The fundamental solutions for fractional evolution equations of parabolic type
type: Item
instance of: scholarly article
title: The fundamental solutions for fractional evolution equations of parabolic type
zbMATH Open document ID: 1081.34053
DOI: 10.1155/S1048953304311020
author: Mahmoud M. El-Borai
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://eudml.org/doc/52084
review text: Let \(0<\alpha\leq 1\), \(T> 0\), \(E\) a Banach space, \(\{A(t); t\in [0,T]\}\) a family of linear closed operators defined on dense set \(D(A)\) in \(E\) with values in \(E\), \(u: E\to E\) a function, \(u_0\in D(A)\), \(f: [0,T]\to E\) a given function and \(B(E)\) be the Banach space of linear bounded operators \(E\to E\) endowed with the topology defined by the operator norm. Assume that \(D(A)\) is independent of \(t\). Suppose that the operator \[ (A(t)+\lambda I)^{-1} \] exists in \(B(E)\) for any \(\lambda\) with \(\text{Re\,}\lambda\geq 0\) and \[ \begin{aligned} &(\exists C> 0)(\forall t\in [0,T])(\forall\lambda\in \mathbb{C})(\|(A(t)+\lambda I)^{-1}\|\leq C(1+ |\lambda|)^{-1},\\ &(\exists C> 0)(\exists\gamma\in (0,1])(\forall(t_1, t_2,S)\in [0, T]^3)(\|(A(t_2)- A(t_1)(A^{-1}(s))\|\leq C|t_2- t_1|^\gamma),\\ &(\exists C> 0)(\exists\beta\in (0,1])(\forall(t_1, t_2)\in [0,T]^2)(\|(f(t_2)- f(t_1)\|\leq C|t_2- t_1|^\beta). \end{aligned} \] The author considers the fractional integral evolution equation \[ u(t)= u_0- (\Gamma(\alpha))^{-1} \int^t_0 (t-\theta)^{\alpha-1} (A(\theta) u(\theta)- f(\theta))\,d\theta,\tag{1} \] where \(\Gamma: (0,+\infty)\to \mathbb{C}\) is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation \[ {d^\alpha v(t)\over dt^\alpha}+ A(t) v(t)= 0,\qquad t> 0,\tag{2} \] and proves the existence and uniqueness of the solution of (2) with the initial condition \(v(0)= u_0\). The author also proves the continuous dependence of the solutions of equation (1) on the element \(u_0\) and the function \(f\) and gives an application to a mixed problem of a parabolic partial differential equation of fractional order.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 35K99
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 2148074
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
OpenAlex ID: W1996995461
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description: scientific article; zbMATH DE number 2148074