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On the solution of some simple fractional differential equations. (English) Zbl 0711.34019

The aim of the author is to show that the classical elementary methods used to solve the ordinary differential equations with constant coefficients or the equations of Euler type can also be used to solve some functional equations in which the fundamental operator consists of a generalization of the classical concept of derivative, namely:

D ν F Dz ν =Γ(1+ν) 2πi (z+) F(ζ)dζ (ζ-z) ν+1 ,ν-1,-2,···,
D ν F Dz ν =1 Γ(-ν) z F(x)dx (z-x) ν+1 ,Reν<0·

Notice that D n F/Dz n is exactly equal to F (n) (z) when n0 is an integer and F(z) is regular near the point z while for n=-1,-2,··. it becomes equal to the n-times repeated integral of F(z). The generalization made by the author relies upon the fact that D ν (e az )/Dz ν =a ν e az , which permits to write the solution of an equation of the form A m D ν m F/Dz ν m =0 as F(z)= m C m z e a m z · Similar results concerning the equations of Euler type as well as those with forcing terms are also given.

Reviewer: M.Idemen

34A99General theory of ODE
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
26A33Fractional derivatives and integrals (real functions)
45D05Volterra integral equations
45J05Integro-ordinary differential equations