On some fractional differential equations in the Hilbert space
Mahmoud M. El-Borai - Alexandria University, Faculty of Science, Egypt (email) Abstract: Let $A$ be a closed linear operator defined on a dense set in the Hilbert space $H$. Fractional evolution equations of the form $\frac{d^\alpha u(t)}{dt^\alpha} = Au(t), 0 < \alpha \leq 1$, are studied in $H$, for a wide class of the operators $A$. Some properties of the solutions of the Cauchy problem for the considered equation are studied under suitable conditions . It is proved also that there exists a dense set $S$ in $H$, such that if the initial condition $u(0)$ is an element of $S$, then there exists a solution $u(t)$ of the considered Cauchy problem. Applications to general partial differential equations of the form $$ \frac{\partial^\alpha u(x,t)}{\partial t^\alpha} = \sum_{|q| \leq m} a_q(x) D^q u(x,t) $$ are given without any restrictions on the characteristic form $\sum_{|q|=m} a_\alpha(x) \xi^q$, where $D^q = D_1^{q_1} ... D_n^{q_n}, x = (x_1, ..., x_n), D_j= \frac{\partial}{\partial x_j}, \xi^q = \xi_1^{q_1}, ..., \xi_n^{q_n}, |q| = q_1 + ... + q_n$, and $q = (q_1, ..., q_n)$ is a multi index.\par}
Keywords: Fractional derivatives and integrals, Closed operators, Cauchy problem,
Hilbert space, General partial differential equations.
Received: September 2004; Revised: May 2005; Published: September 2005. |