Hata Model for Suburban Areas

In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has order one and is A-stable.

Description

Consider the ordinary differential equation

${\frac {\mathrm {d} y}{\mathrm {d} t}}=f(t,y)$

with initial value $y(t_{0})=y_{0}.$ Here the function $f$ and the initial data $t_{0}$ and $y_{0}$ are known; the function $y$ depends on the real variable $t$ and is unknown. A numerical method produces a sequence $y_{0},y_{1},y_{2},\ldots$ such that $y_{k}$ approximates $y(t_{0}+kh)$, where $h$ is called the step size.

The backward Euler method computes the approximations using

$y_{k+1}=y_{k}+hf(t_{k+1},y_{k+1}).$

This differs from the (forward) Euler method in that the latter uses $f(t_{k},y_{k})$ in place of $f(t_{k+1},y_{k+1})$.

The backward Euler method is an implicit method: the new approximation $y_{k+1}$ appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown $y_{k+1}$. Sometimes, this can be done by fixed-point iteration:

$y_{k+1}^{[0]}=y_{k},\quad y_{k+1}^{[i+1]}=y_{k}+hf(t_{k+1},y_{k+1}^{[i]}).$

If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation $y_{k+1}$.

Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation.

Derivation

Integrating the differential equation ${\frac {\mathrm {d} y}{\mathrm {d} t}}=f(t,y)$ from $t_{k}$ to $t_{k+1}=t_{k}+h$ yields

$y(t_{k+1})-y(t_{k})=\int _{t_{k}}^{t_{k+1}}f(t,y(t))\,\mathrm {d} t.$

Now approximate the integral on the right by the right-hand rectangle method (with one rectangle):

$y(t_{k+1})-y(t_{k})\approx hf(t_{k+1},y(t_{k+1})).$

Finally, use that $y_{k}$ is supposed to approximate $y(t_{k})$ and the formula for the backward Euler method follows.

The same reasoning leads to the (standard) Euler method if the left-hand rectangle rule is used instead of the right-hand one.

Analysis

The pink region outside the disk shows the stability region of the backward Euler method.

The backward Euler method has order one. This means that the local truncation error (defined as the error made in one step) is $O(h^{2})$, using the big O notation. The error at a specific time $t$ is $O(h)$.

The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. This includes the whole left half of the complex plane, so the backward Euler method is A-stable, making it suitable for the solution of stiff equations. In fact, the backward Euler method is even L-stable,

Extensions and modifications

The backward Euler method is a variant of the (forward) Euler method. Other variants are the semi-implicit Euler method and the exponential Euler method.

The backward Euler method can be seen as a Runge–Kutta method with one stage, described by the Butcher tableau:

${\begin{array}{c|c}1&1\\\hline &1\\\end{array}}$

The backward Euler method can also be seen as a linear multistep method with one step. It is the first method of the family of Adams–Moulton methods, and also of the family of backward differentiation formulas.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.