/
desolvers.py
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/
desolvers.py
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r"""
Solving ordinary differential equations
This file contains functions useful for solving differential equations
which occur commonly in a 1st semester differential equations
course. For another numerical solver see the :meth:`ode_solver` function
and the optional package Octave.
Solutions from the Maxima package can contain the three constants
``_C``, ``_K1``, and ``_K2`` where the underscore is used to distinguish
them from symbolic variables that the user might have used. You can
substitute values for them, and make them into accessible usable
symbolic variables, for example with ``var("_C")``.
Commands:
- :func:`desolve` - Compute the "general solution" to a 1st or 2nd order
ODE via Maxima.
- :func:`desolve_laplace` - Solve an ODE using Laplace transforms via
Maxima. Initial conditions are optional.
- :func:`desolve_rk4` - Solve numerically an IVP for one first order
equation, return list of points or plot.
- :func:`desolve_system_rk4` - Solve numerically an IVP for a system of first
order equations, return list of points.
- :func:`desolve_odeint` - Solve numerically a system of first-order ordinary
differential equations using ``odeint`` from `scipy.integrate module.
<https://docs.scipy.org/doc/scipy/reference/integrate.html#module-scipy.integrate>`_
- :func:`desolve_system` - Solve a system of 1st order ODEs of any size using
Maxima. Initial conditions are optional.
- :func:`eulers_method` - Approximate solution to a 1st order DE,
presented as a table.
- :func:`eulers_method_2x2` - Approximate solution to a 1st order system
of DEs, presented as a table.
- :func:`eulers_method_2x2_plot` - Plot the sequence of points obtained
from Euler's method.
The following functions require the optional package ``tides``:
- :func:`desolve_mintides` - Numerical solution of a system of 1st order ODEs via
the Taylor series integrator method implemented in TIDES.
- :func:`desolve_tides_mpfr` - Arbitrary precision Taylor series integrator implemented in TIDES.
AUTHORS:
- David Joyner (3-2006) - Initial version of functions
- Marshall Hampton (7-2007) - Creation of Python module and testing
- Robert Bradshaw (10-2008) - Some interface cleanup.
- Robert Marik (10-2009) - Some bugfixes and enhancements
- Miguel Marco (06-2014) - Tides desolvers
"""
##########################################################################
# Copyright (C) 2006 David Joyner <wdjoyner@gmail.com>, Marshall Hampton,
# Robert Marik <marik@mendelu.cz>
#
# Distributed under the terms of the GNU General Public License (GPL):
#
# https://www.gnu.org/licenses/
##########################################################################
from __future__ import division
import shutil
import os
from sage.interfaces.maxima import Maxima
from sage.plot.all import line
from sage.symbolic.expression import is_SymbolicEquation
from sage.symbolic.ring import is_SymbolicVariable
from sage.calculus.functional import diff
from sage.misc.functional import N
from sage.rings.real_mpfr import RealField
maxima = Maxima()
def fricas_desolve(de, dvar, ics, ivar):
r"""
Solve an ODE using FriCAS.
EXAMPLES::
sage: x = var('x')
sage: y = function('y')(x)
sage: desolve(diff(y,x) + y - 1, y, algorithm="fricas") # optional - fricas
_C0*e^(-x) + 1
sage: desolve(diff(y, x) + y == y^3*sin(x), y, algorithm="fricas") # optional - fricas
-1/5*(2*cos(x)*y(x)^2 + 4*sin(x)*y(x)^2 - 5)*e^(-2*x)/y(x)^2
TESTS::
sage: from sage.calculus.desolvers import fricas_desolve
sage: Y = fricas_desolve(diff(y,x) + y - 1, y, [42,1783], x) # optional - fricas
sage: Y.subs(x=42) # optional - fricas
1783
"""
from sage.interfaces.fricas import fricas
from sage.symbolic.ring import SR
if ics is None:
y = fricas(de).solve(dvar.operator(), ivar).sage()
else:
eq = fricas.equation(ivar, ics[0])
y = fricas(de).solve(dvar.operator(), eq, ics[1:]).sage()
if isinstance(y, dict):
basis = y["basis"]
particular = y["particular"]
return particular + sum(SR.var("_C"+str(i))*v for i, v in enumerate(basis))
else:
return y
def fricas_desolve_system(des, dvars, ics, ivar):
r"""
Solve a system of first order ODEs using FriCAS.
