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In [[mathematics]], the '''Fictitious domain method''' is a method to find the solution of a [[partial differential equation]]s on a complicated [[Domain of a function|domain]] <math>D</math>, by substituting  a given  problem
posed on a domain <math>D</math>, with a new problem posed on a simple domain <math>\Omega</math> containing <math>D</math>.
 
==General formulation==
Assume in some area <math>D \subset \mathbb{R}^n </math> we want to find solution <math>u(x)</math> of the [[equation]]:
 
: <math>
  Lu = - \phi(x), x = (x_1, x_2, \dots , x_n) \in D
</math>
 
with [[Boundary value problem|boundary conditions]]:
 
: <math>
lu = g(x), x \in \partial D \,
</math>
 
The basic idea of fictitious domains method is to substitute  a given  problem
posed on a domain <math>D</math>, with a new problem posed on a simple [[shaped domain]] <math>\Omega</math> containing <math>D</math> (<math>D \subset \Omega</math>). For example, we can choose ''n''-dimensional parallelepiped as <math>\Omega</math>.
 
Problem in the [[extended domain]] <math>\Omega</math> for the new solution <math>u_{\epsilon}(x)</math>:
 
: <math>
L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega
</math>
 
: <math>
  l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega
</math>
 
It is necessary to pose the problem in the extended area so that the following condition is fulfilled:
 
: <math>
u_\epsilon (x) \xrightarrow[\epsilon \rightarrow 0]{ } u(x), x \in D \,
</math>
 
== Simple example, 1-dimensional problem ==
 
:<math>
  \frac{d^2u}{dx^2} = -2, \quad 0 < x < 1 \quad (1)
</math>
 
:<math>
u(0) = 0, u(1) = 0 \,
</math>
 
=== Prolongation by leading coefficients ===
 
<math>u_\epsilon(x)</math> solution of problem:
 
: <math>
\frac{d}{dx}k^\epsilon(x)\frac{du_\epsilon}{dx} = - \phi^{\epsilon}(x), 0 < x < 2 \quad (2)
</math>
Discontinuous [[coefficient]] <math>k^{\epsilon}(x)</math> and right part of equation previous equation we obtain from expressions:
 
: <math>
k^\epsilon (x)=\begin{cases} 1, &  0 < x < 1 \\ \frac{1}{\epsilon^2}, & 1 < x < 2
\end{cases}
</math>
: <math>
(3)
</math>
: <math>
\phi^\epsilon (x)=\begin{cases} 2, &  0 < x < 1 \\ 2c_0, & 1 < x < 2
\end{cases}
</math>
 
Boundary conditions:
 
: <math>
u_\epsilon(0) = 0, u_\epsilon(1) = 0
</math>
 
Connection conditions in the point <math>x = 1</math>:
 
: <math>
[u_\epsilon(0)] = 0,\  \left[k^\epsilon(x)\frac{du_\epsilon}{dx}\right] = 0
</math>
 
where <math>[ \cdot ]</math> means:
 
: <math>
[p(x)] = p(x + 0) - p(x - 0) \,
</math>
 
Equation (1) has [[analytical solution]] therefore we can easily obtain error:
 
: <math>
u(x) - u_\epsilon(x) = O(\epsilon^2), \quad 0 < x < 1
</math>
 
===Prolongation by lower-order coefficients===
<math>u_\epsilon(x)</math> solution of problem:
 
: <math>
\frac{d^2u_\epsilon}{dx^2} - c^\epsilon(x)u_\epsilon = - \phi^\epsilon(x), \quad 0 < x < 2 \quad (4)
</math>
 
Where <math>\phi^{\epsilon}(x)</math> we take the same as in (3), and expression for <math>c^{\epsilon}(x)</math>
 
: <math>
c^\epsilon(x)=\begin{cases} 1, & 0 < x < 1 \\ \frac{1}{\epsilon^2}, & 1 < x < 2
  \end{cases}
</math>
 
[[Boundary value problem|Boundary conditions]] for equation (4) same as for (2).
 
