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306 Formula

The document outlines various numerical methods for solving differential equations, including Householder's Method, Euler Method, Modified Euler Method, Runge Kutta methods (both second and fourth order), and Adams-Bashforth and Adams-Moulton methods. Each method is presented with its respective formulas for calculating the next value based on previous values and function evaluations. These methods are essential for approximating solutions to ordinary differential equations in numerical analysis.

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8 views3 pages

306 Formula

The document outlines various numerical methods for solving differential equations, including Householder's Method, Euler Method, Modified Euler Method, Runge Kutta methods (both second and fourth order), and Adams-Bashforth and Adams-Moulton methods. Each method is presented with its respective formulas for calculating the next value based on previous values and function evaluations. These methods are essential for approximating solutions to ordinary differential equations in numerical analysis.

Uploaded by

abdullahshadid18
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Householder’s Method:

α= -sgn(a(k)k+1, k ){∑𝑛𝑗=𝑘+1(𝑎(k)jk )2 }1/2


1 1
r=(2α2 -2α a(k)k+1, k )1/2

w1(k) =w2(k) =…=0


wk+1(k) =(a(k)k+1,k -α)/2r
wj(k) =a(k)jk /2r, for j=k+2,k+3,…
P(k) =I-2w(k) (w(k) )t
A(k+1) =P(k) A(k) P(k)
Then, α= -sgn(a21 )(∑𝑛𝑗=2 𝑎2j1 )1/2 α= -sgn(a32 )(∑𝑛𝑗=3 𝑎 2j2 )1/2
1 1 1 1
r=(2α2 -2α a21)1/2 r=(2α2 -2α a32)1/2

w1 =0 w1= w2 =0
w2 =(a21 -α)/2r w3 =(a32 -α)/2r
wj =aj1 /2r, for j=3,4,…n wj =aj2 /2r, for j=4,…n
P(1) =I-2w(1) (w(1) )t P(2) =I-2w(2) (w(2) )t
A(2) =P(1) A P(1) A(3) =P(2) A(2) P(2)

Euler Method:
w 0= α , h=(b-a)/n
wi+1=wi+hf(ti,wi),for t=0,1,…,n-1

Modified Euler Method :


w 0= α , h=(b-a)/n
ℎ
wi+1=wi+2 [f(ti,wi)+ f(ti+1,wi+ hf(ti,wi))],for t=0,1,…,n-1

Runge Kutta Order Four:


w0=α , h=(b-a)/n
ℎ k1 ℎ k1
k1= hf(ti,wi) k2=hf(ti+2, wi+ 2 ) k3=hf(ti+2, wi+ 2 ) k4= hf(ti+1 ,wi+k3)
1
wi+1= wi+6 ( k1+2k2+2k3+k4)
Runge Kutta Order Two:
w0=α , h=(b-a)/n
k1= hf(ti,wi) k2=hf(ti+ℎ, wi+k1)
1
wi+1= wi+2 ( k1+k2)

Adams Bashforth Two Step:


w0=α ,w1=α1
ℎ
wi+1=wi+2 [3f(ti,wi)- f(ti-1,wi-1)] , for i=1,2,…,n-1

Adams Bashforth Three Step:


w0=α ,w1=α1 , w2=α2
ℎ
wi+1=wi+12 [23f(ti,wi) - 16f(ti-1,wi-1)+5f(ti-2,wi-2)] , for i=2,…,n-1

Adams Bashforth Four Step:


w0=α ,w1=α1 , w2=α2 , w3=α3
ℎ
wi+1=wi+24 [55f(ti,wi) - 59f(ti-1,wi-1)+37f(ti-2,wi-2) -9 f(ti-3,wi-3)] , for i=3,…,n-1

Adams Bashforth Five Step:


w0=α ,w1=α1 , w2=α2 , w3=α3 , w4=α4
ℎ
wi+1=wi+ [1901f(ti,wi) - 2774f(ti-1,wi-1)+2616f(ti-2,wi-2) -1274 f(ti-3,wi-3) +251 f(ti-4,wi-4)]
720

for i=4,…,n-1

Adams Moulton Two Step:


w0=α ,w1=α1
ℎ
wi+1=wi+12 [5f(ti+1,wi+1) +8f(ti,wi)- f(ti-1,wi-1)] , for i=1,2,…,n-1
Adams Moulton Three Step:
w0=α ,w1=α1 , w2=α2
ℎ
wi+1=wi+24 [9f(ti+1,wi+1) +19f(ti,wi) - 5f(ti-1,wi-1)+f(ti-2,wi-2)] , for i=2,…,n-1

Adams Moulton Four Step:


w0=α ,w1=α1 , w2=α2 , w3=α3
ℎ
wi+1=wi+720 [251f(ti+1,wi+1)+646f(ti,wi) - 264f(ti-1,wi-1)+106f(ti-2,wi-2) - 19f(ti-3,wi-3)]

for i=3,…,n-1

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