EXAMPLES::
sage: t = var('t')
sage: x = function('x')(t)
sage: y = function('y')(t)
sage: de1 = diff(x,t) + y - 1 == 0
sage: de2 = diff(y,t) - x + 1 == 0
sage: desolve_system([de1, de2], [x, y], algorithm="fricas") # optional - fricas
[x(t) == _C0*cos(t) + cos(t)^2 + _C1*sin(t) + sin(t)^2,
y(t) == -_C1*cos(t) + _C0*sin(t) + 1]
sage: desolve_system([de1, de2], [x,y], [0,1,2], algorithm="fricas") # optional - fricas
[x(t) == cos(t)^2 + sin(t)^2 - sin(t), y(t) == cos(t) + 1]
TESTS::
sage: from sage.calculus.desolvers import fricas_desolve_system
sage: t = var('t')
sage: x = function('x')(t)
sage: fricas_desolve_system([diff(x,t) + 1 == 0], [x], None, t) # optional - fricas
[x(t) == _C0 - t]
sage: y = function('y')(t)
sage: de1 = diff(x,t) + y - 1 == 0
sage: de2 = diff(y,t) - x + 1 == 0
sage: sol = fricas_desolve_system([de1,de2], [x,y], [0,1,-1], t) # optional - fricas
sage: sol # optional - fricas
[x(t) == cos(t)^2 + sin(t)^2 + 2*sin(t), y(t) == -2*cos(t) + 1]
"""
from sage.interfaces.fricas import fricas
from sage.symbolic.ring import SR
from sage.symbolic.relation import solve
ops = [dvar.operator() for dvar in dvars]
y = fricas(des).solve(ops, ivar).sage()
basis = y["basis"]
particular = y["particular"]
pars = [SR.var("_C"+str(i)) for i in range(len(basis))]
solv = particular + sum(p*v for p, v in zip(pars, basis))
if ics is None:
sols = solv
else:
ics0 = ics[0]
eqs = [val == sol.subs({ivar: ics0}) for val, sol in zip(ics[1:], solv)]
pars_values = solve(eqs, pars, solution_dict=True)
sols = [sol.subs(pars_values[0]) for sol in solv]
return [dvar == sol for dvar, sol in zip(dvars, sols)]
def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False,
algorithm="maxima"):
r"""
Solve a 1st or 2nd order linear ODE, including IVP and BVP.
INPUT:
- ``de`` -- an expression or equation representing the ODE
- ``dvar`` -- the dependent variable (hereafter called `y`)
- ``ics`` -- (optional) the initial or boundary conditions
- for a first-order equation, specify the initial `x` and `y`
- for a second-order equation, specify the initial `x`, `y`,
and `dy/dx`, i.e. write `[x_0, y(x_0), y'(x_0)]`
- for a second-order boundary solution, specify initial and
final `x` and `y` boundary conditions, i.e. write `[x_0, y(x_0), x_1, y(x_1)]`.
- gives an error if the solution is not SymbolicEquation (as happens for
example for a Clairaut equation)
- ``ivar`` -- (optional) the independent variable (hereafter called
`x`), which must be specified if there is more than one
independent variable in the equation
- ``show_method`` -- (optional) if ``True``, then Sage returns pair
``[solution, method]``, where method is the string describing
the method which has been used to get a solution (Maxima uses the
following order for first order equations: linear, separable,
exact (including exact with integrating factor), homogeneous,
bernoulli, generalized homogeneous) - use carefully in class,
see below the example of an equation which is separable but
this property is not recognized by Maxima and the equation is solved
as exact.
- ``contrib_ode`` -- (optional) if ``True``, ``desolve`` allows to solve
Clairaut, Lagrange, Riccati and some other equations. This may take
a long time and is thus turned off by default. Initial conditions
can be used only if the result is one SymbolicEquation (does not
contain a singular solution, for example).
- ``algorithm`` -- (default: ``'maxima'``) one of
* ``'maxima'`` - use maxima
* ``'fricas'`` - use FriCAS (the optional fricas spkg has to be installed)
OUTPUT:
In most cases return a SymbolicEquation which defines the solution
implicitly. If the result is in the form `y(x)=\ldots` (happens for
linear eqs.), return the right-hand side only. The possible
constant solutions of separable ODEs are omitted.
.. NOTE::
Use ``desolve? <tab>`` if the output in the Sage notebook is truncated.
