2. INTRODUCTION
• A number of numerical methods are available
for the solution of first order differential
equation of form:
• dy/ dx = f(x ,y)
• These methods yield solution either as power
series or in x form which the values of y can be
found by direct substitution, or a set of values
of x and y.
3. NUMERICAL METHODS
• The numerical methods for ODE include:
Picard
Taylor series
Euler series
Runge-kutta methods
Milne
Adams-Bashforth
4. RUNGE-KUTTA METHOD
• For the first time, the methods were
presented by Runge and Kutta two German
mathematicians.
• These methods were very accurate and
efficient and instead direct calculation of
higher derivatives only function used for
different values.
5. WORKING RULE
• Working rule to solve the differential equation
by runge’s method is given by:
k1 = hf (x0,y0)
k2 = hf (x0 + 0.5 h, y0 +0.5 k1)
k′ = hf (x0 +h, y0 + k1)
k3 = hf (x0 +h, y0 + k′)
k = 1/6 (k1 +4 k2 + k3 )
7. RUNGE KUTTA
The range kutta is classified as :
first order runge kutta
dy/dx = f (x ,y)
second order runge kutta
third order runge kutta
fourth order runge kutta
9. APPLICATIONS
• If a model rocket is prepared this method can
be used for successive iterations of the
differential equations for lift off the rocket.
• Runge kutta method is used to know the
ballistic missile trajectory of a rocket.
• In the study of self propelled missiles.
10. CONCLUSION
• Rocket trajectories are optimized to achieve
target by either minimum time or control
surfaces or fuel.
• First we calculate using the steepest descent
method and then it is compared to runge
kutta method and results say that the runge
kutta method is more efficient and accurate .
• But takes time compared to others .
