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Zbl 1005.34051
El-Borai, Mahmoud M.
Some probability densities and fundamental solutions of fractional evolution equations. (English)
[J] Chaos Solitons Fractals 14, No.3, 433-440 (2002). ISSN 0960-0779

Summary: Here, if $0<\alpha\le 1$, the author studies the Cauchy problem in a Banach space $E$ for fractional evolution equations of the form $${d^\alpha u\over dt^\alpha} =Au(t)+B(t)u(t),$$ where $A$ is a closed linear operator defined on a dense set in $E$ into $E$, which generates a semigroup and $\{B(t):t\ge 0\}$ is a family of closed linear operators defined on a dense set in $E$ into $E$. The existence and uniqueness of a solution to the considered Cauchy problem is studied for a wide class of the family of operators $\{B(t):t\ge 0\}$. The solution is given in terms of some probability densities. An application is given for the theory of integro-partial differential equations of fractional orders.

MSC 2000:: *34G20 Nonlinear ODE in abstract spaces
35K90 Abstract parabolic evolution equations
45K05 Integro-partial differential equations

Keywords: fractional evolution equations; closed linear operator; probability densities; integro-partial differential equations; fractional orders

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Zentralblatt MATH Berlin [Germany]