Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions
Abstract
:1. Introduction
2. Notions and Preliminaries
3. Theoretical Study
3.1. Position of the Problem
- Reformulation of the problem into a problem with homogeneous conditions.
- The uniqueness of the solution to the problem using the a priori estimate method.
- The existence of the solution of the problem based on the density of the range of the operator generated by the abstract formulation of the problem.
3.2. Reformulation of the Problem
3.3. Energy Inequality Method and Consequences
3.4. Existence of the Solution
4. Numerical Study
4.1. Finite Difference Method
4.1.1. Discretization of the Problem
4.1.2. General Case
4.2. Stability and Convergence
4.2.1. Stability
4.2.2. Convergence
4.3. Applications
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
5 × 10 | 6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 9 × 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 9 × 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
5 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 7 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
6 × 10 | 6 × 10 | 7 × 10 | 8 × 10 | 8 × 10 | 9 × 10 | 10 | 10 | 10 | 10 | 10 |
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Brahimi, S.; Merad, A.; Kılıçman, A. Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions. Mathematics 2021, 9, 1987. https://doi.org/10.3390/math9161987
Brahimi S, Merad A, Kılıçman A. Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions. Mathematics. 2021; 9(16):1987. https://doi.org/10.3390/math9161987
Chicago/Turabian StyleBrahimi, Saadoune, Ahcene Merad, and Adem Kılıçman. 2021. "Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions" Mathematics 9, no. 16: 1987. https://doi.org/10.3390/math9161987
APA StyleBrahimi, S., Merad, A., & Kılıçman, A. (2021). Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions. Mathematics, 9(16), 1987. https://doi.org/10.3390/math9161987