Initial value problems for some classes of linear evolution equations with several fractional derivatives

  • Fedorov Vladimir E., kar@csu.ru Chelyabinsk State University, Mathematical Analysis Department, 129 Brothers Kashirin Street, Chelyabinsk 454021, Russia; South Ural State University, Laboratory of Functional Materials, 76 Lenin Avenue, Chelyabinsk 454080, Russia
  • Boyko Kseniya V., kvboyko@mail.ru Chelyabinsk State University, Mathematical Analysis Department, 129 Brothers Kashirin Street, Chelyabinsk 454021, Russia
  • Phuong Ta D., tdphuong@math.ac.vn Institute of Mathematics of the Vietnamese Academy of Science and Technology, Department of Numerical Analysis and Scientific Computing Hanoi, Vietnam
Keywords: fractional order differential equation, Gerasimov-Caputo fractional derivative, degenerate evolution equation, Cauchy problem, initial-boundary value problem

Abstract

The problems of unique solvability of initial problems for linear inhomogeneous equations of a general form with several Gerasimov-Caputo fractional derivatives in Banach spaces are investigated. The Cauchy problem is considered for an equation solved with respect to the highest fractional derivative containing bounded operators at the lowest derivatives. The solution is presented with the use of Dunford-Taylor type integrals. The obtained result allowed us to study an initial problem for a linear inhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that, with respect to this operator, the operator at the second largest derivative is 0-bounded. Abstract results are applied to the study of a class of initial-boundary value problems for equations with several Gerasimov-Caputo time derivatives and with polynomials with respect to a self-adjoint elliptic differential operator in space variables.

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How to Cite
Fedorov, V., Boyko, K. and Phuong, T. (2021) “Initial value problems for some classes of linear evolution equations with several fractional derivatives”, Mathematical notes of NEFU, 28(3), pp. 85-104. doi: https://doi.org/10.25587/SVFU.2021.75.46.006.
Section
Mathematics