Schottky's theorem: Difference between revisions

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== drawing blood ==


The '''Petrov–Galerkin method ''' is a mathematical method used to obtain approximate solutions of [[partial differential equations]] which contain terms with [[Even and odd numbers|odd]] order.  
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In these type of problems a weak formulation with similar function space for test function and solution function is not possible. Hence the method is used in case the test function and solution function belong to different function spaces.<ref>J. N. Reddy: ''An introduction to the finite element method'', 2006, Mcgraw–Hill</ref>
 
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</ul>


==Overview==
== Nike Roshe Run Women in statute ==
An example of differential equation containing a term with odd order is as follows:


: <math>a(x)\dfrac{\mathrm{d}u}{\mathrm{d}x} + b(x)\dfrac{\mathrm{d}^2u}{\mathrm{d}x^2} = f(x),\quad x\in (0,L) \quad\&\quad u(0)=u_o, \left.\dfrac{\mathrm{d}u}{\mathrm{d}x}\right|_{x=L}=u_L'</math>
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If a test function <math>v(x)</math> is used to obtain the weak form, the final Galerkin formulation will be given as follows:
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: <math>\int_0^L a(x)v(x)\dfrac{\mathrm{d}u}{\mathrm{d}x}\mathrm{d}x-\int_0^L b(x)\dfrac{\mathrm{d}v}{\mathrm{d}x}\dfrac{\mathrm{d}u}{\mathrm{d}x}\mathrm{d}x + \left[b(x)v\dfrac{\mathrm{d}u}{\mathrm{d}x}\right]_0^L = \int_0^L v(x)f(x) \, \mathrm{d}x </math>
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The term with even order (2nd term in LHS) is now symmetric, as the test function and solution function both have same order of differentiation and they both belong to <math>H_0^1</math>. However, there is no way the first term on LHS can be made this way. In this case the solution space<math>H_0^1</math> and test function space <math>L^2</math> are different and hence the usually employed [[Galerkin method|Bubnov Galerkin]] method cannot be used.
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In this problem, the formulation method is the same as in the Bubnov-Galerkin method. However the test function and the solution function approximations can not be same, hence they must be approximated separately. As the problem loses symmetry, the final coefficient matrix is also not symmetric and hence the solver selection must be made carefully. These problems sometime pose challenges in terms of computing power and complexity.
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== See also ==
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* [[Bubnov-Galerkin method]]
 
 
</ul>
==Notes==
{{reflist}}
 
{{DEFAULTSORT:Petrov-Galerkin method}}
[[Category:Partial differential equations]]
 
 
{{Mathanalysis-stub}}

Latest revision as of 22:42, 25 September 2014

drawing blood

But the cofounder of the nonprofit organization that photographed the stillborn baby at the Duggars' request asserted thousands of grieving parents have considered the audience for its free services, since the photographs assistance to console them during the worst duration of their lives. Cheryl Haggard, who had a son who died after birth because of a condition that went undetected during pregnancy, cofounded Now I Lay Me Down To Sleep having a photographer she hired..

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