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Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses
- Source :
- Symmetry, Volume 13, Issue 6, Symmetry, Vol 13, Iss 996, p 996 (2021)
- Publication Year :
- 2021
- Publisher :
- Multidisciplinary Digital Publishing Institute, 2021.
-
Abstract
- The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.
- Subjects :
- Physics and Astronomy (miscellaneous)
Differential equation
General Mathematics
010102 general mathematics
Scalar (physics)
Function (mathematics)
Type (model theory)
01 natural sciences
Symmetry (physics)
010101 applied mathematics
symbols.namesake
Transformation (function)
generalized proportional fractional derivatives
Chemistry (miscellaneous)
Mittag-Leffler function
QA1-939
Computer Science (miscellaneous)
symbols
Initial value problem
Applied mathematics
Mittag–Leffler function
0101 mathematics
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 20738994
- Database :
- OpenAIRE
- Journal :
- Symmetry
- Accession number :
- edsair.doi.dedup.....6dd64e68cdf3f5a9bf1d490acc18e1c8
- Full Text :
- https://doi.org/10.3390/sym13060996