ON THE METHOD FOR CONSTRUCTING THE SOLUTION OF A FRACTIONAL DIFFERENTIAL EQUATION WITH A HADAMARD-TYPE DERIVATIVE

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Abstract

This article presents a comprehensive analysis of fractional-order differential equations that involve the Hadamard derivative and its various modifications. Particular attention is given to the features of the Hadamard–Caputo operators, which substantially broaden the analytical toolkit for studying processes with multiplicative scaling and logarithmic dependence. The paper provides a detailed examination of the method of normalized systems, which is based on the concept of generalized homogeneity and makes it possible to develop a unified and effective approach to constructing solutions. This method was previously applied mainly to integer-order equations; however, in this article it is adapted to a considerably more complex class of Hadamard integro-differential operators.

For the homogeneous equation, an explicit solution formula is derived in the form of a functional series whose coefficients are expressed in terms of the gamma function and the Pochhammer symbol. It is shown that this series possesses absolute convergence and defines an analytic function on the entire complex plane. In the case of a non-homogeneous equation, a method for constructing a particular solution using the right inverse operator is presented, which makes it possible to obtain the solution in a closed form. Conditions ensuring the well-posedness of the problem are established. The results obtained expand the theoretical foundation of fractional calculus and open new perspectives for research in the field of Hadamard-type operators.