Graduate Seminar Report on Non-Integer Order of Differentiation and its Application on the Dirichlet Problem of Laplace Equation
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Date
2002-06
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Addis Ababa University
Abstract
This seminar report basically consists of two chapters. In the first part, the Sobolev
and Nikol Skii Besov functions spaces are investigated and some inclusion theorems
are developed as well as some relations of the two function spaces are stated as a
particular case. As the main objective of the seminar report we formulate the
necessary and sufficient condition for the function (which is equal to the solution of
the Dirichlet problem on the boundary) on the boundary of some region, for the
solution of the generalized formulation of the Dirichlet problem of Laplace equaton
to exist. For the classical formulation of the Dirichlet problem, we, as an assumption,
take the value of the function on the boundary of some region in which the solution
is defined is continuous on the boundary. But, this does not guarantee the exis tence
of the solution of the problem, which is shown by Hadamard's example. Therefore,
as a conclusion, we will see that the formulation of the necessary and sufficient
condition for the generalized formulation of the Dirichlet problem of Laplace
equaton in the second chapter of this semina r report.
Before all, T thank the almighty God with out His help nothing is happened.
I would like to express my special gratitude to my advisor and teacher Dr. Tsegaye
Gedif for his unlimited help for the completion of this seminar report. T am really
lucky for his being here with me. I would also like to thank all my teachers for their
con tribution to bring me to this level.
Finally, I thank my parents for all they have given me and for their prayers. Also,
I would like to thank W /0 Tobiaw Tefera, for typing this manuscript partially.