Сauchy – Euler Equation: Integer and Fractional Orders
DOI:
https://doi.org/10.52575/2687-0959-2025-57-3-172-185Keywords:
Cauchy – Euler Equation, Cauchy – Euler Differential Operator, Stirling Numbers of the Second Kind, Stirling Functions of the Second Kind, Hadamard Fractional Derivatives, Mellin TransformAbstract
In this paper, we study the Cauchy-Euler equations of both integer and fractional orders. We analyze and utilize the fact that the operators involved in these equations are closely related to the Stirling numbers of the second kind and their fractional generalizations. We propose a finite-difference interpretation of the operator (x d/dx)n. Additionally, we consider the application of the Mellin transform for solving inhomogeneous Cauchy – Euler equations of both integer and fractional orders.
Acknowledgements
The work of the second author was carried out with the support of the Ministry of Education and Science of the Russian Federation on a state assignment (project FEGS-2023-0003).
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References
Ross CC. Differential Equations. An Introduction with Mathematica. 2th ed. New York: Springer; 2004. 431 p.
Boyce WE, DiPrima RC. Elementary Differential Equations and Boundary Value Problem. 8th ed. New York: Wiley; 2005. 790 p.
Sadykov T. Graceful bases in solution spaces of differential and difference equations. Journal of Symbolic Computation. 2025;127:1–13. DOI: 10.1016/j.jsc.2024.10235
Takahasi SE, Oka H, Miura T, Takagi H. A Cauchy-Euler Type Factorization of Operators. Tokyo Journal of Mathematics. 2008;31(2):489–493.
Berman G, Fryer KD. Introduction to Combinatorics. New York: Academic Press; 2014. 314 p.
Deza EI. Special combinatorial numbers: From Stirling numbers to Motzkin numbers: everything about the twelve well-known sets of numbers of combinatorial nature (history, classical properties, examples, and problems). Moscow: Lenand; 2018. 504 p. (In Russ.)
Butzer PL, Hauss M, Schmidt M. Factorial functions and Stirling numbers of fractional orders. Results. Math. 1989;16:16–48. DOI: 10.1007/BF03322642
Butzer PL, Kilbas AA, Trujillo JJ. Stirling functions of the second kind in the setting of difference and fractional calculus. Numer. Funct. Anal. Optim. 2003;24(7–8):673–711. DOI: 10.1081/nfa-120026366
Schwatt IJ. An Introduction to the Operations with Series. New York: Chelsea Publishing Co; 1962. 328 p.
Knopf PM. The operator (x d/dx)n and its applications to series. Math. Mag. 2003;76(5):364–371. DOI: 10.1080/0025570X.2003.11953210
Gonz´alez GJR, Plaza Galvez LF. Soluci´on de la ecuaci´on de Cauchy-Euler por medio de la transformada de Mellin. Scientia Et Technica. 2009;2(42):300–303. DOI: 10.22517/23447214.2651
Brychkov Y, Marichev O, Savischenko N. Handbook of Mellin Transforms. New York: Chapman and Hall/CRC; 2018. 609 p.
Balakrishnan AV. An operational calculus for infinitesimal generators of semigroups. Trans. Amer. Math. Soc. 1959;91:330–353. DOI: 10.2307/1993125
Westphal U. Ein Kalk¨ul f¨ur gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren, Teil I: Halbgruppenerzeuger, Teil II : Gruppenerzeuger’. Gompositio Math. 1970;22:67–103, 104–136.
Yosida K. Functional Analysis, 6 th ed. Berlin: Springer-Verlag; 1980. 504 p.
Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. Yverdon: Gordon and Breach; 1993. 1016 p.
Ahmad B, Alsaedi A, Ntouyas SK, Tariboon J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. New York: Springer; 2017. 427 p.
Garra R, Orsingher E, Polito F. A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability. Mathematics. 2018;6(1):1–10. DOI: 10.3390/math6010004
Butzer PL, Kilbas AA, Trujillo JJ. Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 2002;269(1):1–27. DOI: 10.1016/S0022-247X(02)00001-X
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Switzerland: Elsevier Science; 2006. 540 p.
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