On the solutions of fractional order of evolution equations

• Published in: European physical journal plus Vol. 132; no. 1; p. 47
• Main Authors: Morales-Delgado, V. F., Taneco-Hernández, M. A., Gómez-Aguilar, J. F.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-01-2017; Springer Nature B.V
• Subjects: Applied and Technical Physics; Atomic; Calculus; Cauchy problems; Complex Systems; Condensed Matter Physics; Decomposition; Diffusion waves; Evolution; Fourier transforms; Laplace transforms; Mathematical analysis; Mathematical and Computational Physics; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy; Regular Article; Theoretical; Vibration
• ISSN: 2190-5444; 2190-5444
• DOI: 10.1140/epjp/i2017-11341-0

.
In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.