On Monday, 1 March 14:44:25 UTC+11, rancid moth wrote:

I would very much like to solve:

D_1/2[F(t),t] = -F(t)*Log(F(t)) [1]

where D_1/2[f,t] := d^(1/2)f(t)/dt^(1/2) is the fractional 1/2 order differential operator (sometimes known as the semi-derivitive). i.e. [1] is a nonlinear fractional order differential equation.

One possible way i thought of was to invert the equation, literealy, i.e.

D_1/2[t,F] = -1/(F(t)*Log(F(t)))

and then reverse the operator by taking D_-1/2 on both sides -

t = -D_-1/2 [ 1/(F*Log(F)) , F] = -1/Sqrt(pi) * Integral (0,F) 1/(g*Log(g)) *1/Sqrt(F-g) dg (by the Rieman-Liouville defintiion of fractional integration)

but there are problems with this: 1. i dont know if i can simply invert the equation when the operator is of fractional order...and 2) i cant do the above integral anyway...so its probably doesnt get me anywhere even if it is allowed. I fear there is little hope for an explicit solution of [1].

another way of framing this problem is to Solve the following nonlinear weakly singular integral equation (of Abel type: i say "of" because it is like the Abel integral equation however it is worse since the unkown is on both sides of the equality)

F(t) = -1/Sqrt(pi) * Integral(0,t) F(k)*Log(F(k))/Sqrt(t-k) dk

cheers
moth

one click to solve (fractional) differential and integral equation.

input your equation into website www.mathHandbook.com, click the dsolve buttion to solve fractional differential equation, the d/dx button for derivative, or the "semi d/dx" button to calculate the 0.5 order derivative, or the "semi integrate" button for
the semi-integral.

for this problem, its solution is http://www.mathhandbook.com/input/?guess=dsolve%28ds%28y%2Cx%2C0.5%29%3D-y*log%28y%29%29

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