Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 15, Issue 4, Pages 639-668Publisher
WALTER DE GRUYTER GMBH
DOI: 10.2478/s13540-012-0044-x
Keywords
time-fractional differential equations; resolvent families; alpha-times resolvent families; generators
Funding
- Ministry of Science and Technological Development, Republic of Serbia [144016]
- NSFC of China [10971146]
- Program for New Century Excellent Talents in University of China
- RFBR [10-01-00297a, 12-01-90401-Ukra]
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In this paper we investigate Cauchy problem for a class of time-fractional differential equation D(t)(alpha)u(t) + c(1)D(t)(beta 1)u(t) +...+ c(d)D(t)(beta d)u(t) = Au(t), t > 0, u((j))(0) = x(j), j = 0, ..., m-1, where A is a closed densely defined linear operator in a Banach space X, alpha > beta (1) > ... > beta (d) > 0, c (j) are constants and m = aOEI +/- aOES. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations.
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