Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

: 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.


Introduction
Recently, fractional differential equations have appeared strongly in the diffusion process, the process of dynamics, signal and image processing, etc. It has been mainly due to the fact that the mathematical modeling of numerous processes and phenomena in the frame of the fractional operators is capable of tracing the previous effects of the concerned phenomena. For instance, see [1][2][3][4][5][6] and references therein.
In 2014, Khalil et al. [7] introduced an interesting derivative, called the conformable derivative. Later, many researchers argued that this derivative could not be considered as a fractional derivative because it has no memory property. This new definition seems to be a natural extension of the classical derivative. Unfortunately, this new definition has a point of weakness as it does not tend towards the original function when the order approaches zero. Anderson and Ulness [8,9] proposed a modified conformable derivative by utilizing proportional derivatives. Later, Jarad et al. [10] introduced a new generalized proportional derivative which is well-behaved and has several advantages over the classical derivatives such as meaning that it generalizes formerly known derivatives in the literature. For recent contributions relevant to fractional differential equations via generalized proportional derivatives, see [11][12][13][14][15][16].
One of the main problems in differential equations is the solvability. At the same time, the local solvability being the same as the existence of solutions to the initial value problem, it is connected with the symmetry of a transformation of a system of differential equations. This paper is the first work to give an explicit formula for the solutions of the initial value problem for scalar linear fractional differential equation with generalized proportional fractional derivative in terms of the Mittag-Leffler function, which reflects the novelty of the work compared to the aforementioned contributions, which mainly depend on discussing the mild solution of the integral equations corresponding to the differential equation in question.
The rest of the paper is structured as follows: In Section 2, we recall some useful preliminaries and auxiliary results. In Section 3, a scalar linear generalized proportional fractional differential equation with an initial condition expressed by a generalized proportional fractional integral is defined. An explicit formula of the solutions of the studied initial value problem is obtained. In Section 4, a linear generalized proportional fractional differential equation with instantaneous impulses is discussed. Finally, in order to confirm the validity of the theoretical findings, two examples are given in Section 5.

Preliminaries and Auxiliary Results
We provide some basic definitions and properties of the fractional proportional derivative and integral (see for example, [10]).
The aforesaid amended conformable derivative (1) is said to be a proportional derivative. For more details, see [8].
We will provide some results which will be used in our further considerations.
then also exists a limit then if there exists the limit lim t→a+ e Proof. Let the limit (8) be satisfied and > 0 be an arbitrary number. From (8) there exists a number η > 0 such that Moreover, since the exponential function is continuous Then according to (7) Then applying (12)- (14) we obtain which proves the claim (i) of Lemma 2.
Assume the limit lim t→a+ e exists and is equal to c. Then , according to Lemma 2(i) the equality holds and, hence, in accordance with (10) the validity of (11) follows.

Remark 3.
Note in the case ρ = 1 the generalized proportional fractional integral and the generalized proportional fractional derivative are reduced to Riemann-Liouville fractional integral and derivatives, respectively, and the formula (17) is reduced to formula (4.1.14) [1] for the linear Riemann-Liouville fractional differential equation.
The impulse at a point τ means that there is a jump of the solution at this point and after the jump for t > τ the solution is determined by the same differential equation but with a new initial value. Therefore, we need an initial condition at the impulsive point τ. Following the idea of Section 3 we will define two equivalent types of the impulsive conditions at the point τ (see Remark 2): integral form of the impulsive condition where P, G : R → R are given functions.

Conclusions
In this paper a scalar linear fractional differential equation with a generalized proportional fractional derivative of Riemann-Liouville type (LFDEGD) on a finite interval is studied. Two different cases are investigated. The object of investigation in the first case is the initial value problem of LFDEGD with an initial condition expressed by a generalized proportional fractional integral. An explicit formula of the solution of the studied initial value problem is obtained. In the second case the case when instantaneous impulses occur at fixed initially given points is considered. We study the case of a changeable lower limit of the generalized proportional fractional derivative at each impulsive time. It is reasonable because each impulsive time is considered as an initial time of the fractional differential equation. An appropriate impulsive conditions by generalized proportional fractional integrals are set up. An explicit solution is given.
Note that in the case of ordinary derivatives, the impulsive case is a generalization of the case without impulses. However, it is not the situation of the generalized proportional fractional derivative of Riemann-Liouville type. It is mainly because the solution has a singularity at each impulsive point. It requires the study of both cases, impulsive and non-impulsive, neither of which is a partial case of the other one.