EXAMPLES::
sage: x = var('x')
sage: y = function('y')(x)
sage: desolve(diff(y,x) + y - 1, y)
(_C + e^x)*e^(-x)
::
sage: f = desolve(diff(y,x) + y - 1, y, ics=[10,2]); f
(e^10 + e^x)*e^(-x)
::
sage: plot(f)
Graphics object consisting of 1 graphics primitive
We can also solve second-order differential equations::
sage: x = var('x')
sage: y = function('y')(x)
sage: de = diff(y,x,2) - y == x
sage: desolve(de, y)
_K2*e^(-x) + _K1*e^x - x
::
sage: f = desolve(de, y, [10,2,1]); f
-x + 7*e^(x - 10) + 5*e^(-x + 10)
::
sage: f(x=10)
2
::
sage: diff(f,x)(x=10)
1
::
sage: de = diff(y,x,2) + y == 0
sage: desolve(de, y)
_K2*cos(x) + _K1*sin(x)
::
sage: desolve(de, y, [0,1,pi/2,4])
cos(x) + 4*sin(x)
::
sage: desolve(y*diff(y,x)+sin(x)==0,y)
-1/2*y(x)^2 == _C - cos(x)
Clairaut equation: general and singular solutions::
sage: desolve(diff(y,x)^2+x*diff(y,x)-y==0,y,contrib_ode=True,show_method=True)
[[y(x) == _C^2 + _C*x, y(x) == -1/4*x^2], 'clairault']
For equations involving more variables we specify an independent variable::
sage: a,b,c,n=var('a b c n')
sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True)
[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]]
::
sage: desolve(x^2*diff(y,x)==a+b*x^n+c*x^2*y^2,y,ivar=x,contrib_ode=True,show_method=True)
[[[y(x) == 0, (b*x^(n - 2) + a/x^2)*c^2*u == 0]], 'riccati']
Higher order equations, not involving independent variable::
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y).expand()
1/6*y(x)^3 + _K1*y(x) == _K2 + x
::
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3]).expand()
1/6*y(x)^3 - 5/3*y(x) == x - 3/2
::
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,1,3],show_method=True)
[1/6*y(x)^3 - 5/3*y(x) == x - 3/2, 'freeofx']
Separable equations - Sage returns solution in implicit form::
sage: desolve(diff(y,x)*sin(y) == cos(x),y)
-cos(y(x)) == _C + sin(x)
::
sage: desolve(diff(y,x)*sin(y) == cos(x),y,show_method=True)
[-cos(y(x)) == _C + sin(x), 'separable']
::
sage: desolve(diff(y,x)*sin(y) == cos(x),y,[pi/2,1])
-cos(y(x)) == -cos(1) + sin(x) - 1
Linear equation - Sage returns the expression on the right hand side only::
sage: desolve(diff(y,x)+(y) == cos(x),y)
1/2*((cos(x) + sin(x))*e^x + 2*_C)*e^(-x)
::
sage: desolve(diff(y,x)+(y) == cos(x),y,show_method=True)
[1/2*((cos(x) + sin(x))*e^x + 2*_C)*e^(-x), 'linear']
::
sage: desolve(diff(y,x)+(y) == cos(x),y,[0,1])
1/2*(cos(x)*e^x + e^x*sin(x) + 1)*e^(-x)
This ODE with separated variables is solved as
exact. Explanation - factor does not split `e^{x-y}` in Maxima
into `e^{x}e^{y}`::
sage: desolve(diff(y,x)==exp(x-y),y,show_method=True)
[-e^x + e^y(x) == _C, 'exact']
You can solve Bessel equations, also using initial
conditions, but you cannot put (sometimes desired) the initial
condition at `x=0`, since this point is a singular point of the
equation. Anyway, if the solution should be bounded at `x=0`, then
``_K2=0``. ::
sage: desolve(x^2*diff(y,x,x)+x*diff(y,x)+(x^2-4)*y==0,y)
_K1*bessel_J(2, x) + _K2*bessel_Y(2, x)
Example of difficult ODE producing an error::
sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y) # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."
Another difficult ODE with error - moreover, it takes a long time::
sage: desolve(sqrt(y)*diff(y,x)+e^(y)+cos(x)-sin(x+y)==0,y,contrib_ode=True) # not tested
Some more types of ODEs::
sage: desolve(x*diff(y,x)^2-(1+x*y)*diff(y,x)+y==0,y,contrib_ode=True,show_method=True)
[[y(x) == _C + log(x), y(x) == _C*e^x], 'factor']
::
sage: desolve(diff(y,x)==(x+y)^2,y,contrib_ode=True,show_method=True)
[[[x == _C - arctan(sqrt(t)), y(x) == -x - sqrt(t)], [x == _C + arctan(sqrt(t)), y(x) == -x + sqrt(t)]], 'lagrange']
These two examples produce an error (as expected, Maxima 5.18 cannot
solve equations from initial conditions). Maxima 5.18
returns false answer in this case! ::
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2]).expand() # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."
::
sage: desolve(diff(y,x,2)+y*(diff(y,x,1))^3==0,y,[0,1,2],show_method=True) # not tested
Traceback (click to the left for traceback)
...
NotImplementedError, "Maxima was unable to solve this ODE. Consider to set option contrib_ode to True."
