Supposing $d\u2192(t)=[d1(t),d2(t),\cdots ,dn(t)]T$ is a $n\xd71$ vector, $f\u2192(t):=[f1(t),f2(t),\cdots ,fn(t)]T$ is a $n\xd71$ function vector, $A=(aij)n\xd7m$ is a $n\xd7m$ matrix and $u(x,y)=[uij(x,y)]n\xd7m$ is a $n\xd7m$ function matrix. $D\alpha $ is $\alpha $th ($\alpha \u2208R+$) fractional derivative operator, then we define: Display Formula
$d\u2192A:=(d1a11d1a12\cdots d1a1\u2009\u2009md2a21d2a22\cdots d2a2\u2009\u2009m\vdots \vdots \ddots \vdots dnan1dnan2\cdots dnanm),f\u2192(A)(t):=(f1(a11)(t)f1(a12)(t)\cdots f1(a1\u2009\u2009m)(t)f2(a21)(t)f2(a22)(t)\cdots f2(a2\u2009\u2009m)(t)\vdots \vdots \ddots \vdots fn(an1)(t)fn(an2)(t)\cdots fn(anm)(t)),DA:=(Da11Da12\cdots Da1\u2009\u2009mDa21Da22\cdots Da2\u2009\u2009m\vdots \vdots \ddots \vdots Dan1Dan2\cdots Danm),DAu(x,y):=(Da11u11(x,y)Da12u12(x,y)\cdots Da1\u2009\u2009mu1\u2009\u2009m(x,y)Da21u21(x,y)Da22u22(x,y)\cdots Da2\u2009\u2009mu2\u2009\u2009m(x,y)\vdots \vdots \ddots \vdots Dan1un1(x,y)Dan2un2(x,y)\cdots Danmunm(x,y)),$(3)
we call $DA$ as fractional-varying-order differential operator.