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Fractional Differential Equation

The solution to the differential equation

[D^(2v)+alphaD^v+betaD^0]y(t)=0
(1)

is

y(t)={e_alpha(t)-e_beta(t) for alpha!=beta; te^(alphat)sum_(k=-(q-1))^(q-1)alpha^k(q-|k|)D^(1-(k+1)v)(te^(alpha^qt)) for alpha=beta!=0; (t^(2nu-1))/(Gamma(2v)) for alpha=beta=0,
(2)

where

q=1/v
(3)
e_beta(t)=sum_(k=0)^(q-1)beta^(q-k-1)E_t(-kv,beta^q),
(4)

E_t(a,x) is the Et-function, and Gamma(n) is the gamma function.

See also

Fractional Calculus, Fractional Derivative

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References

Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183-192, 1995.

Referenced on Wolfram|Alpha

Fractional Differential Equation

Cite this as:

Weisstein, Eric W. "Fractional Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FractionalDifferentialEquation.html

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