A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term
Abstract
:1. Introduction
2. Some Notions on Fractional Calculus
3. Main Results
- (i)
- , ;
- (ii)
- (the support of φ with respect to the variable x is a compact subset of Q);
- (iiii)
- , .
- (i)
- (ii)
- if and , , , then for all
4. Preliminaries
4.1. A Priori Estimate
4.2. Test Functions
4.3. Preliminary Estimates
5. Proofs of the Obtained Results
- (I)
- For the case , we discuss two sub-cases.
- (i)
- If , in this case, one has
- (ii)
- (I)
- (11) is satisfied. Theorem 1 applies.
- (II)
- For the case and , let
- (III)
6. Conclusions
- if and , , , then for all
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Aldawish, I.; Jleli, M.; Samet, B. A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term. Symmetry 2023, 15, 1071. https://doi.org/10.3390/sym15051071
Aldawish I, Jleli M, Samet B. A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term. Symmetry. 2023; 15(5):1071. https://doi.org/10.3390/sym15051071
Chicago/Turabian StyleAldawish, Ibtisam, Mohamed Jleli, and Bessem Samet. 2023. "A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term" Symmetry 15, no. 5: 1071. https://doi.org/10.3390/sym15051071
APA StyleAldawish, I., Jleli, M., & Samet, B. (2023). A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term. Symmetry, 15(5), 1071. https://doi.org/10.3390/sym15051071