Second order linear ODE::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y)
(_K2*x + _K1)*e^(-x) + 1/2*sin(x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,show_method=True)
[(_K2*x + _K1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1])
1/2*(7*x + 6)*e^(-x) + 1/2*sin(x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,1],show_method=True)
[1/2*(7*x + 6)*e^(-x) + 1/2*sin(x), 'variationofparameters']
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2])
3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == cos(x),y,[0,3,pi/2,2],show_method=True)
[3*(x*(e^(1/2*pi) - 2)/pi + 1)*e^(-x) + 1/2*sin(x), 'variationofparameters']
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y)
(_K2*x + _K1)*e^(-x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,show_method=True)
[(_K2*x + _K1)*e^(-x), 'constcoeff']
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1])
(4*x + 3)*e^(-x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,1],show_method=True)
[(4*x + 3)*e^(-x), 'constcoeff']
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2])
(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x)
::
sage: desolve(diff(y,x,2)+2*diff(y,x)+y == 0,y,[0,3,pi/2,2],show_method=True)
[(2*x*(2*e^(1/2*pi) - 3)/pi + 3)*e^(-x), 'constcoeff']
Using ``algorithm='fricas'`` we can invoke the differential
equation solver from FriCAS. For example, it can solve higher
order linear equations::
sage: de = x^3*diff(y, x, 3) + x^2*diff(y, x, 2) - 2*x*diff(y, x) + 2*y - 2*x^4
sage: Y = desolve(de, y, algorithm="fricas"); Y # optional - fricas
(2*x^3 - 3*x^2 + 1)*_C0/x + (x^3 - 1)*_C1/x
+ (x^3 - 3*x^2 - 1)*_C2/x + 1/15*(x^5 - 10*x^3 + 20*x^2 + 4)/x
The initial conditions are then interpreted as `[x_0, y(x_0),
y'(x_0), \ldots, y^(n)(x_0)]`::
sage: Y = desolve(de, y, ics=[1,3,7], algorithm="fricas"); Y # optional - fricas
1/15*(x^5 + 15*x^3 + 50*x^2 - 21)/x
FriCAS can also solve some non-linear equations::
sage: de = diff(y, x) == y / (x+y*log(y))
sage: Y = desolve(de, y, algorithm="fricas"); Y # optional - fricas
1/2*(log(y(x))^2*y(x) - 2*x)/y(x)
TESTS:
:trac:`9961` fixed (allow assumptions on the dependent variable in desolve)::
sage: y=function('y')(x); assume(x>0); assume(y>0)
sage: sage.calculus.calculus.maxima('domain:real') # needed since Maxima 5.26.0 to get the answer as below
real
sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y == 0, y, contrib_ode=True)
[x - arcsinh(y(x)/x) == _C]
:trac:`10682` updated Maxima to 5.26, and it started to show a different
solution in the complex domain for the ODE above::
sage: forget()
sage: sage.calculus.calculus.maxima('domain:complex') # back to the default complex domain
complex
sage: assume(x>0)
sage: assume(y>0)
sage: desolve(x*diff(y,x)-x*sqrt(y^2+x^2)-y == 0, y, contrib_ode=True)
[x - arcsinh(y(x)^2/(x*sqrt(y(x)^2))) - arcsinh(y(x)/x) + 1/2*log(4*(x^2 + 2*y(x)^2 + 2*sqrt(x^2*y(x)^2 + y(x)^4))/x^2) == _C]
:trac:`6479` fixed::
sage: x = var('x')
sage: y = function('y')(x)
sage: desolve( diff(y,x,x) == 0, y, [0,0,1])
x
::
sage: desolve( diff(y,x,x) == 0, y, [0,1,1])
x + 1
:trac:`9835` fixed::
sage: x = var('x')
sage: y = function('y')(x)
sage: desolve(diff(y,x,2)+y*(1-y^2)==0,y,[0,-1,1,1])
Traceback (most recent call last):
...
NotImplementedError: Unable to use initial condition for this equation (freeofx).
:trac:`8931` fixed::
sage: x=var('x'); f=function('f')(x); k=var('k'); assume(k>0)
sage: desolve(diff(f,x,2)/f==k,f,ivar=x)
_K1*e^(sqrt(k)*x) + _K2*e^(-sqrt(k)*x)
:trac:`15775` fixed::
sage: forget()
sage: y = function('y')(x)
sage: desolve(diff(y, x) == sqrt(abs(y)), dvar=y, ivar=x)
integrate(1/sqrt(abs(y(x))), y(x)) == _C + x
AUTHORS:
- David Joyner (1-2006)
- Robert Bradshaw (10-2008)
- Robert Marik (10-2009)
"""
if is_SymbolicEquation(de):
de = de.lhs() - de.rhs()
if is_SymbolicVariable(dvar):
raise ValueError("You have to declare dependent variable as a function evaluated at the independent variable, eg. y=function('y')(x)")
# for backwards compatibility
if isinstance(dvar, list):
dvar, ivar = dvar
elif ivar is None:
ivars = de.variables()
ivars = [t for t in ivars if t is not dvar]
if len(ivars) != 1:
raise ValueError("Unable to determine independent variable, please specify.")
