Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations
Abstract
:1. Introduction
2. Finite Difference Scheme
3. Stability and Convergence of the Numerical Method
3.1. Stability
- (i)
- (ii)
- (i)
- This is straightforward since ;
- (ii)
- Regarding (ii), note that:By the mean value theorem, there must exist a such that:Analogously, there must exist a such that:Then,
3.2. Convergence
- For Mesh 1:
- For Mesh 2:
- For Mesh 3:
- For Mesh 4:
4. Numerical Results and Discussion
- For Mesh 1 and , we have that ;
- For Mesh 2 and , we have that ;
- For Mesh 3 and , we have that (since );
- For Mesh 3 and , we have that (since ).
- For Mesh 1 and , we have that ;
- For Mesh 2 and , we have that (since );
- For Mesh 3 and , we have that (since );
- For Mesh 3 and , we have that (since ).
- For Mesh 1 and , we have that (since );
- For Mesh 2 and , we have that (since );
- For Mesh 3 and , we have that (since );
- For Mesh 3 and , we have that (since ).
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Abbreviations
EOC | Experimental order of convergence |
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Uniform Mesh | Graded Mesh— | Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 4 | 1.67 | − | − | − | − | − | − | − | − | − | − | − | − | |||||
4 | 16 | ||||||||||||||||||
8 | 64 | ||||||||||||||||||
16 | 256 | ||||||||||||||||||
32 | 1024 |
Uniform Mesh | Graded Mesh— | Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 4 | − | − | − | − | − | − | − | − | − | − | − | − | ||||||
4 | 16 | ||||||||||||||||||
8 | 64 | ||||||||||||||||||
16 | 256 | ||||||||||||||||||
32 | 1024 |
Uniform Mesh | Graded Mesh— | Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 4 | − | − | − | − | − | − | − | − | − | − | − | − | ||||||
4 | 16 | ||||||||||||||||||
8 | 64 | ||||||||||||||||||
16 | 256 | ||||||||||||||||||
32 | 1024 |
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Morgado, M.L.; Rebelo, M.; Ferrás, L.L. Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations. Mathematics 2021, 9, 1975. https://doi.org/10.3390/math9161975
Morgado ML, Rebelo M, Ferrás LL. Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations. Mathematics. 2021; 9(16):1975. https://doi.org/10.3390/math9161975
Chicago/Turabian StyleMorgado, M. Luísa, Magda Rebelo, and Luís L. Ferrás. 2021. "Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations" Mathematics 9, no. 16: 1975. https://doi.org/10.3390/math9161975
APA StyleMorgado, M. L., Rebelo, M., & Ferrás, L. L. (2021). Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations. Mathematics, 9(16), 1975. https://doi.org/10.3390/math9161975