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| '''Numerov's method''' is a numerical method to solve [[ordinary differential equation]]s of second order in which the first-order term does not appear. It is a fourth-order [[linear multistep method]]. The method is implicit, but can be made explicit if the differential equation is linear. | | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. Medical word press themes give you the latest medical designs. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. s ultimately easy to implement and virtually maintenance free. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. If a newbie missed a certain part of the video then they could always rewind. Which is perfect for building a mobile site for business use. You can up your site's rank with the search engines by simply taking a bit of time with your site. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Websites using this content based strategy are always given top scores by Google. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. If you beloved this report and you would like to acquire far more facts regarding [http://www.kutscherhof-broock.de/index.php?mod=users&action=view&id=124992 backup plugin] kindly visit the web site. The company gains commission from the customers' payment. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. I'm a large fan of using Word - Press to create pretty much any sort of web page. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. In addition, Word - Press design integration is also possible. The 2010 voting took place from July 7 through August 31, 2010. |
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| Numerov's method was developed by [[Boris Vasil'evich Numerov]].
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| ==The method==
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| The Numerov method can be used to solve differential equations of the form
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| :<math> | |
| \left ( \frac{d^2}{dx^2} + f(x) \right ) y(x) = 0
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| </math>
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| The function <math>y(x)</math> is sampled in the interval [a..b] at equidistant positions <math>x_n</math>. Starting from function values at two consecutive samples <math>x_{n-1}</math> and <math>x_n</math> the remaining function values can be calculated as
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| :<math>
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| y_{n+1} = \frac {\left( 2-\frac{5 h^2}{6} f_n \right) y_n - \left( 1+\frac{h^2}{12}f_{n-1} \right)y_{n-1}}{1+\frac{h^2}{12}f_{n+1}}
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| </math> | |
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| where <math>f_n=f(x_n)</math> and <math>y_n=y(x_n)</math> are the function values at the positions <math>x_n</math> and <math>h=x_n-x_{n-1}</math> is the distance between two consecutive samples.
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| === Nonlinear equations ===
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| For nonlinear equations of the form
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| :<math> \frac{d^2}{dt^2} y = f(t,y) </math>
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| the method is given by | |
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| :<math> y_{n+1} = 2y_n - y_{n-1} + \tfrac{1}{12} h^2 (f_{n+1} + 10f_n + f_{n-1}). </math>
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| This is an implicit [[linear multistep method]], which reduces to the explicit method given above if the function ''f'' is linear in ''y''. It achieves order 4 {{harv|Hairer|Nørsett|Wanner|1993|loc=§III.10}}.
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| ==Application==
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| In numerical physics the method is used to find solutions of the radial [[Schrödinger Equation]] for arbitrary potentials.
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| :<math>
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| \left [ -{\hbar^2 \over 2\mu} \left ( \frac{1}{r} {\partial^2 \over \partial r^2} r- {l(l+1) \over r^2} \right ) + V(r) \right ] R(r) = E R(r)
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| </math> | |
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| The above equation can be rewritten in the form
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| :<math>
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| \left [ {\partial^2 \over \partial r^2} - {l(l+1) \over r^2} + { 2\mu \over \hbar^2} \left( E - V(r)\right) \right ] u(r) = 0
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| </math>
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| with <math> u(r) = r R(r)</math>. If we compare this equation with the defining equation of the Numerov method we see
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| :<math>
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| f(x) = \frac{2\mu}{\hbar^2} \left(E - V(x) \right) - \frac{l(l+1)}{x^2}
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| </math>
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| and thus can numerically solve the radial Schrödinger equation.
