is given by Fourier analysis. For the inverse problem, we first obtain the fractional order sensitive matrix, and then introduce the Levenberg-Marquardt iterative method, and give the optimal estimation of the order of the two fractional derivative. The effect of the selection of initial parameter values on the estimation results. Numerical examples verify the effectiveness of our algorithm. In the fifth chapter, we construct a time fractional thermal wave model for the double layer spherical structure, which contains the heat conduction in the tumor (0 < R < < < < < R) and the healthy tissue (Rr < a). The equation is as follows: we use the implicit difference method, we give For the inverse problem, for the inverse problem, with the aid of the thermotherapy experimental data, we use the nonlinear parameter estimation method to give the optimal estimation of the unknown fractional derivative and relaxation time parameters, and discuss the influence of the selection of the initial parameter values on the estimation results. The time fractional heat wave model proposed by us is suitable for the heat conduction behavior in the simulated thermotherapy test, and the method of estimating the numerical parameters is effective for estimating the parameters in the fractional heat wave model in the composite medium. In the sixth chapter, based on the fractional order calculus theory, the fractional order Fick's law is used. Under the common action of concentration gradient and potential gradient, the anomalous transport of sodium ions across the intestinal wall has been established. An anomalous transport model with the fractional derivative of Riemann-Liouville space is established. The numerical solution of this problem is obtained by the finite difference method. We have studied the two layers of sodium ions inside and outside the cells during the transport of the intestinal wall. The concentration of the concentration and the average concentration of sodium ions at the capillaries change with time, and according to the different values of different parameters, describe the different trend of the change of sodium ion concentration, and discuss the change process of the sodium ion concentration curve and the sodium ion in the process of the change of the spatial fractional derivative in the process of taking different values. The research results show that the fractional order anomalous diffusion model is suitable for describing the trans intestinal transport process of sodium ions. The seventh chapter, we give a summary of this paper and the possible future research direction.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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