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Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients

  • Ravshan Ashurov EMAIL logo , Alberto Cabada and Batirkhan Turmetov


One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of B. Bondarenko for construction of solutions of differential equations of integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and with Caputo fractional derivatives. Then the fundamental solutions are used to obtain the unique solution of the Cauchy problem, where the initial conditions are given in terms of the unknown function and its derivatives of integer order. Comparison is made with the use of Mikusinski operational calculus for solving similar problems.


This work has been partially supported by the Ministry of Higher and Secondary Special Education of Uzbekistan under Research Grant F4-FAF010, FEDER and by Ministerio de Ciencia y Tecnología, Spain, and FEDER, Projects MTM2010-15314 and MTM2013-43014-P, and by the Ministry of Education and Science of the Republic of Kazakhstan through the project 0819/GF4.

We would also like to express our special thanks to the reviewers for their remarks, which considerably improved the content of this paper.


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  1. Please cite to this paper as published in:

    Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 229–252, DOI: 10.1515/fca-2016-0013

Received: 2015-3-20
Revised: 2015-11-15
Published Online: 2016-3-9
Published in Print: 2016-2-1

© 2016 Diogenes Co., Sofia

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