| Fri, May 04, 2018
|
Senior Honors Thesis Defense
2:30 PM
MSCS 514 | | On partial differential equations modified with fractional operators and integral transformations
Nicholas H. Nelsen, OSU
| | | Abstract: We present the theory and applications of nonlocal operators in the setting of partial differential equations (PDE), with emphasis placed on the fractional Laplacian, $ \Lambda^{\alpha}=(-\Delta)^{\frac{\alpha}{2}} $, which generalizes the Laplacian differential operator, $ \Delta $ (also denoted by $ \nabla^2 $). By extending the heat equation
$
\partial_t u - \nu\Delta u =f
$
into its fractional counterpart
$
\partial_t u + \nu\Lambda^{\alpha} u =f ,
$
we can study a whole family of PDE with parameter $ \alpha \in [0,2] $ all at once. This mindset naturally motivates the investigation of $ \alpha's $ influence on the solution to these nonlocal PDE; it is not at all obvious what effect or meaning these colloquially named ``fractional derivatives'' have, especially at small values of $ \alpha $. We detail some decay bounds and perform numerical experiments to provide further insight into our theorems and observe how these results hold up in practice. Since our focus is on the corresponding initial value problems for these evolutionary PDE, we primarily perform 1D simulations on a periodic domain using pseudo-spectral methods. We conclude this work by incorporating advection (also known as ``transport'' or ``convection'') terms into the PDE, with both nonlinear and nonlocal modifications, e.g., integral transformations such as the Hilbert transform. These equations are more physically realistic and are often considered models for viscous incompressible fluid flow and related phenomena, particularly in the case of Burgers' equation which exhibits shock wave behavior. |
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