Connection conditions in the point <math>x = 1</math>:
 
: <math>
[u_\epsilon(0)] = 0,\ \left[\frac{du_\epsilon}{dx}\right] = 0
</math>
 
Error:
 
: <math>
u(x) - u_\epsilon(x) = O(\epsilon), \quad 0 < x < 1
</math>
 
==Literature==
 
* P.N. Vabishchevich, The Method of Fictitious Domains in Problems of Mathematical Physics, Izdatelstvo Moskovskogo Universiteta, Moskva, 1991.
* Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Preprint CC SA USSR, 68, 1979.
* Bugrov A.N., Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Mathematical model of fluid flow, Novosibirsk, 1978, p.&nbsp;79–90
 
{{Numerical PDE}}
 
{{DEFAULTSORT:Fictitious Domain Method}}
[[Category:Domain decomposition methods]]
[[Category:Applied mathematics]]

Revision as of 11:06, 14 February 2014



Sunlight supplies the greatest type of light for your plants, but developing your plants outdoors in complete sun is not always an choice. Some claim the LED grow light technology might finally place the cultivation of a handful of marijuana plants within reach of anyone who has a few extra bucks and a spare closet. On paper, LED grow lights look to outperform HPS develop lights (the current common for developing marijuana) in virtually every way. The truth that HPS lights produce excessive heat is one particular of the significant motives why there's so significantly interest in LED grow lights. LED grow lights are totally a amazing factor.

Cherry tomatoes are in particular effortless to develop in a household or apartment for the reason that the plants do not root as deeply as varieties such as like Massive Boy and Early Girl. Indoor gardeners can spot the lights close to the plants due to the fact LED lights emit tiny heat, unlike incandescent develop lights. Given that the lights run cool, the area where the tomatoes grow will not have to have unique ventilation, while tomatoes do not thoughts warmth. Strawberries, one more plant that needs direct sunshine, develop nicely under LED lights.

For about $500, you can obtain the similar Licor 190 PAR meter that Utah State University Crop Physiology Laboratory faculty utilized to evaluate HID and LED plant lights , and measure how lots of μmoles/m²/s at typical intervals to get a PAR map, like the outstanding examples at BML It can be a little tedious, but it would not take a lot more than an hour or two. I won't complain about significantly less energy consumption as extended as the plants grow nicely.

Unlike most other varieties of develop lights, such as metal halide lights and higher-pressure sodium lights, tiny energy taken in by LEDs is offered off in the form of heat. According to LED Grow , most grow lights include traces of toxic elemental mercury. In terms of cost and heat, fluorescent lights are a greater option than incandescent bulbs. Every of the LED grow lights pointed out have been shown to proficiently mimic the light spectrum that is equivalent to the sun.

A warranty may well not appear also significant when LED develop lights have the lengthy life that they possess, but that doesn't imply you should take their word for it. When searching at a develop light, be sure to take a look at what sort of warranty they have in location. The extended term savings of LED develop lights are very useful, but you have to be certain that you aren't going to be dealing with additional expenses soon after the acquire.

The LED grow lights commonly make it much a lot easier for any person to grow the plants they have to have at a considerably faster price, because they encourage enhanced photosynthesis and also make sure that nutrients supplied to the plants are applied optimally. Which means, you can count on to harvest a great deal greater crops simply because of lighting them with these LED grow lights and even if you evaluate them with other plants, they are more satisfying and larger. Today, LED lights have come to be well known for folks.

I was doing some looking and I located that this firm do industrial develop web pages for health-related marijuana in the US and makes use of a triband systems which consists of 1 watt led and three watt led The benefits that I have received from my plants using these have been wonderful. I will have to say it did take slightly longer to grow that your typical hps set up but the bills exactly where less costly ! At the finish of 500000 hours a LED maintains 70% of its light output.

Unlike other LED develop lights you can be assured that with the G8LED you are obtaining lights that do not use an aluminum that hides inferior diodes. You are receiving lights that are configured in parallel that present Zener protection. That is what separates these distinct develop lights from many of the other individuals on the industry. You can even grow broccoli, peppers, tomatoes and wheatgrass as nicely. You can also utilize it assist develop your indoor plants.

Do folks knowledge shading concerns at the base of the plants due to led being to narrow of a spectrum. I tested LED develop lights and i can confirm you that LED is nevertheless way below HPS lightning for flowering stage…. I was incorrect pondering that HPS is problematic since of temperature, point is that photosynthesis is at its peak at 29-31C and LED cannot delivered sufficient lumens and temperature ! So comparing MH/HPS to LED based on lumens is a waste of time.

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