Maple Dsolve Numeric

7/23/2019 Maple Dsolve Numeric

1/36

dsolve/numeric/DAE- find numerical solution of differential-algebraic initialvalue problems

Calling Sequence

dsolve(daesys, numeric, vars, options)

Parameters

daesys - set or list; ordinary differential equation(s), algebraic equation(s) and

initial conditions

numeric - name; instruct dsolveto find a numerical solution

vars - (optional) dependent variable or a set or list of dependent variables for

daesys

options - (optional) equations of the form keyword = value

Description

The dsolvecommand with the numericor type=numericoption and a real-valueddifferential-algebraic initial value problem (DA !"#) finds a numerical solution forthe DA !"#$ !f the optional equation method=numericmethodis provided (where

numericmethodis one of r%f&'dae, rosenbroc%dae, or mebdfi, dsolveuses that

method to obtain the numerical solution$

!n most cases dsolveis able to detect if the problem is a DA system, as opposed toan D system, namely the cases in which pure algebraic equations in the dependent

variables are present$ !f the input is a DA system containing no purely algebraic

equations, the method mustbe included to specify that the system is a DA system$

*onstrained mechanical systems often give rise to DA problems$ (+ee the pendulumeample below$)

The return value of dsolveand the following high level or common options arediscussed in dsolvenumeric.and dsolverror*ontrol.$

'output' = keyword or array

'range' = numeric..numeric

'abserr' = numeric or list

'relerr' = numeric

'stiff' = boolean

'known' = name or list of names

'optimize' = boolean

/


2/36

The eception is that for the DA solvers the absolute error tolerance can be specified

as a per-component list$ 0or epected behavior, the variables of the problem must also

be specified as a list, and the entries of 'aserr'must correspond /-/ to the variablesin the list, or to the variables in the system converted to first order using the order of

the variables in the list$ 0or more information, see the 1dsolvenumeric,!"#.help

page$ The default DA !"# method is a modifiedRunge-Kutta Fehlbergmethod, which

uses a base order &-' method, but has been modified to find solutions for DA

problems$ The default sti!!method is aRosenbrockmethod, which uses a base order

2-& method$ 0or a description of the modifications done to these methods inetending them to DA solution, see the 1daeetensionhelp page$ The other method

available for DA !"# is the dsolvemebdfi.method, which is short for 3odified

tended 4ac%ward-Differentiation 0ormula !mplicit method$

!n general, the DA !"# solvers are very similar to the standard differential !"#solvers, so this page is primarily concerned with outlining the differences between

them$

The DA solvers are currently restricted to finding solutions for real-valuedproblems$

0or use of any of the methods, the specified initial conditions must satisfy all hiddenconstraints of the system (that is, they must be a consistent set of initial conditions

with respect to the DA)$ !n the event that they do not, an error results, and

information is provided on the unsatisfied condition$

!n some cases, it may be necessary to use fsolveto compute consistent initialconditions for the problem$

!f the 1daeetensionmethods are in use, the differential option is set to true, and thesystem is sufficiently linear in the algebraic variables (i$e$, variables which have noderivatives appearing in the input system), it is possible to s%ip initial conditions for

those variables$ !f the initial conditions are s%ipped when they are required, an error

will be produced$

The following options are also available for some or all of the DA methods5'minstep' = numeric

'maxstep' = numeric

'initstep' = numeric

'startinit' = boolean

'events' = list

'event_pre' = keyword

'event_maxiter' = integer

'projection' = boolean

'differential' = boolean

6


3/36

'implicit' = boolean

'parameters' = list of names

'minstep', 'ma"step', and 'initstep'

These options are discussed in dsolverror*ontrol., and are only available for the

dsolvemebdfi.method$

'startinit'= oolean

This option controls the behavior of the numerical integration with respect to

consecutive calculations$ This option is described in dsolvenumeric,!"#.$

0or the default methods, r%f&'daeand rosenbroc%dae, when a 'range'has been

specified, the solution is recomputed only when the new value of the independentvariable is not between the initial point and any other previously requested solution

point$ This has the effect of never reversing the direction of integration, and ma%ing

evaluation of the solution for an already computed interval quite inepensive (the

solution values are obtained by interpolation)$ The storage of the solution can also beenabled by using the storageargument$ The startinitparameter also forces these

methods to recompute the solution whenever a solution value is desired$

'events'= list

'event#pre'= keyword

'event#ma"iter'= integer

These options are available only for the r%f&'daeand rosenbroc%daemethods, and

are used to specify and control the behavior of events$ These are the same as forstandard !"# problems$ 0or a description, see the 1dsolvenumeric,!"#.and 1

dsolvevents.help pages$

'pro$ection', 'di!!erential', and 'implicit'

