Question

give a simple odes system with a time-dependent coefficients

Answer 1

time-dependent ordinary differential equations:-

Let x(t) be the position of a particle moving on the x-axis at time t.

1st order ODE
dx/dt = f(x) : velocity is a function of position
x(0) : initial position
The problem is to find the position x(t) for t > 0.
ex
1.
dx/dt = x, x(0) = 1 ) x(t) = et

2.
dx/dt = x2 , x(0)= 1 x(t) = 1/1-t

3.
dx/dt = sin x, x(0)=1 x(t)=?

The simplest numerical method is Euler’s method.
choose Δt: time step

define wn : numerical solution at time tn = ndefine wn:nΔt

wn+1-wn/Δt= f(Wn)

Wn+1=wn + Δtf(Wn)

Given W0, W1, W2.......

questions : accuracy , stability , efficiency

2nd order ODE

d2x/dt2 = f(x) : acceleration is a function of position (Newton’s equation)
x(0) , x0
(0) : initial position , velocity

(wn+1 - 2wn+wn-1)/(Δt)2 = f(wn)

given w0 and w1, compute w2, w3,...