1. Formula-1 & Example-1
Formula
2. Second order R-K method
Method-1 :
`k_1=hf(x_0,y_0,z_0)`
`l_1=hg(x_0,y_0,z_0)`
`k_2=hf(x_0+h,y_0+k_1,z_0+l_1)`
`l_2=hg(x_0+h,y_0+k_1,z_0+l_1)`
`y_1=y_0+(k_1+k_2)/2`
Method-2 :
`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)`
`y_1=y_0+k_2`
|
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 2 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)`
Method-1 : Using formula `k_2=hf(x_0+h,y_0+k_1,z_0+l_1)`
Second 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,y_0+k_1,z_0+l_1)=(0.1)*f(0.1,1,0.1)=(0.1)*(0.1)=0.01`
`l_2=hg(x_0+h,y_0+k_1,z_0+l_1)=(0.1)*g(0.1,1,0.1)=(0.1)*(1.199)=0.1199`
`y_1=y_0+(k_1+k_2)/2=1+0.005=1.005`
`:.y(0.1)=1.005`
`:.y(0.1)=1.005`
Method-2 : Using formula `k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`
Second 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`
`y_1=y_0+k_2=1+0.005=1.005`
`:.y(0.1)=1.005`
`:.y(0.1)=1.005`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
|
