On asymptotics of some fractional differential equations (Q2846821)

Some asymptotic results are given for fractional differential equations as \(x\to\infty\). A typical result is the following one: Let \(y(x)\) solve (\(D^\alpha\) is the fractional derivative of order \(\alpha\)) NEWLINE\[NEWLINED^\alpha y(x)=\lambda q(x) y(x),NEWLINE\]NEWLINE where \(0<\alpha< 1\), \(\lambda> 0\), and \(q(x)\) behaves asymptotically as \(cx^\mu\), \(\mu> 0\). Then \(y(x)\) behaves asymptotically as \(c\exp(\lambda^{{1\over\alpha}}\int q(x)^{{2\over\nu}} dx)\). Some numerical results are also presented.