how to solve problem of RK order 4?
Answers
In the last section, Euler's Method gave us one possible approach for solving differential equations numerically.
The problem with Euler's Method is that you have to use a small interval size to get a reasonably accurate result. That is, it's not very efficient.
The Runge-Kutta Method produces a better result in fewer steps.
Mechanics
Springs and dampeners on cars (This spring applet uses RK4.)
Biology
Predator-prey models
Fisheries collapses
Drug delivery
Epidemic prediction
Physics
Climate change models
Ozone protection
Aviation
On-board computers
Aerodynamics
\displaystyle{y}{\left({x}+{h}\right)}y(x+h) \displaystyle={y}{\left({x}\right)}+=y(x)+ \displaystyle\frac{1}{{6}}{\left({F}_{{1}}+{2}{F}_{{2}}+{2}{F}_{{3}}+{F}_{{4}}\right)}
6
1
(F
1
+2F
2
+2F
3
+F
4
)
where
\displaystyle{F}_{{1}}={h} f{{\left({x},{y}\right)}}F
1
=hf(x,y)
\displaystyle{F}_{{2}}={h} f{{\left({x}+\frac{h}{{2}},{y}+\frac{{F}_{{1}}}{{2}}\right)}}F
2
=hf(x+
2
h
,y+
2
F
1
)
\displaystyle{F}_{{3}}={h} f{{\left({x}+\frac{h}{{2}},{y}+\frac{{F}_{{2}}}{{2}}\right)}}F
3
=hf(x+
2
h
,y+
2
F
2
)
\displaystyle{F}_{{4}}={h} f{{\left({x}+{h},{y}+{F}_{{3}}\right)}}F
4
=hf(x+h,y+F
3