分数阶微分方程的新方法及其应用

A precise integration method for Mittag-Leffler matrix function is presented in this project. Based on this a series of highly effective and accurate methods are provided for solving varieties of fractional differential equations with constant coefficients. Using the technique of isolating large value from small value, the peturbation method and transformation method, a high order multiplication perturbation method for the initial value problems of linear fractional differential equations with variable coefficients will be built up, which will be promoted to solve nonlinear problems. Since the transfer matrix is the product of a series of Mittag-Leffler matrix functions, which can be worked out through precise computaion, this peturbation method is of high accuracy and efficiency and its accuracy improves with the increase of peturbation order. In addition, combine high order multiplication peturbation method with the reduction method and build up a highly efficient and precise method for the two-point boundary value problems of fractional differential equation with variable coefficients, to solve relevant variable coefficient probems and linear problems. Finally, combine high order multiplication peturbation method with the substructure method and apply it to nonlinear structural dynamic equation, non-linear fractional heat conduction equation, fractional Schrodinger equation, etc. This project enriches the computation methods and contents of fractional differential equations and is of great theoretical significance. At the same time, it provides highly accurate solutions for many engineering problems and is thus of a bright prospect for wide application.

提出Mittag-Leffler矩阵函数的精细积分法，在此基础上建立各类常系数分数阶微分方程组的高效精细解法。利用大量小量分离技术及摄动变换，建立线性变系数分数阶微分方程组初值问题的高阶乘法摄动法，并推广到非线性问题。由于传递矩阵为一系列Mittag-Leffler矩阵函数之积，可通过精细计算给出，故此摄动法具有很好的精度和效率，且其精度随摄动阶数的增加而不断提高。此外，将高阶乘法摄动法和递推消元法结合，建立变系数分数阶微分方程两点边值问题的一种高效精细解法，以求解相关变系数及非线性问题。最后，将高阶乘法摄动方法与子结构方法相结合，应用于非线性结构动力方程、非线性分数阶热传导方程、分数阶薛定谔方程、功能梯度材料板热力分析等问题。本项目丰富了分数阶微分方程的计算方法和内容，具有重要的理论意义；同时还为许多工程问题提供了一种高精度方法，因此也具有广泛的应用前景。

分数阶微分方程以其对大量客观问题的精确描述，在物理学、生物学、力学、生物物理、生物工程、电气工程、化学、医学、经济学、控制论、航空动力学等许多领域都产生了深远的影响。相关高性能算法的研究具有重要的科学意义和应用价值。将分数阶微分方程转化为代数方程进行求解的过程极易出现病态问题。针对这种病态问题，提出了求解病态线性方程组的范数均衡预处理精细积分法和简化精细迭代方法，有效提高了病态方程的求解精度和效率。针对分数阶积分的奇异性问题，提出了相关奇异方程的精细求解方法。为建立粘弹性蜂窝拉胀材料的分数阶本构方程，构造了几种新型的蜂窝拉胀结构。此外还建立了结构的非概率损伤预测模型。本项研究为深入研究分数阶微分方程问题打下了良好的基础，不仅具有重要的理论意义，同时还在航天、压电传感器、生物医学等领域具有良好的应用前景。