1. Introduction
Recently, fractional differential equations have been used as more adequate models of real world problems in engineering, physics, finance, etc. ([
1,
2]). Usually, fractional differential equations are considered as a generalization of ordinary differential equations and delay equations. Research findings have revealed that many models based only on integer order derivatives do not provide enough information to describe the complexity of real world phenomena. In comparison to integer-order derivatives, there are several kinds of definitions for fractional derivatives. These definitions are generally not equivalent to each other; the main ones in the literature are the Caputo fractional derivative and the Riemann-Liouville (RL) fractional derivative. Comparing both definitions we mention that the set of functions which are differentiable in the RL sense is wider than the set of functions which are differentiable in the Caputo sense. A good overview of properties of fractional derivatives and potential applications of the RL derivative for modeling in science and engineering is given in [
3] and physical and geometric interpretations for the RL derivative can be found in [
1]. In addition, many real world processes and phenomena are characterized by the influence of past values of the state variable on the recent one and this leads to the includion of delays (finite, variable, state dependent, etc.) in the models. The analysis of delay fractional differential equations can be rather complex (analytical solution computation, controllability analysis, etc.). Recently, there were developments on seeking the explicit formula of solutions to linear delay fractional differential equations. Li and Wang [
4] studied the linear homogenous Caputo fractional delay differential equations and gave a representation of the solution. Also, in [
5,
6] representations of the solution of linear non-homogeneous Caputo fractional delay differential equations are provided.
However, this is not the situation with Riemann-Louisville (RL) fractional differential equations with delays. The RL fractional differential equations with delays are not well studied. We mention the papers [
7,
8] where the explicit formula for the linear RL fractional equations is given but the initial condition does not correspond to the idea of the case of ordinary differential equations. In these papers the lower bound of the RL fractional derivative coincides with the left side end of the initial interval.
In this paper we study initial value problems of scalar linear RL fractional differential equations with constant delay. Similar to the case of the ordinary derivative, the differential equation is given to the right of the initial time interval. It requires the lower bound of the RL fractional derivative to coincide with the right side end of the initial interval (usually this point is zero). Note that in this case any solution of an initial problem (IVP) with RL fractional derivatives is not continuous at this point. That is why RL fractional delay differential equations are convenient for modeling process with impulsive types of initial conditions. This type of processes can be found in physics, chemistry, engineering, biology, and economics. To determine the law of the initial impulsive reaction we need to add to the usual initial condition (for example,
on the initial interval
,
is the delay) a fractional condition. This conclusion is based on the results obtained in [
1,
9] concerning the physical interpretation of the RL fractional derivatives and initial conditions which include derivatives of the same kind. Based on the above we set up appropriate IVPs for RL linear fractional differential equations with lower limit of the RL derivative equal to the right side point of the initial interval. Explicit formulas for the solutions of the initial value problems with both zero and nonzero initial functions are obtained. Also, the cases of homogeneous as well as non-homogeneous equations are studied.
2. Preliminary Notes on Fractional Derivatives and Equations
Let and . In this paper we will use the following definitions for fractional derivatives and integrals:
- -
Riemann-Liouville fractional integral of order
([
10,
11])
where
is the Gamma function.
Note sometimes the notation is used.
- -
Riemann-Liouville fractional derivative of order
([
10,
11])
We will give fractional integrals and RL fractional derivatives of some elementary functions which will be used later:
Proposition 1. The following equalities are true: The definitions of the initial condition for fractional differential equations with RL-derivatives are based on the following result:
Proposition 2. (Lemma 3.2 [12]). Let , and . - (a)
If there exists a.e. a limit , then there also exists a limit - (b)
If there exists a.e. a limit and if there exists the limit then
In the case of a scalar linear RL fractional differential equation we have the following result:
Proposition 3. (Example 4.1 [12]) The solution of the Cauchy type problemhas the following form (formula 4.1.14 [12])where is the Mittag-Leffler function with two parameters (see, for example, [10]). From Proposition 3 and Proposition 2 (a) we obtain the following result for the weighted form of the initial condition:
Proposition 4. The solution of the Cauchy type problemhas the following form 3. Explicit Formula for the Solutions of Scalar Linear Rl Fractional Equations with Delays and Zero Initial Values
Throughout the paper we will assume for the integers .
3.1. Homogeneous Linear RL Fractional Differential Equation
Consider scalar linear Riemann-Liouville fractional differential equations with a constant delay (HFrDE):
where
,
B,
are real constants.
We will consider the zero initial value
and
or
Remark 1. According to Proposition 2 the conditions (5) and (6) are equivalent. According to Remark 1 we will consider only the fractional initial condition (
5).
Remark 2. Note the IVP for HFrDE Equations (3) and (4) with zero fractional initial condition, i.e., , has only a zero solution. Theorem 1. The solution of the IVP (3)–(5) is given by Remark 3. The proof of Theorem 1 can be done by the application of Proposition 4 with and but instead we will give a direct proof.