ivar = ivars[0]
if algorithm == "fricas":
return fricas_desolve(de, dvar, ics, ivar)
elif algorithm != "maxima":
raise ValueError("unknown algorithm %s" % algorithm)
de00 = de._maxima_()
P = de00.parent()
dvar_str=P(dvar.operator()).str()
ivar_str=P(ivar).str()
de00 = de00.str()
def sanitize_var(exprs):
return exprs.replace("'"+dvar_str+"("+ivar_str+")",dvar_str)
de0 = sanitize_var(de00)
ode_solver="ode2"
cmd="(TEMP:%s(%s,%s,%s), if TEMP=false then TEMP else substitute(%s=%s(%s),TEMP))"%(ode_solver,de0,dvar_str,ivar_str,dvar_str,dvar_str,ivar_str)
# we produce string like this
# ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y(x),x)
soln = P(cmd)
if str(soln).strip() == 'false':
if contrib_ode:
ode_solver="contrib_ode"
P("load('contrib_ode)")
cmd="(TEMP:%s(%s,%s,%s), if TEMP=false then TEMP else substitute(%s=%s(%s),TEMP))"%(ode_solver,de0,dvar_str,ivar_str,dvar_str,dvar_str,ivar_str)
# we produce string like this
# (TEMP:contrib_ode(x*('diff(y,x,1))^2-(x*y+1)*'diff(y,x,1)+y,y,x), if TEMP=false then TEMP else substitute(y=y(x),TEMP))
soln = P(cmd)
if str(soln).strip() == 'false':
raise NotImplementedError("Maxima was unable to solve this ODE.")
else:
raise NotImplementedError("Maxima was unable to solve this ODE. Consider to set option contrib_ode to True.")
if show_method:
maxima_method=P("method")
if (ics is not None):
if not is_SymbolicEquation(soln.sage()):
if not show_method:
maxima_method=P("method")
raise NotImplementedError("Unable to use initial condition for this equation (%s)."%(str(maxima_method).strip()))
if len(ics) == 2:
tempic=(ivar==ics[0])._maxima_().str()
tempic=tempic+","+(dvar==ics[1])._maxima_().str()
cmd="(TEMP:ic1(%s(%s,%s,%s),%s),substitute(%s=%s(%s),TEMP))"%(ode_solver,de00,dvar_str,ivar_str,tempic,dvar_str,dvar_str,ivar_str)
cmd=sanitize_var(cmd)
# we produce string like this
# (TEMP:ic2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,'diff(y,x)=1),substitute(y=y(x),TEMP))
soln=P(cmd)
if len(ics) == 3:
#fixed ic2 command from Maxima - we have to ensure that %k1, %k2 do not depend on variables, should be removed when fixed in Maxima
P("ic2_sage(soln,xa,ya,dya):=block([programmode:true,backsubst:true,singsolve:true,temp,%k2,%k1,TEMP_k], \
noteqn(xa), noteqn(ya), noteqn(dya), boundtest('%k1,%k1), boundtest('%k2,%k2), \
temp: lhs(soln) - rhs(soln), \
TEMP_k:solve([subst([xa,ya],soln), subst([dya,xa], lhs(dya)=-subst(0,lhs(dya),diff(temp,lhs(xa)))/diff(temp,lhs(ya)))],[%k1,%k2]), \
if not freeof(lhs(ya),TEMP_k) or not freeof(lhs(xa),TEMP_k) then return (false), \
temp: maplist(lambda([zz], subst(zz,soln)), TEMP_k), \
if length(temp)=1 then return(first(temp)) else return(temp))")
tempic=P(ivar==ics[0]).str()
tempic=tempic+","+P(dvar==ics[1]).str()
tempic=tempic+",'diff("+dvar_str+","+ivar_str+")="+P(ics[2]).str()
cmd="(TEMP:ic2_sage(%s(%s,%s,%s),%s),substitute(%s=%s(%s),TEMP))"%(ode_solver,de00,dvar_str,ivar_str,tempic,dvar_str,dvar_str,ivar_str)
cmd=sanitize_var(cmd)
# we produce string like this
# (TEMP:ic2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,'diff(y,x)=1),substitute(y=y(x),TEMP))
soln=P(cmd)
if str(soln).strip() == 'false':
raise NotImplementedError("Maxima was unable to solve this IVP. Remove the initial condition to get the general solution.")
