# 381 - A15 - Differential Equations - Taylor's Methods

## 252 days ago by Professor381

Solve numerically the initial value problem:

$y' = f(t,y)$

$y(a) = y_0$

So, below we will need be given $f(t,y)$ and the starting point $(a,y_0)$.

var('t,y') example = 2 if example == 1: f = 2*y t0 = 0 y0 = 2 # for higher order methods....the derivative of f with respect to t treating any y terms with implicit differentiation and noting that y' = f fp = 2*f # actually fp = 2y' = 2f = 4y fpp = 4*f # actually fpp = 4y' = 4f = 8y exact = y0*exp(2*t) end if example == 2: f = t - y t0 = 0 y0 = 2 fp = 1 - f # actually fp = 1 - y' = 1 - f = 1 - (t-y) = 1 - t + y fpp = -1 + f # actually fpp = -1 + y' exact = ((t-1)*exp(t)+3)*exp(-t) end # set up discrete time steps for each method below steps = 8 delt = 1/8 ts = [t0] for k in range(steps+1): # create uniform steps to make things easy. Not using any adjustable step size ts.append(ts[k] + delt) end
# var('t') y = function('y')(t) exact = desolve(diff(y,t) == t-y, y, ics=[t0,y0]) show(exact) # Need to make this automatic using f and not having to explicitly enter f's formula # exact = desolve(diff(z,x) == f(t=t,y=z), z, ics=[t0,y0]) # show(exact)
# Euler's Method doing 3 explicit steps t1 = t0 + delt y1 = y0 + f(t=t0,y=y0)*delt pretty_print((t1,y1)) t2 = t1 + delt y2 = y1 + f(t=t1,y=y1)*delt pretty_print((t2,y2)) t3 = t2 + delt y3 = y2 + f(t=t2,y=y2)*delt pretty_print((t3,y3))
 Traceback (click to the left of this block for traceback) ... NameError: name 't0' is not defined Traceback (most recent call last): pretty_print((t2,y2)) File "", line 1, in File "/tmp/tmpyBlgKa/___code___.py", line 3, in t1 = t0 + delt NameError: name 't0' is not defined
# Euler's method using a loop. ys = [y0] euler_pts = [(t0,y0)] for k in range(steps): tempy = ys[k] + f(t=ts[k],y=ys[k])*delt ys.append(tempy) euler_pts.append((ts[k+1],tempy)) end G = points(euler_pts,color='magenta') G += plot(exact,(t,t0,t0+steps*delt),color='green') S = plot(spline(euler_pts),(t,t0,t0+steps*delt),color='blue') show(G+S)
# Second order Taylors method ys = [y0] tay2_pts = [(t0,y0)] for k in range(steps): tempy = ys[k] + f(t=ts[k],y=ys[k])*delt + fp(t=ts[k],y=ys[k])*delt^2/2 ys.append(tempy) tay2_pts.append((ts[k+1],tempy)) end G = points(tay2_pts,color='magenta') G += plot(exact,(t,t0,t0+steps*delt),color='green') S = plot(spline(tay2_pts),(t,t0,t0+steps*delt),color='blue') show(G+S)
# Third order Taylors method ys = [y0] tay2_pts = [(t0,y0)] for k in range(steps): tempy = ys[k] + f(t=ts[k],y=ys[k])*delt + fp(t=ts[k],y=ys[k])*delt^2/2 + fpp(t=ts[k],y=ys[k])*delt^3/6 ys.append(tempy) tay2_pts.append((ts[k+1],tempy)) end G = points(tay2_pts,color='magenta') G += plot(exact,(t,t0,t0+steps*delt),color='green') S = plot(spline(tay2_pts),(t,t0,t0+steps*delt),color='blue') show(G+S)