Application of Residue Theorem to evaluate real integrations.pptx
Runge Kutta Method
1. G. H. Patel College of
Engineering & Technology
Subject : Numerical and Statistical Method for Computer Engineering
(2140706)
Topic : Runge Kutta Methods
Guided by : Prof. Tejas Jani
Made by : Vashi Bhavik (160110116061)
Shivam Zala (160110116062)
Aakash Godhani (160110116063)
Harshal Dankhara (160110116064)
Shreya Patel (160110116065)
3. Runge Kutta Method : Introduction
Developed by two German mathematicians Runge and kutta .
It is also called R-K method.
Runge-kutta method distinguished by their order
4. First Order Runge –Kutta Method
Considering the differential equation
dy/dx=f(x,y)
With the initial condition y(x0)=y0
By the Euler’s method
yn+1=yn + h*f(xn,yn)
yn+1=yn + h*y’n + (h2/2!)y’’n + ... (by Taylor’s series)
5. Second Order Runge –Kutta Method
Yn+1=yn + k
Where k=(1/2)(k1+k2)
K1=h*f(xn , yn)
K2=h*f(xn + h , yn + k1)
6. Third Order Runge –Kutta Method
Yn+1=yn + k
Where k=(1/6)(k1+4k2+k3)
K1=h*f(xn , yn)
K2=h*f(xn + (h/2) , yn + (k1/2) )
K3=h*f(xn + h , yn + 2k2 – k1 )
7. Fourth Order Runge –Kutta Method
Yn+1=yn + k
Where k=(1/6)(k1+2k2+2k3+k4)
K1=h*f(xn , yn)
K2=h*f(xn + (h/2) , yn + (k1/2) )
K3=h*f(xn + (h/2) , yn + (k2/2) )
K4=h*f(xn + h, yn + k3 )
8. Example
Solve the differential equation dy/dx=x+y , with the fourth order Runge-Kutta
method ,where y(0)=1 with x=0.2 with h=0.1.
Given data y(0)=1 and h=0.1
dy/dx = x+y
f(x,y)=dy/dx=x+y