The 'pro$ection', 'di!!erential'and 'implicit'options are specific to the etensionmethods, so they are discussed there$

!n addition to the computation of solution values for the given input problem, theprocedure returns (that is, output7procedurelist, listprocedureor operator) provide

additional interactive features, including the ability to specify parameters$!nformation on these features is provided on the dsolvenumeric,interactive.page$

The eception is that the med!imethod cannot wor% with parametri8ed problems$

E"amples

As a first eample, we consider the problem of modeling the dynamics of a mass on astring of unit length in 6-D *artesian coordinates (the pendulum problem)$ 9e let rbethe position of the mass on the string, and vthe velocity5

:VectorCalculus[SetCoordinates](cartesian):

:VectorCalculus[BasisFormat](false):

: r := VectorCalculus[Vector]([x(t),y(t)]);

2


4/36

57r

( )t

( )yt

: v := VectorCalculus[diff](r,t);

57v

ddt

( )t

d

d

t( )yt

+ince the mass is on a string of fied length l, we have the constraint5

: con := VectorCalculus[ot!roduct](r,r)"l#$;

57con + ( )t 6 ( )yt 6 l6

ow we want to construct the DA system using the uler-


5/36

57Q ? @, , 6 ( )t ( )t m d

d

t6

( )t +( )t 6 ( )yt 6 l6 m g 6 ( )t ( )yt m d

d

t6

( )yt

: .x := isolate(select(-as,,diff(x(t),t,t))[0],diff(x(t),t,t));

57EQx =d

d

t6

( )t 6( )t ( )t

m

: .y := isolate(select(-as,,diff(y(t),t,t))[0],diff(y(t),t,t));

57EQy =d

d

t6

( )yt +m g 6 ( )t ( )yt

m

: .c := remove(-as,,diff)[0];

57EQc +( )t 6 ( )yt 6 l6

ow we have the equations of motion for the pendulum$ et, we need to determine

consistent initial conditions$ To do so, we must identify any hidden constraints of thesystem$ These are easy to find, as we have only one constraint$

: con := diff(.c,t);

57Dcon 6 ( )t d

d

t( )t 6 ( )yt

dd

t( )yt

: con := eval(diff(con,t),1.x,.y2);

57DDcon + +6 d

d

t( )t &

( )t 6 ( )tm

6 d

d

t( )yt 6

( )yt ( )+m g 6 ( )t ( )ytm

ur initial conditions must satisfy Ec, Dcon, and DDconat the initial point, leavingonly 6 degrees of freedom for the conditions$ +o for a pendulum starting at the minimum

value of y()* = +lhaving an initial hori8ontal velocity of D("*()* = v", we get5

: sys := 1y(3)="l,(x)(3)=vx2 union

eval(convert(1.c,con,con2,),t=3);

sys 6 ( ) ( )( )Dx 6 ( )y ( )( )Dy +( ) ( )y l, ,?57

+ +6 ( )( )Dx 6 &( )

6( )

m6 ( )( )Dy

66

( )y ( )+m g 6 ( ) ( )y m

=( )y l, ,

=( )( )Dx vx@

'


6/36

: ini := [solve(sys,1x(3),y(3),(x)(3),(y)(3),lam+da(3)2)][0];

57n ? @, , , ,=( ) /

6

m ( )+vx6 l gl6

=( ) =( )y l =( )( )Dx vx =( )( )Dy

+o we consider the above with a pendulum of unit length l = ,having unit mass m = ,

and an initial hori8ontal velocity of v" = ,/,), giving us the DA system and initial

conditions5

: dsys := eval(1.x,.y,.c2,1l=0,m=0,=456,vx=0&032);

57dsys ? @, , +( )t 6 ( )yt 6 / =d

d6

t6

( )t 6 ( )t ( )t =d

d6

t6

( )yt =$> 6 ( )t ( )yt

: dini := eval(ini,1l=0,m=0,=456,vx=0&032);

57dn ? @, , , ,=( ) &$=' =( ) =( )y -/ =( )( )Dx /

/=( )( )Dy

9e can then obtain the solution as5

: dsol0 := dsolve(dsys union dini, numeric);

57dsol! proc( ) $$$ end prox"rk#$%"dae

: dsol0(0&$);

=t $' =( )t &$>=B'62B&&6'>>B =( )t $2/=2=6'&B&'&B>, , ,

=d

d

t( )t $'C&'2/&&=CCC'>/=/ =( )yt -$===&>=>//>=2>CB/, ,

=d

d

t( )yt $/>2>6B2C=&C22&2

: dsol0(0);

