A solution to the fundamental linear complex-order differential equation

作者：Adams Jay L^*; Hartley Tom T; Adams Lynn I

来源：Advances in Engineering Software, 2010, 41(1): 70-74.

DOI：10.1016/j.advengsoft.2008.12.014

• Fractional calculus
• Fractional-order systems
• Fractional-order differential equations
• Complex-order derivatives
• Complex-order systems
• Complex-order differential equations

This paper provides the solution to the complex-order differential equation, (0)d(t)(q)x(t) = kx(t) + bu(t), where both q and k are complex. The time-response solution is shown to be a series that is complex-valued. Combining this system with its complex conjugate-order system yields the following generalized differential (0)d(t)(2Re(q))x(t) - (k) over bar (0)d(t)(q)x(t) - k(0)d(t)((q) over bar)x(t) + k (k) over barx(t) = p(0)d(t)(q)u(t) + (p) over bar (0)d(t)((q) over bar)u(t) - (k + (k) over bar )u(t). The transfer function of this system is p(s(q) - k)(-1) + (p) over bar (s((q) over bar) - (k) over bar)(-1), having a time-response 2 Sigma(infinity)(n=0)t((n+1)u-1) (Re(pk(n)/Gamma((n+1)q)) cos((n + 1)v ln t) - lm(pk(n)/Gamma((n+1)q)) sin((n + 1)v ln t). The transfer function has an infinite number of complex-conjugate pole pairs. Bounds on the parameters u = Re(q), v = lm(q), and k are determined for system stability.

• 出版日期2010-1