Proof. Let
. Then from (
3) we have the RL fractional differential equation
whose solution is given by
because from Proposition 1 we have
, i.e.,
and
Let
. Then from (
3) and (
8) we have the following RL fractional equation
Therefore, the solution is given by
Indeed, from Proposition 1 with
we have
Let
. Then from (
3), (
8) and (
11) we have
Therefore,
because from Proposition 1 with
and the equality
we have
Continue this process and the claim is established. □
3.2. Non-Homogeneous Linear RL Fractional Differential Equation
Consider non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay (NFrDE):
with the zero initial condition (
4) and fractional condition
or
where
, and
B,
are real constants.
According to Remark 1 we will consider only the initial condition (
19).
Using a direct proof we will obtain an explicit formula for the solution of the IVP (
4), (
17) and (
19).
Theorem 2. The solution of the IVP (4), (17) and (19) is given by Proof. Let
. Use the variation of constants method we will search for solutions in the form
where
is the unknown function to be obtained.
Then according to (
17) we have
Also, applying
we obtain
From (
22) and (
23) we obtain
, i.e., the solution is given by
Note that it is easy to check the validity of condition (
19) for the solution defined by (
24).
Let
. Use the variation of constants method we will search for solutions in the form
where
is the unknown function to be obtained.
Then according to (
17) and (
24) we have
Also, applying
, equality
(see Proposition 1), (
15) with
, and
we obtain
i.e., from (
26) and (
28) we get
and
Let
. Use the variation of constants method we will search for solutions in the form
Then according to (
17), (
24) and (
29) we have
Similar to (
28) we obtain
i.e., from (
31) and (
32) we get
and
Continue this process and the claim is established. □
Remark 4. Note the formula for the solution in the homogeneous case does not follow from the one in the non-homogeneous case because of the fractional conditions (5) and (18) (respectively, (6) and (19)). 4. Explicit Formula for the Solutions of Scalar Linear Rl Fractional Equations with Delays and Non-Zero Initial Values
Consider the linear non-homogeneous RL fractional differential Equation (
17) with nonzero initial value:
where
.
Remark 5. Note that the function is not applicable in this case as an initial function.
Remark 6. According to Remark 1 the fractional condition in (34) could be replaced by . Theorem 3. The solution of the IVP (17), (34) is given by Proof. Let
. Then from the Equation (
3) and the initial condition (
34) we have
According to Proposition 3 with
and the equality
, the solution is
Let
. Then from (
17), (
34) and (
37) we have
According to Proposition 3 and Equation
the solution is
Let
. Then from (
17), (
34), (
37) and (
38) we have
According to Proposition 3 the solution is
Continue the process, and the proof is complete. □
Special Case: In the homogeneous case of
the solution of the IVP (
3), (
34) is given by the function
5. Explicit Formula for the Solutions of the General Scalar Linear Rl Fractional Equations with Delays and Non-Zero Initial Values
5.1. Zero Initial Function
Consider the non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay:
with the initial conditions
where
,
,
are real constants. Define the function
Theorem 4. The solution of the IVP (40) and (41), (42) is given by Proof. Let
. Then from the Equation (
40) we have
with initial condition (
42).
According to Proposition 3 with
the solution is
Let
. Then from (
40)–(
42) and (
45) we have
According to Proposition 3 the solution is
Let
. Then from (
40)–(
42) and (
46) we have
According to Proposition 3 the solution is
Continue this process, and the claim is established. □
5.2. Non-Zero Initial Function
Consider non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay (
40) with the initial conditions
where
,
.
Theorem 5. The solution of the IVP (40) and (48), (49) is given by Proof. The proof is similar to the one of Theorem 3 so we omit it. □
6. Conclusions
In fractional models finding exact solutions is an important question and it can be quite complicated even in the linear scalar case when considering RL fractional differential equations. In this paper we study initial value problems of scalar linear RL fractional differential equations with constant delay and an initial value problem is set up in an appropriate way. On the one hand, it is similar to the case of the integer order derivative, and on the other hand, it is related to the definition and properties of the RL derivative. This allows the corresponding initial value problem to be used for modeling processes with impulsive type initial conditions which can be found in physics, chemistry, engineering, biology, and economics. Explicit formulas for the solutions of initial value problems with both zero and nonzero initial functions are obtained and homogeneous and non-homogeneous equations are studied. The formulas given will be very helpful in the theoretical study of linear scalar fractional models, for linearization of nonlinear models, and for the monotone-iterative technique for RL fractional differential equations.
Author Contributions
Conceptualization, R.A., S.H. and D.O’R.; methodology, R.A., S.H. and D.O’R.; validation, R.A., S.H. and D.O’R.; formal analysis, R.A., S.H. and D.O’R.; writing–original draft preparation, R.A., S.H. and D.O’R.; writing–review and editing R.A., S.H. and D.O’R.; funding acquisition, S.H. All authors have read and agreed to the published version of the manuscript.
Funding
Research was partially supported by Fund Scientific Research FP19-FMI-002, Plovdiv University.
Conflicts of Interest
The authors declare no conflict of interest.
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