反常动力学与回火反常动力学的模型与计算方法

发布时间：2018-04-18 01:40

本文选题：反常动力学 + 预估-校正法　；参考：《兰州大学》2016年博士论文

【摘要】：自然界中存在着大量的反常扩散现象,反常扩散在物理、化学、生物及工程等领域有着广泛应用.分数阶微积分和分数阶微分方程能够用于描述这些反常扩散现象,反常动力学和回火反常动力学模型的建立有助于我们更好地理解反常扩散现象.本文的具体研究内容如下:一、通过反常动力学和回火反常动力学的微观和宏观模型研究扩散过程,微观方面的研究对象是随机游走模型,具体包括连续时间随机游走(Continuous Time Random Walk,CTRW),莱维游走(Lévy walk)和莱维飞行(Lévy flight).基于这些模型我们详细讨论了粒子随机游走轨迹的统计性质,并进行数值模拟,提出了生成随机变量的高效算法.宏观方面的研究对象是粒子满足幂律分布的确定性方程,即时间分数阶偏微分方程.二、讨论有关时间动力学演化方程的数值计算方法.首先挖掘分数阶算子的短记忆原理的潜在性,并运用等分布网格的思想计算初值问题,提出了等分布网格预估-校正法,通过伪代码对算法进行详细描述.数值实验结果证实,在保证不损失算法精度的同时,我们提出的算法大大减少了计算量,该算法能有效解决相应发展方程的数值解问题.然后,介绍了等分布网格技术,给出了时间动力学演化方程的数值格式,并给出了详细的误差分析.最后,通过数值例子验证了算法的有效性和可行性.三、在第二部分的研究基础上,我们介绍了回火分数阶算子,研究回火时间动力学演化方程的数值计算方法,挖掘出回火分数阶算子的短记忆原理的潜力,结合等分布网格思想提出了回火分数阶微分方程的预估-校正法,并证明了算法的收敛阶.数值结果表明在不损失计算精度的同时减少了计算量,尤其是时间越长,本文提出的算法优点越突出.四、导出了回火分数阶导数的比较原理,然后利用Lyapunov直接法研究回火分数阶系统,并扩展了Lyapunov直接法,导出了Mittag-Leffler稳定性并得到了验证系统稳定性的方法,通过本文提出的方法验证实际系统的稳定性是比较方便的.
[Abstract]:There are a lot of abnormal diffusion phenomena in nature, which are widely used in physics, chemistry, biology and engineering.Fractional calculus and fractional differential equations can be used to describe these anomalous diffusion phenomena. The establishment of anomalous dynamics and tempering anomalous dynamics models can help us to better understand anomalous diffusion phenomena.The main contents of this paper are as follows: first, the diffusion process is studied by microscopic and macroscopic models of anomalous dynamics and tempering anomalous dynamics.These include continuous Time Random walkthrough, Levi walk and L 茅 vy flight.Based on these models, we discuss the statistical properties of random walk trajectories of particles in detail, carry out numerical simulation, and propose an efficient algorithm for generating random variables.The macroscopical research object is the deterministic equation of the particle satisfying the power law distribution, that is, the fractional partial differential equation of time.Secondly, the numerical calculation method of time dynamics evolution equation is discussed.Firstly, the potential of the short memory principle of fractional order operator is explored, and the initial value problem is calculated by using the idea of equal distribution grid. An equal distribution mesh predictor-correction method is proposed, and the algorithm is described in detail by pseudo code.The results of numerical experiments show that the proposed algorithm can effectively solve the problem of numerical solution of the corresponding evolution equation without losing the accuracy of the algorithm.Then, the equal-distributed grid technique is introduced, the numerical scheme of time-dynamics evolution equation is given, and the error analysis is given in detail.Finally, the effectiveness and feasibility of the algorithm are verified by numerical examples.Thirdly, on the basis of the second part, we introduce the tempering fractional order operator, study the numerical calculation method of the tempering time dynamics evolution equation, and excavate the potential of the short memory principle of the tempering fractional order operator.Based on the idea of equal distribution grid, a predictor-correction method for tempering fractional differential equations is proposed, and the convergence order of the algorithm is proved.Numerical results show that the computational complexity is reduced without loss of accuracy, especially the longer the time is, the more outstanding the advantages of the proposed algorithm are.Fourth, the comparison principle of the tempering fractional derivative is derived, and then the Lyapunov direct method is used to study the tempering fractional order system, and the Lyapunov direct method is extended. The Mittag-Leffler stability is derived and the method to verify the stability of the system is obtained.It is convenient to verify the stability of the practical system by the method proposed in this paper.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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