解Riemann-Liouville分数阶导数微分方程两点边值问题(英文)

Solving Two-Point Boundary Value Problems of Fractional Differential Equations with Riemann-Liouville Derivatives

研究了两类含Riemann-Liouville分数阶导数的分数阶微分方程两点边值问题。理论上,通过引入分数阶Green函数将含有Riemann-Liouville分数阶导数的两点边值问题等价转换成一个积分方程;并用Lipschitz条件和压缩映射原理给出了含有Riemann-Liouville分数阶导数的两点边值问题的解存在唯一的充分条件;数值上,设计了单打靶法,把含Riemann-Liouville分数阶导数的两点边值问题转化为含Riemann-Liouville分数阶导数的初值问题进行求解,并给出了较为精确的数值解。仿真结果表明:单打靶法是数值求解此类分数阶微分方程两点边值问题的有效工具。

Two kinds of two-point boundary value problems of fractional differential equations with Riemalm-Liouville derivatives （FBVPs） were studied. Analytically, via fractional Green functions, FBVPs were transformed into equivalent integral equations, and then existence and uniqueness of the solutions were proved according to the Lipschitz conditions and the contractive mapping principle. Numerically, the single shooting methods were designed, and solving FBVPs was transformed into solving initial value problems of fractional differential equations with Riemann-Liouville derivatives （FIVPs） in order to get approximation solutions. Simulation results show that the single shooting methods are natural and efficient in numerically solving these FBVPs.