Behaviour of the boundary potentials and boundary integral solution of the time fractional diffusion equation 

Author:  Kemppainen, Jukka 
Organizations:  University of Oulu, Faculty of Science, Department of Mathematical Sciences 
Format:  eBook 
Online Access:  PDF Full Text (PDF, 0.6 MB) 
Persistent link:  http://urn.fi/urn:isbn:9789514261329 
Language:  English 
Published: 
2010

Publish Date:  20100331 
Thesis type:  Doctoral Dissertation 
Defence Note:  Academic dissertation to be presented with the assent of the Faculty of Science of the University of Oulu for public defence in OPsali (Auditorium L10), Linnanmaa, on 10 April 2010, at 12 noon 
Reviewer: 
Professor Martin Costabel Professor Dorina Mitrea 
Description: 
AbstractThe dissertation considers the time fractional diffusion equation (TFDE) with the Dirichlet boundary condition in the subdiffusion case, i.e. the order of the time derivative is α ∈ (0,1). In the thesis we have studied the solvability of TFDE by the method of layer potentials. We have shown that both the single layer potential and the double layer potential approaches lead to integral equations which are uniquely solvable. The dissertation consists of four articles and a summary section. The first article presents the solution for the time fractional diffusion equation in terms of the single layer potential. In the second and third article we have studied the boundary behaviour of the layer potentials for TFDE. The fourth paper considers the spline collocation method to solve the boundary integral equation related to TFDE. In the summary part we have proved that TFDE has a unique solution and the solution is given by the double layer potential when the lateral boundary of a bounded domain admits C^{1} regularity. Also, we have proved that the solution depends continuously on the datum in the sense that a nontangential maximal function of the solution is norm bounded from above by the datum in L^{2}(Σ_{T}). If the datum belongs to the space H^{1,α/2}(Σ_{T}), we have proved that the nontangential function of the gradient of the solution is norm bounded from above by the datum in H^{1,α/2}(Σ_{T}). 
Series: 
Acta Universitatis Ouluensis. A, Scientiae rerum naturalium 
ISSNE:  1796220X 
ISBN:  9789514261329 
ISBN Print:  9789514261312 
Issue:  548 
Subjects:  
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