Numerical solution of fractional evolution equations of Caputo type
In this project, we investigate Caputo-type fractional derivatives, time-fractional evolution equations similar to the heat equation, and their numerical solutions using a Feynman-Kac type representation. We discuss several applications of PDEs involving the Caputo derivative. We review results b...
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| Format: | Final Year Project |
| Language: | English |
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Nanyang Technological University
2024
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| Online Access: | https://hdl.handle.net/10356/175651 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | In this project, we investigate Caputo-type fractional derivatives, time-fractional evolution equations
similar to the heat equation, and their numerical solutions using a Feynman-Kac type representation.
We discuss several applications of PDEs involving the Caputo derivative. We review
results by Meerschaert et al. and Bonaccorsi et al., and implement them in R to attain Monte
Carlo estimates to solutions of the PDEs involving the Caputo derivative. We compare the results
of our simulation to those by Meerschaert, and apply the Monte Carlo method on a point initial
distribution and uniform initial distribution. We perform some analysis on the errors and runtime
of the Monte Carlo scheme. Finally, we discuss future work that can be done in the same spirit
of this project, performing some simple exploratory work on a simple variation of the heat-like
equation. |
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