Finite difference

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A different to is a mathematical expression of the form f ( x Ã, b ) -Ã, < i> f ( x Ã, a ) . If the finite difference is shared by b Ã, - a , one gets the result for the difference. The derivative approximation by difference to plays a central role in the difference method to for numerical solutions of differential equations, particularly the problem of boundary values.

A particular repetition relation can be written as a difference equation by changing the iteration notation with a finite difference.

Today, the term "difference to" is often taken as a synonym for estimating the finite difference of derivatives, especially in the context of numerical methods. The approximation of the finite difference is the finite difference with the difference in the terminology used above.

The limited difference has also been the subject of research as an abstract stand-alone mathematical object, for example in works by George Boole (1860), LM Milne-Thomson (1933), and KÃÆ'¡roly Jordan (1939), tracing its origins back to false one Jost BÃÆ'¼rgi (ca. 1592) algorithm and the other including Isaac Newton. In this viewpoint, the formal calculus of finite difference is an alternative to the infinitesimals calculus.

Video Finite difference

Perbedaan maju, mundur, dan pusat

Three forms are generally considered: differences forward, back, and center.

A perbedaan maju adalah ekspresi bentuk

${\ displaystyle \ Delta _ {h} [f] (x) = f (x h) -f (x).}  ÂÂ$

Tergantung pada aplikasi, jarak h mungkin variabel atau konstan. Ketika dihilangkan, h dianggap sebagai 1: ${\ displaystyle \ Delta [f] (x) = \ Delta _ {1} [f] (x)}  ÂÂ$ .

A perbedaan terbalik menggunakan nilai fungsi pada x dan x Â- h , sebagai ganti nilai pada < i> x  h dan x :

${\ displaystyle \ nabla _ {h} [f] (x) = f (x) -f (x-h).}  ÂÂ$

Akhirnya, perbedaan pusat diberikan oleh

${\ displaystyle \ delta _ {h} [f] (x) = f (x {\ tfrac {1} {2}} h) -f (x - {\ tfrac {1} {2}} h) ~.}  ÂÂ$

Maps Finite difference

Hubungan dengan turunan

Turunan dari fungsi f pada suatu titik x ditentukan oleh batas.

${\ displaystyle f '(x) = \ lim _ {h \ to 0} {\ frac {f (x h) -f (x)} {h}}.}  ÂÂ$

Jika h memiliki nilai tetap (bukan nol) daripada mendekati nol, maka sisi kanan persamaan di atas akan ditulis

${\ displaystyle {\ frac {f (x h) -f (x)} {h}} = {\ frac {\ Delta _ {h} [f] ( x)} {h}}.}  ÂÂ$

Oleh karena itu, perbedaan ke depan dibagi dengan h mendekati turunan ketika h kecil. Kesalahan dalam pendekatan ini dapat berasal dari teorema Taylor. Dengan asumsi bahwa f dapat terdiferensiasi, kita memiliki

${\ displaystyle {\ frac {\ Delta _ {h} [f] (x)} {h}} - f '(x) = O (h) \ ke 0 \ quad {\ text {as}} (h \ to 0).}  ÂÂ$

Formula yang sama berlaku untuk perbedaan mundur:

${\ displaystyle {\ frac {\ nabla _ {h} [f] (x)} {h}} - f '(x) = O (h) \ ke 0 \ quad {\ text {as}} (h \ to 0).}  ÂÂ$

Namun, perbedaan pusat (juga disebut terpusat) menghasilkan pendekatan yang lebih akurat. Jika f dua kali terdiferensiasi,

${\ displaystyle {\ frac {\ delta _ {h} [f] (x)} {h}} - f '(x) = O (h ^ {2} ).}  ÂÂ$

The main problem with the central difference method, however, is that the oscillating function can produce a zero derivative. If f ( nh ) = 1 for strange n and f nh ) = 2 for n even, then f ' ( nh ) = 0 if calculated with the central difference schema. This is very inconvenient if the domain f is discrete.

The author for whom the finite difference means different estimates to determine the difference of forward/backward/center as quotients given in this section (rather than using the definitions given in the previous section).

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High order difference

In an analogous way, one can obtain approximations of difference to high order derivatives and differential operators. For example, using the above center difference formula for f ' ( x h /2) and f ' x - h /2) and apply the central difference formula for the f' at x , we derive the approximate central difference from the second derivative f :

Urutan pusat kedua

${\ displaystyle f '' (x) \ kira {\ frac {\ delta _ {h} ^ {2} [f] (x)} {h ^ {2} }} = {\ frac {f (x h) -2f (x) f (xh)} {h ^ {2}}}.}  ÂÂ$

Similarly we can apply other recursive differencing formulas.

Urut urutan ke-2

${\ displaystyle f '' (x) \ kira {\ frac {\ Delta _ {h} ^ {2} [f] (x)} {h ^ {2} }} = {\ frac {f (x 2h) -2f (x h) f (x)} {h ^ {2}}}.}  ÂÂ$

Urutan ke-2 mundur

${\ displaystyle f '' (x) \ kira {\ frac {\ Delta _ {h} ^ {2} [f] (x)} {h ^ {2} }} = {\ frac {f (x) -2f (xh) f (x-2h)} {h ^ {2}}}.}  ÂÂ$

In general, the difference n -th in order to advance, backward, and center is provided by, respectively,

Teruskan

${\ displaystyle \ Delta _ {h} ^ {n} [f] (x) = \ jumlah _ {i = 0} ^ {n} (- 1) ^ { i} {\ binom {n} {i}} f (x (ni) h),}  ÂÂ$

atau untuk h = 1,

${\ displaystyle \ Delta ^ {n} [f] (x) = \ jumlah _ {k = 0} ^ {n} {\ binom {n} {k}} (-1) ^ {nk} f (x k)}  ÂÂ$

Mundur

${\ displaystyle \ nabla _ {h} ^ {n} [f] (x) = \ jumlah _ {i = 0} ^ {n} (- 1) ^ { i} {\ binom {n} {i}} f (x-ih),}  ÂÂ$

Pusat

${\ displaystyle \ delta _ {h} ^ {n} [f] (x) = \ jumlah _ {i = 0} ^ {n} (- 1) ^ { i} {\ binom {n} {i}} f \ kiri (x \ kiri ({\ frac {n} {2}} - i \ right) h \ right).}  ÂÂ$

Persamaan ini menggunakan koefisien binomial setelah tanda penjumlahan ditampilkan sebagai ${\ displaystyle \ {\ binom {n} {i}}}  ÂÂ$ . Setiap baris segitiga Pascal menyediakan koefisien untuk setiap nilai i .

Perhatikan bahwa perbedaan pusat akan, untuk aneh n , memiliki h dikalikan dengan non-bilangan bulat. Ini sering menjadi masalah karena ini mengubah interval diskretisasi. Masalahnya dapat diperbaiki dengan mengambil rata-rata ${\ displaystyle \ delta ^ {n} [f] (x-h/2)}  ÂÂ$ dan $            {\displaystyle         {}^{}}$

Source of the article : Wikipedia