euler
t
t =
Columns 1 through 3
0 0.300000000000000 0.600000000000000
Columns 4 through 6
0.900000000000000 1.200000000000000 1.500000000000000
Columns 7 through 9
1.800000000000000 2.100000000000000 2.400000000000000
Columns 10 through 11
2.700000000000000 3.000000000000000
euler
{Error: File: ode1.m Line: 5 Column: 4
This statement is not inside any function.
(It follows the END that terminates the definition of the function
"ode1".)
Error in euler (line 9)
u(i+1) = u(i) + h * ode1( u(i), t(i) );
}
euler
plot(t,u)
plot(t,u)
euler
plot(t,u)
help ode45
ode45 Solve non-stiff differential equations, medium order method.
[TOUT,YOUT] = ode45(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates
the system of differential equations y' = f(t,y) from time T0 to TFINAL
with initial conditions Y0. ODEFUN is a function handle. For a scalar T
and a vector Y, ODEFUN(T,Y) must return a column vector corresponding
to f(t,y). Each row in the solution array YOUT corresponds to a time
returned in the column vector TOUT. To obtain solutions at specific
times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN =
[T0 T1 ... TFINAL].
[TOUT,YOUT] = ode45(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
integration properties replaced by values in OPTIONS, an argument created
with the ODESET function. See ODESET for details. Commonly used options
are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
of absolute error tolerances 'AbsTol' (all components 1e-6 by default).
If certain components of the solution must be non-negative, use
ODESET to set the 'NonNegative' property to the indices of these
components.
ode45 can solve problems M(t,y)*y' = f(t,y) with mass matrix M that is
nonsingular. Use ODESET to set the 'Mass' property to a function handle
MASS if MASS(T,Y) returns the value of the mass matrix. If the mass matrix
is constant, the matrix can be used as the value of the 'Mass' option. If
the mass matrix does not depend on the state variable Y and the function
MASS is to be called with one input argument T, set 'MStateDependence' to
'none'. ODE15S and ODE23T can solve problems with singular mass matrices.
[TOUT,YOUT,TE,YE,IE] = ode45(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events'
property in OPTIONS set to a function handle EVENTS, solves as above
while also finding where functions of (T,Y), called event functions,
are zero. For each function you specify whether the integration is
to terminate at a zero and whether the direction of the zero crossing
matters. These are the three column vectors returned by EVENTS:
[VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function:
VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration
is to terminate at a zero of this event function and 0 otherwise.
DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only
zeros where the event function is increasing, and -1 if only zeros where
the event function is decreasing. Output TE is a column vector of times
at which events occur. Rows of YE are the corresponding solutions, and
indices in vector IE specify which event occurred.
SOL = ode45(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
used with DEVAL to evaluate the solution or its first derivative at
any point between T0 and TFINAL. The steps chosen by ode45 are returned
in a row vector SOL.x. For each I, the column SOL.y(:,I) contains
the solution at SOL.x(I). If events were detected, SOL.xe is a row vector
of points at which events occurred. Columns of SOL.ye are the corresponding
solutions, and indices in vector SOL.ie specify which event occurred.
Example
[t,y]=ode45(@vdp1,[0 20],[2 0]);
plot(t,y(:,1));
solves the system y' = vdp1(t,y), using the default relative error
tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
component, and plots the first component of the solution.
Class support for inputs TSPAN, Y0, and the result of ODEFUN(T,Y):
float: double, single
See also ode23, ode113, ode15s, ode23s, ode23t, ode23tb, ode15i,
odeset, odeplot, odephas2, odephas3, odeprint, deval,
odeexamples, rigidode, ballode, orbitode, function_handle.
Reference page in Help browser
doc ode45
ode45( @ode1, [0 3], 1 )
ode45( @ode1, [0 3], 1 )
hold on
plot(t,u,'+r')
feuler( @ode1, 1, 0, 3, 10 )
ans =
Columns 1 through 3
1.000000000000000 0.700000000000000 -0.618443691672424
Columns 4 through 6
-0.549083037934104 0.211793004434057 0.274492599267961
Columns 7 through 9
-0.121799308067884 -0.187443606786559 0.030376076322544
Columns 10 through 11
0.094278448793071 -0.015033202622791
[fu, ft ] = feuler( @ode1, 1, 0, 3, 1000 );
plot( ft, fu, 'b')
close all
plot( ft, fu, 'b')
hold
Current plot held
hold on
ode45( @ode1, [0 3], 1 )
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 10 )
fu =
Columns 1 through 3
2.718281828459046 3.533766376996759 4.539773443530052
Columns 4 through 6
5.774499187659123 7.282994814895872 9.118241839096839
Columns 7 through 9
11.342387458469149 14.028162191422943 17.260505010152450
Columns 10 through 11
21.138424678639183 25.777129906895595
ft =
Columns 1 through 3
1.000000000000000 1.100000000000000 1.200000000000000
Columns 4 through 6
1.300000000000000 1.400000000000000 1.500000000000000
Columns 7 through 9
1.600000000000000 1.700000000000000 1.800000000000000
Columns 10 through 11
1.900000000000000 2.000000000000000
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 1000 );
plot( ft, fu, 'r')
hold on
ode45( @ode2, [1 2], exp(1) )
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 100 );
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 100 );
ode45( @ode2, [1 2], exp(1) )
hold on
plot( ft, fu, 'r')
close all
ode45( @ode2, [1 4], exp(1) )
[fu, ft ] = feuler( @ode2, exp(1), 1, 4, 100 );
hold on
plot( ft, fu, 'r')
[fu, ft ] = feuler( @ode2, exp(1), 1, 4, 1000 );
plot( ft, fu, 'g')
target = -0.037822634055742
target =
-0.037822634055742
[fu, ft ] = feuler( @ode1, 1, 0, 3, 10 );
abs( target - fu(end) )/abs(target)
ans =
0.602534223273937
[fu, ft ] = feuler( @ode1, 1, 0, 3, 100 );
abs( target - fu(end) )/abs(target)
ans =
0.001946729465011
[fu, ft ] = feuler( @ode1, 1, 0, 3, 1000 );
abs( target - fu(end) )/abs(target)
ans =
7.272437623209290e-04
[fu, ft ] = feuler( @ode1, 1, 0, 3, 10000 );
abs( target - fu(end) )/abs(target)
ans =
7.799831062937499e-05
[fu, ft ] = feuler( @ode1, 1, 0, 3, 100000 );
abs( target - fu(end) )/abs(target)
ans =
7.852521557373846e-06
[fu, ft ] = feuler( @ode1, 1, 0, 3, 10000000 );
abs( target - fu(end) )/abs(target)
ans =
7.858365598684429e-08
diary off