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- 저자 : Anatoly A.Kilbas (검색결과 2 건)
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HADAMARD-TYPE FRACTIONAL CALCULUS
Anatoly A.Kilbas Korean Mathematical Society 2001 대한수학회지 Vol.38 No.6
The paper is devoted to the study of fractional integration and differentiation on a finite interval [a, b] of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int_{a}^{x}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation ${\delta}+{\mu}({\delta}=xD,D=d/dx)$ with real $\mu$, being taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X^{p}_{c}$(a, b) of Lebesgue measurable functions f on $R_{+}=(0,{\infty})$ such that for c${\in}R=(-{\infty}{\infty})$, in particular in the space $L^{p}(0,{\infty})\;(1{\le}{\le}{\infty})$. The existence almost every where is established for the coorresponding Hadamard-type fractional derivative for a function g(x) such that $x^{p}$g(x) have $\delta$ derivatives up to order n-1 on [a, b] and ${\delta}^{n-1}[x^{\mu}$g(x)] is absolutely continuous on [a, b]. Semigroup and reciprocal properties for the above operators are proved.
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Solution in Closed Form of Boundary Value Problem for Degenerate Equation of Hyperbolic Typ
Anatoly A. Kilbas ...et al KYUNGPOOK UNIVERSITY 1996 Kyungpook mathematical journal Vol.36 No.2
The paper deals with a boundary value problem for a partial differential equation ofhyperbolic type with boundary conditions involving generalized fractional integrals andderivatives with the Gauss hypergeometric function in the kernel. By using the propertiesof the fractional calculus operators in Hölder spaces it is shown that this problem canbe reduced to a singular integral equation with the Cauchy kernel, and its solvabilityconditions and a solution in a closed form are given.
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