1. Formula-1 & Example-1
Formula
4. Forth order R-K method
`k_1=hf(x_0,y_0,z_0)`
`l_1=hg(x_0,y_0,z_0)`
`k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`
`l_2=hg(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`
`k_3=hf(x_0+h/2,y_0+k_2/2,z_0+l_2/2)`
`l_3=hg(x_0+h/2,y_0+k_2/2,z_0+l_2/2)`
`k_4=hf(x_0+h,y_0+k_3,z_0+l_3)`
`l_4=hg(x_0+h,y_0+k_3,z_0+l_3)`
`y_1=y_0+1/6(k_1+2k_2+2k_3+k_4)`
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Examples
1. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Runge-Kutta 4 method (2nd order derivative)
Solution:
Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, h=0.1, y(0.1)=?`
put `(dy)/(dx)=z` and differentiate w.r.t. x, we obtain `(d^2y)/(dx^2)=(dz)/(dx)`
We have system of equations
`(dy)/(dx)=z=f(x,y,z)`
`(dz)/(dx)=1+2xy-x^2z=g(x,y,z)`
Forth order R-K method for second order differential equation
`k_1=hf(x_0,y_0,z_0)=(0.1)*f(0,1,0)=(0.1)*(0)=0`
`l_1=hg(x_0,y_0,z_0)=(0.1)*g(0,1,0)=(0.1)*(1)=0.1`
`k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)=(0.1)*f(0.05,1,0.05)=(0.1)*(0.05)=0.005`
`l_2=hg(x_0+h/2,y_0+k_1/2,z_0+l_1/2)=(0.1)*g(0.05,1,0.05)=(0.1)*(1.09988)=0.10999`
`k_3=hf(x_0+h/2,y_0+k_2/2,z_0+l_2/2)=(0.1)*f(0.05,1.0025,0.05499)=(0.1)*(0.05499)=0.0055`
`l_3=hg(x_0+h/2,y_0+k_2/2,z_0+l_2/2)=(0.1)*g(0.05,1.0025,0.05499)=(0.1)*(1.10011)=0.11001`
`k_4=hf(x_0+h,y_0+k_3,z_0+l_3)=(0.1)*f(0.1,1.0055,0.11001)=(0.1)*(0.11001)=0.011`
`l_4=hg(x_0+h,y_0+k_3,z_0+l_3)=(0.1)*g(0.1,1.0055,0.11001)=(0.1)*(1.2)=0.12`
Now,
`y_1=y_0+1/6(k_1+2k_2+2k_3+k_4)`
`y_1=1+1/6[0+2(0.005)+2(0.0055)+(0.011)]`
`y_1=1.00533`
`:.y(0.1)=1.00533`
`:.y(0.1)=1.00533`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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