Abstract
In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation.
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Acknowledgements
The authors are grateful to the editor and referees for the valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China [Grant No. 11871064]; and the Graduate Research and Innovation Projects of Jiangsu Province [Grant No. XKYCX20_010].
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Liu, L., Dong, Q. & Li, G. Exact solutions of fractional oscillation systems with pure delay. Fract Calc Appl Anal 25, 1688–1712 (2022). https://doi.org/10.1007/s13540-022-00062-y
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DOI: https://doi.org/10.1007/s13540-022-00062-y