if len(ics) == 4:
#fixed bc2 command from Maxima - we have to ensure that %k1, %k2 do not depend on variables, should be removed when fixed in Maxima
P("bc2_sage(soln,xa,ya,xb,yb):=block([programmode:true,backsubst:true,singsolve:true,temp,%k1,%k2,TEMP_k], \
noteqn(xa), noteqn(ya), noteqn(xb), noteqn(yb), boundtest('%k1,%k1), boundtest('%k2,%k2), \
TEMP_k:solve([subst([xa,ya],soln), subst([xb,yb],soln)], [%k1,%k2]), \
if not freeof(lhs(ya),TEMP_k) or not freeof(lhs(xa),TEMP_k) then return (false), \
temp: maplist(lambda([zz], subst(zz,soln)),TEMP_k), \
if length(temp)=1 then return(first(temp)) else return(temp))")
cmd="bc2_sage(%s(%s,%s,%s),%s,%s=%s,%s,%s=%s)"%(ode_solver,de00,dvar_str,ivar_str,P(ivar==ics[0]).str(),dvar_str,P(ics[1]).str(),P(ivar==ics[2]).str(),dvar_str,P(ics[3]).str())
cmd="(TEMP:%s,substitute(%s=%s(%s),TEMP))"%(cmd,dvar_str,dvar_str,ivar_str)
cmd=sanitize_var(cmd)
# we produce string like this
# (TEMP:bc2(ode2('diff(y,x,2)+2*'diff(y,x,1)+y-cos(x),y,x),x=0,y=3,x=%pi/2,y=2),substitute(y=y(x),TEMP))
soln=P(cmd)
if str(soln).strip() == 'false':
raise NotImplementedError("Maxima was unable to solve this BVP. Remove the initial condition to get the general solution.")
soln=soln.sage()
if is_SymbolicEquation(soln) and soln.lhs() == dvar:
# Remark: Here we do not check that the right hand side does not depend on dvar.
# This probably will not happen for solutions obtained via ode2, anyway.
soln = soln.rhs()
if show_method:
return [soln,maxima_method.str()]
else:
return soln
#def desolve_laplace2(de,vars,ics=None):
## """
## Solves an ODE using laplace transforms via maxima. Initial conditions
## are optional.
## INPUT:
## de -- a lambda expression representing the ODE
## (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)")
## vars -- a list of strings representing the variables
## (eg, vars = ["x","f"], if x is the independent
## variable and f is the dependent variable)
## ics -- a list of numbers representing initial conditions,
## with symbols allowed which are represented by strings
## (eg, f(0)=1, f'(0)=2 is ics = [0,1,2])
## EXAMPLES::
## sage: from sage.calculus.desolvers import desolve_laplace
## sage: x = var('x')
## sage: f = function('f')(x)
## sage: de = lambda y: diff(y,x,x) - 2*diff(y,x) + y
## sage: desolve_laplace(de(f(x)),[f,x])
## #x*%e^x*(?%at('diff('f(x),x,1),x=0))-'f(0)*x*%e^x+'f(0)*%e^x
## sage: desolve_laplace(de(f(x)),[f,x],[0,1,2]) ## IC option does not work
## #x*%e^x*(?%at('diff('f(x),x,1),x=0))-'f(0)*x*%e^x+'f(0)*%e^x
## AUTHOR: David Joyner (1st version 1-2006, 8-2007)
## """
# ######## this method seems reasonable but doesn't work for some reason
# name0 = vars[0]._repr_()[0:(len(vars[0]._repr_())-2-len(str(vars[1])))]
# name1 = str(vars[1])
# #maxima("de:"+de+";")
# if ics is not None:
# ic0 = maxima("ic:"+str(vars[1])+"="+str(ics[0]))
# d = len(ics)
# for i in range(d-1):
# maxima(vars[0](vars[1])).diff(vars[1],i).atvalue(ic0,ics[i+1])
# de0 = de._maxima_()
# #cmd = "desolve("+de+","+vars[1]+"("+vars[0]+"));"
# #return maxima.eval(cmd)
# return de0.desolve(vars[0]).rhs()
def desolve_laplace(de, dvar, ics=None, ivar=None):
"""
Solve an ODE using Laplace transforms. Initial conditions are optional.
INPUT:
- ``de`` - a lambda expression representing the ODE (e.g. ``de =
diff(y,x,2) == diff(y,x)+sin(x)``)
- ``dvar`` - the dependent variable (e.g. ``y``)
- ``ivar`` - (optional) the independent variable (hereafter called
`x`), which must be specified if there is more than one
independent variable in the equation.
- ``ics`` - a list of numbers representing initial conditions, (e.g.
