Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses
Abstract
:1. Introduction
2. Some Preliminary Results From Fractional Calculus
- -
- Riemann–Liouville fractional integral of order ([13])This is called by some authors the left Riemann–Liouville fractional integral of order q.Note sometimes the notation is used.
- -
- Riemann–Liouville fractional derivative of order ([13])This is also called the left Riemann–Liouville fractional derivative.
- -
- Caputo fractional derivative of order ([13])
- (a)
- If there exists a limit , then there also exists a limit
- (b)
- If there exists a limit and if the limit exists, then
- (i)
- If , then for any point
- (ii)
- If and , then for any point
3. Non-Instantaneous Impulses in RL Fractional Differential Equations
3.1. Fixed Lower Bound of the RL Fractional Derivative at the Given Initial Time
3.1.1. Integral Form of the Initial Value Problem
- (i)
- the function x satisfies the linear problem
- (ii)
- The function x is continuous at .
- 1.
- The function for any and the inequality holds for all and , where .
- 2.
- For all the functions for any and the inequality holds for and where .
- 3.
- The inequality holds where
3.1.2. Weighted Initial Value Problem
- (i)
- the function x satisfies the linear problem
- (ii)
- The function x is continuous at .
3.2. Changed Lower Bounds of the RL Fractional Derivative at the Impulsive Points
3.2.1. Integral form of the Initial Conditions and Impulses
- (i)
- the function x satisfies the linear problem
- (ii)
- The function x satisfies for .
- 1.
- The function for any and the inequality holds for all and , where .
- 2.
- For all the functions for any and the inequality holds for and where .
- 3.
- The inequality holds where
3.2.2. Weighted Form of the Initial Conditions and Impulses
3.2.3. Mixed Forms of the Initial Conditions and Impulses
3.3. Examples
- Case 1.
- Let for , .
- Case 1.1.
- Consider (8)–(10) and (16) to obtain the solution:The solution depends on the initial value only on the interval . Therefore, two solutions with different initial values will coincide for all .
- Case 1.2.
- Consider (25), (27) and apply (33) to obtain the solution:Similarly to Case 1.1 we obtain that two solutions with different initial values coincide for all .
- Case 2.
- Let for , where are constants.
- -
- If , then the impulsive condition (9) is reduced to the condition for , and obviously the IVP for NIRLFrDE (8)–(10), respectively (25), (27), will have an infinite number of solutions.
- -
- If , then the impulsive condition (9) is reduced to the condition for , which has no solution and the IVP for NIRLFrDE (8)–(10), respectively (25), (27), will have no solution.
- -
- If , then the impulsive condition (9) is reduced to the condition for , which has only the zero solution, and therefore any solution of IVP for NIRLFrDE (8)–(10), respectively (25), (27) will be zero on , . In this case from (16) we obtain the solution for (8)–(10):
- -
- If , then the impulsive condition (9) is reduced to the condition for , which has an unique solution and we can talk about uniqueness of the solution IVP for NIRLFrDE (8)–(10), respectively (25), (27).
- Case 3.
- Let for , . Then the impulsive condition (9) is reduced to the algebraic equation which could have more than one solution (for example if , then there are 5 constant solutions), i.e., we do not have uniqueness for the IVP for NIRLFrDE (8)–(10), respectively (25), (27).
4. Brief Overview of RL Fractional Equations with Instantaneous Impulses
Author Contributions
Funding
Conflicts of Interest
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Agarwal, R.; Hristova, S.; O’Regan, D. Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses. Symmetry 2019, 11, 614. https://doi.org/10.3390/sym11050614
Agarwal R, Hristova S, O’Regan D. Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses. Symmetry. 2019; 11(5):614. https://doi.org/10.3390/sym11050614
Chicago/Turabian StyleAgarwal, Ravi, Snezhana Hristova, and Donal O’Regan. 2019. "Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses" Symmetry 11, no. 5: 614. https://doi.org/10.3390/sym11050614