非线性分数阶微分方程初值问题的高效数值算法及其应用

In recent years, many phenomena in materials science, computational biology, chemistry kinetics, control theory and other sciences can be described successfully by the mathematical models with fractional calculus, i.e. the theory of derivatives and integrals of non-integer order. Along with the development of the corresponding theories, the numerical solutions of fractional differential equations (FDEs) have been an important and hot topic in the world. . In this project, we will propose the spline collocation methods (for example, the cubic spline collocation method) for solving the IVPs of nonlinear FDEs and the IVPs of nonlinear multi-order fractional differential equations. The theorems of the local truncation error of those methods will be given. And the results of the convergence and the stability of those methods for the fractional order integral equation which is equivalent to the IVPs of nonlinear FDEs will be obtained respectively. We will also obtain some results of the convergence and the stability of those methods for the IVPs of nonlinear FDEs by using the relationship between those methods for solving the IVPs of nonlinear FDEs and those methods for solving the corresponding equivalent fractional order integral equation. In order to verify our theoretical results, some illustrative examples will be given. Moreover, the comprehensive numerical comparison of those methods will be made. The numerical results will be very useful for further research on numerical solutions of FDEs.

近年来，随着分数阶微分方程在生物学、材料科学、电磁学、传输(扩散)、自动控制等许多科学领域中日益广泛的应用，分数阶微分方程的相关理论逐步发展完善，分数阶微分方程的数值解也成为了非常重要的研究热点。本项目利用样条配置方法求解几类分数阶非线性常微分方程初值问题(包括分数阶动力系统初值问题以及高阶的非线性分数阶多阶微分方程初值问题)。不同于过去的大多数分数阶微分方程数值方法的研究， 是基于把分数阶微分方程转化为等价的积分方程，再利用数值方法求解。本项目将在不需要将分数阶微分方程转化为积分方程的情况之下，直接离散分数阶微分方程初值问题，获得高精度的数值解，并给出相应的相容性、收敛性、稳定性定理，利用若干的数值实验来验证数值方法的可靠性，将详细地比较分析本项目所得到的数值方法与以往的数值算法的优劣，获得的结果将具有非常大的实际应用价值。

近年来，随着分数阶微分方程在许多科学领域中日益广泛的应用，分数阶微分方程的相关理论逐步发展完善，分数阶微分方程的数值解也成为了重要的研究热点。本项目利用三次样条配置方法求解了几类分数阶非线性常微分方程初值问题。在不将分数阶微分方程转化为积分方程的情况之下，直接离散分数阶微分方程初值问题，获得高精度的数值解，并给出相应的相容性、收敛性、稳定性定理，利用若干的数值实验验证了数值方法的可靠性。另外，本项目还利用三次样条配置方法采用直接法求解了几类分数阶延迟微分方程初值问题，数值结果表明该方法是非常高效的。本项目所获得的结果对于未来研究分数阶微分方程的数值方法具有重要的意义。