``f(0)=1``, ``f'(0)=2`` corresponds to ``ics = [0,1,2]``)
OUTPUT:
Solution of the ODE as symbolic expression
EXAMPLES::
sage: u=function('u')(x)
sage: eq = diff(u,x) - exp(-x) - u == 0
sage: desolve_laplace(eq,u)
1/2*(2*u(0) + 1)*e^x - 1/2*e^(-x)
We can use initial conditions::
sage: desolve_laplace(eq,u,ics=[0,3])
-1/2*e^(-x) + 7/2*e^x
The initial conditions do not persist in the system (as they persisted
in previous versions)::
sage: desolve_laplace(eq,u)
1/2*(2*u(0) + 1)*e^x - 1/2*e^(-x)
::
sage: f=function('f')(x)
sage: eq = diff(f,x) + f == 0
sage: desolve_laplace(eq,f,[0,1])
e^(-x)
::
sage: x = var('x')
sage: f = function('f')(x)
sage: de = diff(f,x,x) - 2*diff(f,x) + f
sage: desolve_laplace(de,f)
-x*e^x*f(0) + x*e^x*D[0](f)(0) + e^x*f(0)
::
sage: desolve_laplace(de,f,ics=[0,1,2])
x*e^x + e^x
TESTS:
Check that :trac:`4839` is fixed::
sage: t = var('t')
sage: x = function('x')(t)
sage: soln = desolve_laplace(diff(x,t)+x==1, x, ics=[0,2])
sage: soln
e^(-t) + 1
::
sage: soln(t=3)
e^(-3) + 1
AUTHORS:
- David Joyner (1-2006,8-2007)
- Robert Marik (10-2009)
"""
#This is the original code from David Joyner (inputs and outputs strings)
#maxima("de:"+de._repr_()+"=0;")
#if ics is not None:
# d = len(ics)
# for i in range(0,d-1):
# ic = "atvalue(diff("+vars[1]+"("+vars[0]+"),"+str(vars[0])+","+str(i)+"),"+str(vars[0])+"="+str(ics[0])+","+str(ics[1+i])+")"
# maxima(ic)
#
#cmd = "desolve("+de._repr_()+","+vars[1]+"("+vars[0]+"));"
#return maxima(cmd).rhs()._maxima_init_()
## verbatim copy from desolve - begin
if is_SymbolicEquation(de):
de = de.lhs() - de.rhs()
if is_SymbolicVariable(dvar):
raise ValueError("You have to declare dependent variable as a function evaluated at the independent variable, eg. y=function('y')(x)")
# for backwards compatibility
if isinstance(dvar, list):
dvar, ivar = dvar
elif ivar is None:
ivars = de.variables()
ivars = [t for t in ivars if t is not dvar]
if len(ivars) != 1:
raise ValueError("Unable to determine independent variable, please specify.")
ivar = ivars[0]
## verbatim copy from desolve - end
dvar_str = str(dvar)
def sanitize_var(exprs): # 'y(x) -> y(x)
return exprs.replace("'"+dvar_str,dvar_str)
de0=de._maxima_()
P = de0.parent()
i = dvar_str.find('(')
dvar_str = dvar_str[:i+1] + '_SAGE_VAR_' + dvar_str[i+1:]
cmd = sanitize_var("desolve("+de0.str()+","+dvar_str+")")
soln=P(cmd).rhs()
if str(soln).strip() == 'false':
raise NotImplementedError("Maxima was unable to solve this ODE.")
soln=soln.sage()
if ics is not None:
d = len(ics)
for i in range(0,d-1):
soln=eval('soln.substitute(diff(dvar,ivar,i)('+str(ivar)+'=ics[0])==ics[i+1])')
return soln
def desolve_system(des, vars, ics=None, ivar=None, algorithm="maxima"):
r"""
Solve a system of any size of 1st order ODEs. Initial conditions
are optional.
One dimensional systems are passed to :meth:`desolve_laplace`.
