[Turkmath:231] Hacettepe Matematik Seminer (18.02.2015, Çarşamba, Doç. Dr. Burak Aksoylu (TOBB ETÜ ve Wayne State University))

[Mesut Şahin]

 Değerli liste üyeleri,

aşağıda ve ekteki afişte detayları verilen bölüm seminerimize ilgilenenleri
bekliyoruz.


En iyi dileklerimle,

mesut


*HACETTEPE ÜNİVERSİTESİ MATEMATİK BÖLÜMÜ GENEL SEMİNERİ*



*(HACETTEPE MATHEMATICS GENERAL SEMINAR)*



*Tarih (Date) :* 18.02.2015, Çarşamba (Wednesday)

*Saat (Time):* 15:00

*Yer (Place):* Yaşar ATAMAN Seminer Salonu



*Konuşmacı (Speaker):* Doç. Dr. Burak Aksoylu (TOBB ETÜ ve Wayne State
University)

*Başlık (Title) :* Incorporating local boundary conditions into nonlocal
theories.

*Özet (Abstract) : *We study nonlocal equations from the area of
peridynamics on bounded domains. In our companion paper, we discover that,
on Rn, the governing operator in peridynamics, which involves a
convolution, is a bounded function of the classical (local) governing
operator.  Building on this, we define an abstract convolution operator on
bounded domains which is a generalization of the standard convolution based
on integrals.  The abstract convolution operator is a function of the
classical operator, defined by a Hilbert basis available due to the purely
discretespectrum of the latter.  As governing operator of the
nonlocalequation we use a function of the classical operator, this allows
us to incorporate local boundary conditions into nonlocal theories. The
governing operator is determined by what we call the regulating
function. By choosing different regulating functions, we can define
governing operators tailored to the needs of the underlying application.



For the homogeneous wave equation with the considered boundary conditions,
we prove that continuity is preserved by time evolution. Namely, if the
initial data is continuous, then the solution is continuous for real t.
This is due to the fact that the solution has a unique decomposition into
two parts.  The first part is the product of a function of time with the
initial data.  The second part is continuous.  This decomposition is
induced by the fact that the governing operator has a unique decomposition
into multiple of the identity and a Hilbert-Schmidt operator.  The
decomposition also implies that discontinuities remain stationary.



We present a numerical study of the solutions of the wave equation. For
discretization, we employ a weak formulation based on a Galerkin projection
and use piecewise polynomials on each element which allows discontinuities
of the approximate solution at the element borders. We study convergence
order of solutions with respect to polynomial order and observe optimal
convergence.  We depict the solutions for each boundary condition.



NOT: Konuşma sonunda çay ve pasta ikramı olacaktır.

(P.S.  Tea and cookies will be served after the talk.)


  Mesut Sahin
  Associate Professor
  Department of Mathematics
  Hacettepe University
  TR 06800 Beytepe
  ANKARA - TURKEY
 http://yunus.hacettepe.edu.tr/~mesut.sahin
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