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| ==Derivation==
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| Start with the [[Taylor expansion]] of <math>y(x)</math> about a point <math>x_0</math>:
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| :<math>
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| y(x) = y(x_0) + (x-x_0)y'(x_0) + \frac{(x-x_0)^2}{2!}y''(x_0) + \frac{(x-x_0)^3}{3!}y'''(x_0) + \frac{(x-x_0)^4}{4!}y''''(x_0) + \frac{(x-x_0)^5}{5!}y'''''(x_0) + \mathcal{O} (h^6)
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| </math>
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| Denote the distance from <math>x</math> to <math>x_0</math> by <math>h=x-x_0</math> and, noting that this means <math>x=x_0+h</math>, we can write the above equation as
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| :<math>
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| y(x_0+h) = y(x_0) + hy'(x_0) + \frac{h^2}{2!}y''(x_0) + \frac{h^3}{3!}y'''(x_0) + \frac{h^4}{4!}y''''(x_0) + \frac{h^5}{5!}y'''''(x_0) + \mathcal{O} (h^6)
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| </math>
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| Computationally, this amounts taking a step ''forward'' by an amount h. If we want to take a step ''backwards'', replace every h with -h for the equation of <math>y(x_0-h)</math>:
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| :<math>
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| y(x_0-h) = y(x_0) - hy'(x_0) + \frac{h^2}{2!}y''(x_0) - \frac{h^3}{3!}y'''(x_0) + \frac{h^4}{4!}y''''(x_0) - \frac{h^5}{5!}y'''''(x_0) + \mathcal{O} (h^6)
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| </math>
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| Note that only the odd powers of h experienced a sign change. On an evenly spaced grid, the ''n''th site on a computational grid corresponds to position <math>x_n</math> if the step-size between grid points are of length <math>h</math> (hence h should be small for the computation to be accurate). This means we have sampling points <math>(x_{n-1},y_{n-1})</math> and <math>(x_{n+1},y_{n+1})</math>. Taking the equations for <math>y_{n-1}</math> and <math>y_{n+1}</math> from continuous space to discrete space, we see that
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| :<math>
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| y_{n+1} = y(x_n+h) = y(x_n) + hy'(x_n) + \frac{h^2}{2!}y''(x_n) + \frac{h^3}{3!}y'''(x_n) + \frac{h^4}{4!}y''''(x_n) + \frac{h^5}{5!}y'''''(x_n) + \mathcal{O} (h^6)
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| </math>
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| :<math>
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| y_{n-1} = y(x_n-h) = y(x_n) - hy'(x_n) + \frac{h^2}{2!}y''(x_n) - \frac{h^3}{3!}y'''(x_n) + \frac{h^4}{4!}y''''(x_n) - \frac{h^5}{5!}y'''''(x_n) + \mathcal{O} (h^6)
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| </math>
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| The sum of those two equations gives
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| :<math> | |
| y_{n-1} + y_{n+1} = 2y_n + {h^2}y''_n + \frac{h^4}{12}y''''_n + \mathcal{O} (h^6)
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| </math>
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| We solve this equation for <math>y''_n</math> and replace it by the expression <math>y''_n = -f_n y_n</math> which we get from the defining differential equation.
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| :<math>
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| h^2 f_n y_n = 2y_n-y_{n-1} - y_{n+1} + \frac{h^4}{12}y''''_n + \mathcal{O} (h^6)
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| </math>
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| We take the second derivative of our defining differential equation and get
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| :<math>
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| y''''(x) = - \frac{d^2}{d x^2} \left[ f(x) y(x) \right]
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| </math>
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| We replace the second derivative <math>\frac{d^2}{d x^2}</math> with the second order [[difference quotient]] and insert this into our equation for <math> f_n y_n</math> (note that we take the mixed forward and backward finite difference, '''not''' the double forward difference or the double backward difference)
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| :<math>
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| h^2 f_n y_n = 2y_n-y_{n-1} - y_{n+1} - \frac{h^4}{12} \frac{f_{n-1} y_{n-1} -2 f_{n} y_{n} + f_{n+1} y_{n+1}}{h^2} + \mathcal{O} (h^6)
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| </math>
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| We solve for <math>y_{n+1}</math> to get
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| :<math>
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| y_{n+1} = \frac {\left( 2-\frac{5 h^2}{6} f_n \right) y_n - \left( 1+\frac{h^2}{12}f_{n-1} \right)y_{n-1}}{1+\frac{h^2}{12}f_{n+1}} + \mathcal{O} (h^6).
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| </math> | |
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| This yields Numerov's method if we ignore the term of order <math>h^6</math>. It follows that the order of convergence (assuming stability) is 4.
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| == References ==
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| * {{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}. <br> This book includes the following references:
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| * {{Citation | last1=Numerov | first1=Boris Vasil'evich | author1-link=Boris Vasil'evich Numerov | title=A method of extrapolation of perturbations | year=1924 | journal=[[Monthly Notices of the Royal Astronomical Society]] | volume=84 | pages=592–601|bibcode = 1924MNRAS..84..592N }}.
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| * {{Citation | last1=Numerov | first1=Boris Vasil'evich | author1-link=Boris Vasil'evich Numerov | title=Note on the numerical integration of d<sup>2</sup>''x''/d''t''<sup>2</sup> = ''f''(''x'',''t'') | year=1927 | journal=[[Astronomische Nachrichten]] | volume=230 | pages=359–364|bibcode = 1927AN....230..359N }}.
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| ==External links==
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| *[http://www.jkrieger.de/download/computerphysik.pdf Lecture notes: Computerphysik und Numerik] - by Jan Krieger
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| *[http://magnet.atp.tuwien.ac.at/scholz/projects/numerov/node2.html#SECTION00020000000000000000 Lecture notes of Werner Scholz] - At Vienna University of Technology
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| *[http://www.stellarcom.org/aw/Physik/cp/node48.html Lecture notes of Alexander Wagner]
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| [[Category:Numerical differential equations]]
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