User s. mabuza - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:13:57Z http://mathoverflow.net/feeds/user/20631 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87769/variational-formulation-of-boundary-value-problems-with-unknown-on-the-boundary Variational Formulation of Boundary Value Problems With Unknown on the boundary S. Mabuza 2012-02-07T06:41:52Z 2012-02-07T06:41:52Z <p>Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,</p> <p>\begin{eqnarray} Lu &amp;=&amp; \frac{\partial u}{\partial t}, u(x,0) &amp;=&amp; u_0 \; \; \mbox{in}\; \Omega \ \end{eqnarray}</p> <p>where $L$ is a Parabolic operator. Furthermore, we consider a situation whereby on a portion of the boundary, say $\Gamma \subset \partial \Omega$ we have Robin conditions with an unknown function $\tilde{u} :\partial \Omega \times [0,T] \rightarrow \mathbb{R}$ posed in the following way</p> <p>$$\tilde{u}_t =\frac{\partial u}{\partial \nu} = cu + \tilde{u}, \; \mbox{on}\; \partial \Omega.$$</p> <p>Question: Whats the best variational formulation for this problem?</p> <p>I have investigated a weak - strong formulation, whereby the main operator equation is transformed to its weak form using the second equality of the boundary condition and coupling that with a strong part( classical ) form of the first equality of the boundary condition. Moreover on $\partial \Omega \cap \Gamma^c$ we have Dirichlet boundary data. It is worth noting that we assume suitable regularity for both unknown function.</p> <p>I have consulted literature on Free boundary problems including the celebrated Stefan Problem, however I am still in the woods as far as figuring out a concrete formulation.</p> http://mathoverflow.net/questions/86943/is-there-a-pde-for-this-phenomenon/87065#87065 Answer by S. Mabuza for Is there a PDE for this phenomenon? S. Mabuza 2012-01-30T20:38:07Z 2012-01-30T20:38:07Z <p>I might be wrong but this seems to be somewhat related to the phenomena of sinks and sources, in that case the stream functions $\psi$ obey the law $d \psi = v_r ds$ where $v_r$ is radial velocity, and $ds$ is arclength measure.</p> http://mathoverflow.net/questions/85511/numerical-solution-to-fisher-kolmogorov-equation/86117#86117 Answer by S. Mabuza for Numerical solution to Fisher-Kolmogorov equation S. Mabuza 2012-01-19T17:32:12Z 2012-01-19T17:32:12Z <p>Another approach especially in higher dimensions would be to use a Fourier - Spectral method for spatial discretization followed by a fourth order Runge - Kutta algorithm to the algebraic differential system which is in "Fourier Space". The Inverse Fast Fourier Transform can then be used to recover the solution in the original domain. This technique is useful especially with coupled reaction diffusion equations.</p> http://mathoverflow.net/questions/86079/reynolds-number-in-multiple-scales Reynolds Number in Multiple Scales S. Mabuza 2012-01-19T07:53:47Z 2012-01-19T07:53:47Z <p>Consider the Navier - Stokes equations in a bounded region $\Omega_t \subset \mathbb{R}^2$ with a Lipschitz boundary $\partial \Omega_t$ and the domain is time dependent. Can one introduce the notion of the Reynolds number if this problem is posed in a periodic media with scaling parameter $\varepsilon &lt;&lt; 1$. To be precise I consider $\Omega_t = (0, \infty)\times (-H - \eta(x,t), H + \eta(x,t))$ where $\eta \in L^1 (I; L^{\infty}(\mathbb{R}_+ ))$, where $I = (0,T)$ and let $\varepsilon = H/L$ where $L$ is an appropriately chosen characteristic length. A non-dimensionalization of the Navier - Stokes is performed and possibly two Reynolds numbers are obtained, one corresponding to the longitudinal characteristic length $L$ and one corresponding to the transversal characteristic length $H$. Is such a characterization physically reasonable?</p> http://mathoverflow.net/questions/89249/monge-ampere-and-calculus Comment by S. Mabuza S. Mabuza 2013-03-18T03:45:46Z 2013-03-18T03:45:46Z I'm not sure about all else, but the $j$ indices are sums over $j$, in the Einstein way. http://mathoverflow.net/questions/105864/how-to-apply-lagrange-multipliers-to-bcs-of-time-dependent-problems-using-finite Comment by S. Mabuza S. Mabuza 2013-03-18T03:30:20Z 2013-03-18T03:30:20Z Hi, what type of BC's do you have? I think if you use backward Euler scheme, you would still have a steady type (at each time step) of problem with a different source term but same Lagrange multipliers.