Analysis of fractional differential equations with multi-orders

作者：Deng Weihua; Li Changpin; Guo Qian

来源：Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 2007, 15(2): 173-182.

DOI：10.1142/S0218348X07003472

• caputo fractional derivative
• multi-order fractional differential systems
• stability

In this paper, we study two kinds of fractional differential systems with multi- orders. One is a system of fractional differential equations with multi- order, D-*((alpha) over bar)(x) over bar (t)=(f) over bar (t,(x) over bar), (x) over bar (0)=(x) over bar (0); the other is a multi-order fractional differential equation, D(*)(beta*)y(t)=g(t,y(t)), D(*)(beta 1)y(t),...,D-*(beta n) y(t)). By the derived technique, such two kinds of fractional differential equations can be changed into equations with the same fractional orders providing that the multi-orders are rational numbers, so the known theorems of existence, uniqueness and dependence upon initial conditions are easily applied. And asymptotic stability theorems for their associate linear systems, D-*((alpha) over bar)(x) over bar (t)=A (x) over bar (t), (x) over bar (0)=(x) over bar (0), and D-*(beta*) y(t)=a(0)y(t)+Sigma(n)(i=1) a(i)D(*)(beta i)y(t), y((k))(0)=y(0)((k)), k=0, 1, ..., [beta*]-1, are also derived.

• 出版日期2007-6
• 单位上海大学; 上海师范大学; 兰州大学