$\mu$-Caputo type time-delay Langevin equations with
two general fractional orders
Abstract
{In the present paper, a $\mu$-delayed Mittag-Leffler
type function is introduced as a fundamental function. By means of
$\mu$-delayed Mittag-Leffler type function, an exact
analytical solution formula to non-homogeneous linear delayed Langevin
equations involving two distinct $\mu$-Caputo type
fractional derivatives of general orders is given. Also, a global
solution of nonlinear version of delayed Langevin equations is inferred
from the findings on hand and is verified with the aid of the
functional(substitutional) operator. In terms of exponential function,
we estimate $\mu$-delayed Mittag-Leffler type function.
Existence uniqueness of solutions to nonlinear delayed Langevin
fractional differential equations are obtained with regard to the
weighted norm defined in accordance with exponential function. The
notion of stability analysis in the sense of solutions to the described
Langevin equations is discussed on the grounds of the fixed point
approach. Numerical and simulated examples are shared to exemplify the
theoretical findings. This paper provides novel results.