• A K Nandakumaran

      Articles written in Proceedings – Mathematical Sciences

    • Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

      A K Nandakumaran M Rajesh

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      In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains$$\begin{gathered} \partial _t b(\tfrac{x}{\varepsilon },u_\varepsilon ) - diva(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon \times (0,T), \hfill \\ a(\tfrac{x}{\varepsilon },u_\varepsilon ,\nabla u_\varepsilon ) \cdot v_\varepsilon = 0 on \partial S_\varepsilon \times (0,T), \hfill \\ u_\varepsilon = 0 on \partial \Omega \times (0,T), \hfill \\ u_\varepsilon (x,0) = u_0 (x) in \Omega _\varepsilon \hfill \\ \end{gathered} $$. Here, ΩɛSɛ is a periodically perforated domain. We obtain the homogenized equation and corrector results. The homogenization of the equations on a fixed domain was studied by the authors [15]. The homogenization for a fixed domain and$$b(\tfrac{x}{\varepsilon },u_\varepsilon ) \equiv b(u_\varepsilon )$$ has been done by Jian [11].

    • Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition

      A K Nandakumaran M Rajesh

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      In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains$$\begin{gathered} \partial _t b(\tfrac{x}{{d_\varepsilon }},u_\varepsilon ) - div a(u_\varepsilon , \nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon x (0, T), \hfill \\ u_\varepsilon ) = 0 on \partial \Omega _\varepsilon x (0, T), \hfill \\ u_\varepsilon (x, 0) = u_0 (x) in \Omega _\varepsilon . \hfill \\ \end{gathered} $$. Here, Ωɛ= ΩSε is a periodically perforated domain anddε is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization for a fixed domain and$$b(\frac{x}{{d_\varepsilon }},u_\varepsilon ) \equiv b(u_\varepsilon )$$ has been done by Jian. We also obtain certain corrector results to improve the weak convergence.

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