INPUT:
- ``des`` -- list of ODEs
- ``vars`` -- list of dependent variables
- ``ics`` -- (optional) list of initial values for ``ivar`` and ``vars``;
if ``ics`` is defined, it should provide initial conditions for each
variable, otherwise an exception would be raised
- ``ivar`` -- (optional) the independent variable, which must be
specified if there is more than one independent variable in the
equation
- ``algorithm`` -- (default: ``'maxima'``) one of
* ``'maxima'`` - use maxima
* ``'fricas'`` - use FriCAS (the optional fricas spkg has to be installed)
EXAMPLES::
sage: t = var('t')
sage: x = function('x')(t)
sage: y = function('y')(t)
sage: de1 = diff(x,t) + y - 1 == 0
sage: de2 = diff(y,t) - x + 1 == 0
sage: desolve_system([de1, de2], [x,y])
[x(t) == (x(0) - 1)*cos(t) - (y(0) - 1)*sin(t) + 1,
y(t) == (y(0) - 1)*cos(t) + (x(0) - 1)*sin(t) + 1]
The same system solved using FriCAS::
sage: desolve_system([de1, de2], [x,y], algorithm='fricas') # optional - fricas
[x(t) == _C0*cos(t) + cos(t)^2 + _C1*sin(t) + sin(t)^2,
y(t) == -_C1*cos(t) + _C0*sin(t) + 1]
Now we give some initial conditions::
sage: sol = desolve_system([de1, de2], [x,y], ics=[0,1,2]); sol
[x(t) == -sin(t) + 1, y(t) == cos(t) + 1]
::
sage: solnx, solny = sol[0].rhs(), sol[1].rhs()
sage: plot([solnx,solny],(0,1)) # not tested
sage: parametric_plot((solnx,solny),(0,1)) # not tested
TESTS:
Check that :trac:`9823` is fixed::
sage: t = var('t')
sage: x = function('x')(t)
sage: de1 = diff(x,t) + 1 == 0
sage: desolve_system([de1], [x])
-t + x(0)
Check that :trac:`16568` is fixed::
sage: t = var('t')
sage: x = function('x')(t)
sage: y = function('y')(t)
sage: de1 = diff(x,t) + y - 1 == 0
sage: de2 = diff(y,t) - x + 1 == 0
sage: des = [de1,de2]
sage: ics = [0,1,-1]
sage: vars = [x,y]
sage: sol = desolve_system(des, vars, ics); sol
[x(t) == 2*sin(t) + 1, y(t) == -2*cos(t) + 1]
::
sage: solx, soly = sol[0].rhs(), sol[1].rhs()
sage: RR(solx(t=3))
1.28224001611973
::
sage: P1 = plot([solx,soly], (0,1))
sage: P2 = parametric_plot((solx,soly), (0,1))
Now type ``show(P1)``, ``show(P2)`` to view these plots.
Check that :trac:`9824` is fixed::
sage: t = var('t')
sage: epsilon = var('epsilon')
sage: x1 = function('x1')(t)
sage: x2 = function('x2')(t)
sage: de1 = diff(x1,t) == epsilon
sage: de2 = diff(x2,t) == -2
sage: desolve_system([de1, de2], [x1, x2], ivar=t)
[x1(t) == epsilon*t + x1(0), x2(t) == -2*t + x2(0)]
sage: desolve_system([de1, de2], [x1, x2], ics=[1,1], ivar=t)
Traceback (most recent call last):
...
ValueError: Initial conditions aren't complete: number of vars is different from number of dependent variables. Got ics = [1, 1], vars = [x1(t), x2(t)]
AUTHORS:
- Robert Bradshaw (10-2008)
- Sergey Bykov (10-2014)
"""
if ics is not None:
if len(ics) != (len(vars) + 1):
raise ValueError("Initial conditions aren't complete: number of vars is different from number of dependent variables. Got ics = {0}, vars = {1}".format(ics, vars))
if len(des) == 1 and algorithm == "maxima":
return desolve_laplace(des[0], vars[0], ics=ics, ivar=ivar)
ivars = set([])
for i, de in enumerate(des):
if not is_SymbolicEquation(de):
des[i] = de == 0
ivars = ivars.union(set(de.variables()))
if ivar is None:
ivars = ivars - set(vars)
if len(ivars) != 1:
raise ValueError("Unable to determine independent variable, please specify.")
ivar = list(ivars)[0]
if algorithm == "fricas":
return fricas_desolve_system(des, vars, ics, ivar)
elif algorithm != "maxima":
raise ValueError("unknown algorithm %s" % algorithm)
dvars = [v._maxima_() for v in vars]
if ics is not None:
ivar_ic = ics[0]
for dvar, ic in zip(dvars, ics[1:]):
dvar.atvalue(ivar==ivar_ic, ic)
soln = dvars[0].parent().desolve(des, dvars)
if str(soln).strip() == 'false':
raise NotImplementedError("Maxima was unable to solve this system.")
soln = list(soln)
for i, sol in enumerate(soln):
soln[i] = sol.sage()
if ics is not None:
ivar_ic = ics[0]
for dvar, ic in zip(dvars, ics[:1]):
dvar.atvalue(ivar==ivar_ic, dvar)
return soln
def eulers_method(f,x0,y0,h,x1,algorithm="table"):
r"""
This implements Euler's method for finding numerically the
solution of the 1st order ODE `y' = f(x,y)`, `y(a)=c`. The ``x``
column of the table increments from `x_0` to `x_1` by `h` (so
`(x_1-x_0)/h` must be an integer). In the ``y`` column, the new
`y`-value equals the old `y`-value plus the corresponding entry in the
last column.
.. NOTE::
This function is for pedagogical purposes only.
EXAMPLES::
sage: from sage.calculus.desolvers import eulers_method
sage: x,y = PolynomialRing(QQ,2,"xy").gens()
sage: eulers_method(5*x+y-5,0,1,1/2,1)
x y h*f(x,y)