pkg load symbolic

syms y(t) t ;
f=@(t,y) sin(t)-y ;
egyenlet=diff(y(t),t)==sin(t)-y ;
sol=dsolve(egyenlet,y(0)==1) ;
fsol=matlabFunction(rhs(sol)) ;
tb=0:0.5:3;
yb=-1:0.5:5;
[T,Y]=meshgrid(tb,yb);
dY=f(T,Y);
dT=ones(size(dY));
quiver(T,Y,dT,dY);
hold on;
xx=linspace(tb(1),tb(end));
plot(xx,fsol(xx));
axis([tb(1),tb(end),yb(1),yb(end)])

Symbolic pkg v2.8.0: Python communication link active, SymPy v1.5.1.
warning: worked around octave bug #42735
warning: called from
    minus at line 42 column 5

fixpont-iteráció¶

y^{'} = f (t, y) y (a) = b

$y'=f(t,y)\ \ y(a)=b$

y (t) = b + \int_{a}^{t} f (s, y (s)) d s

$y(t)=b+\int_a^t f(s,y(s))ds$

y \to T (y), ahol T (y) (t) = b + \int_{a}^{t} f (s, y (s)) d s

$y \to T(y), \ \text{ahol }\ T(y)(t)=b+\int_a^t f(s,y(s))ds$

y_{n} \to y_{n + 1} = T (y_{n}), ahol T (y_{n}) (t) = b + \int_{a}^{t} f (s, y_{n} (s)) d s

$y_{n} \to y_{n+1}=T(y_n), \ \text{ahol }\ T(y_n)(t)=b+\int_a^t f(s,y_n(s))ds$

f (t, y) = \sin (t) - y, y_{0} (t) = 1

$f(t,y)=\sin(t)-y,\ \ y_0(t)=1$

y_{1} (t) = 1 + \int_{0}^{t} (\sin (s) - 1) d s = 1 - \cos (t) + 1 - t = 2 - t - \cos (t)

$y_1(t)=1+\int_0^t (\sin(s)-1) ds= 1-\cos(t)+1-t=2-t-\cos(t)$

y_{2} (t) = 1 + \int_{0}^{t} (\sin (s) - 2 + s + \cos (s)) d s = 2 - 2 t + \frac{t^{2}}{2} - \cos (t) + \sin (t)

$y_2(t)=1+\int_0^t (\sin(s)-2+s+\cos(s)) ds= 2-2t+\frac{t^2}{2}-\cos(t)+\sin(t)$

y_{3} (t) = 1 + \int_{0}^{t} (\sin (s) - 2 + 2 s - \frac{s^{2}}{2} + \cos (s) - \sin (s)) d s = 1 - 2 t + t^{2} - \frac{t^{3}}{6} + \sin (t)

$y_3(t)=1+\int_0^t (\sin(s)-2+2s-\frac{s^2}{2}+\cos(s)-\sin(s)) ds= 1-2t+t^2-\frac{t^3}{6}+\sin(t)$

y_{4} (t) = 1 + \int_{0}^{t} (\sin (s) - 1 + 2 s - s^{2} + \frac{s^{3}}{6} - \sin (s)) d s = 1 - t + t^{2} - \frac{t^{3}}{3} + \frac{t^{4}}{24}

$y_4(t)=1+\int_0^t (\sin(s)-1+2s-s^2+\frac{s^3}{6}-\sin(s)) ds= 1-t+t^2-\frac{t^3}{3}+\frac{t^4}{24}$

quiver(T,Y,dT,dY);
hold on;
xx=linspace(tb(1),tb(end));
plot(xx,fsol(xx),'.-');
uj0=@(t) ones(size(t));
plot(xx,uj0(xx),';0;');
uj1=@(t) 2-cos(t)-sin(t);
plot(xx,uj1(xx),';1;');
uj2=@(t) 2-2*t+1/2*t.^2-cos(t)+sin(t);
plot(xx,uj2(xx),';2;');
uj3=@(t) 1-2*t+t.^2-1/6*t.^3+sin(t);
plot(xx,uj3(xx),';3;');
uj4=@(t) 1-t+t.^2-1/3*t.^3+1/24*t.^4;
plot(xx,uj4(xx),';4;');

axis([tb(1),tb(end),yb(1),yb(end)])