=t /$ =( )t &$=&===&=&='C26'= =( )t $2C>=2&&B>2&, , ,

=d

d

t( )t -$====2CBB=6&>>>B2 =( )yt -$=======2&>B62/&2, ,

=d

d

t( )yt -$2C>B>&6'2==&=

C


7/36

+olution with rosenrock#dae5

: dsol$ := dsolve(dsys union dini, numeric,met-od=rosen+roc78dae);

57dsol& proc( ) $$$ end prox"rosenbrock"dae

: dsol$(0&$);

=t $' =( )t &$>=B'/=//>'>C>> =( )t $2/=2=62C&BC2C=2B, , ,

=d

d

t( )t $'C&C6C&C>C>=BB/ =( )yt -$===&>=>/2'&2222&2=, ,

=d

d

t( )yt $/>&>6B6>B/BC26

: dsol$(0);

=t /$ =( )t &$=&===C6B>/B6/ =( )t $2C=C>&C26C2/&=, , ,

=d

d

t( )t -$====2CBB=&'6/C2>& =( )yt -$=======2CC=/2/>/, ,

=d

d

t( )yt -$2C>=B=>CC>>'B

+olution with med!i5

: dsol9 := dsolve(dsys union dini, numeric, met-od=me+dfi);

57dsol' proc( ) $$$ end prox"mebd#

: dsol9(0&$);

=t $' =( )t &$>=B'6&'/=&2''= =( )t $2/=2>>&2C=&>''/, , ,

=d

d

t

( )t $'C&>B/62&666'B' =( )yt -$===&>=>6'/>='=/6=, ,

=d

d

t( )yt $/>CB&'&C2C=/2

: dsol9(0);

B


8/36

=t /$ =( )t &$=&=B=&&B6&&6==2' =( )t $2C/=B2/6/=&'&=>, , ,

=d

d

t( )t -$====6'2=/=&BC&BB' =( )yt -$=======2&6C/>B=', ,

=ddt

( )yt -$2'=>>/C//'=2/B=

ow consider a similar problem as above, but in addition add a second mass supported

from the first by another string, this one of length /6 (the double pendulum)$ The system

can be obtained and solved as5

: dsysd := 1

diff(x0(t),t,t) $'lam+da0(t)'x0(t)$'lam+da$(t)'(x0(t)"x$(t)),

diff(y0(t),t,t)456$'lam+da0(t)'y0(t)$'lam+da$(t)'(y0(t)"y$(t)),

diff(x$(t),t,t)"$'lam+da$(t)'(x0(t)"x$(t)),

diff(y$(t),t,t)456"$'lam+da$(t)'(y0(t)"y$(t)),

x0(t)#$y0(t)#$"0,

(x0(t)"x$(t))#$(y0(t)"y$(t))#$"0&

2;

dsysd

d

d

t6

( )6t 6 ( )2t ( )( )/t ( )6t + ( )( )/t ( )6t 6 ( )( )y/t ( )y6t 6/

&, ,?57

+ ( )/t 6 ( )y/t 6 / + +d

d6

t6

( )/t 6 ( )1t ( )/t 6 ( )2t ( )( )/t ( )6t, ,

+

d

d6

t6( )y6t =$> 6 ( )

2t ( )

( )y/t ( )y6t ,

+ + +d

d6

t6( )y/t =$> 6 ( )1t ( )y/t 6 ( )2t ( )( )y/t ( )y6t @

: ics := 1x0(3)=3,(x0)(3)="9,y0(3)="0,(y0)(3)=3,

x$(3)=3,(x$)(3)=,y$(3)="9&$,(y$)(3)=32;

>


9/36

cs =( )/ =( )6 =( )y/ -/ =( )y6 -2

6=( )( )Dx! -2 =( )( )Dx& & =( )( )Dy! , , , , , , ,?57

=( )( )Dy& @

: dsold := dsolve(dsysd union ics,numeric);

57dsold proc( ) $$$ end prox"rk#$%"dae

The traEectory of the second mass can be plotted via5

:


10/36

rosenbroc%,bvp, r%f&'dae, rosenbroc%dae, dver%B>, lsode, gear, taylorseries,

mebdfi, or classical), dsolveuses that method to obtain the numerical solution$

4oth initial and boundary value problems can be numerically solved, as well as initialdifferential algebraic problems$ 0or a system input, the type of problem is

automatically detected, but for some DA problems it may be necessary to specify

the method$4y default, a procedure is returned that can be used to obtain solution values if giventhe value of the independent variable$

The default for initial value problems (!"#) is a Funge-Gutta 0ehlberg method thatproduces a fifth order accurate solution (0or more information, see r%f&'ornumeric,!"#)$ 0or boundary value problems (4"#), a finite difference technique with

Fichardson etrapolation is used (0or more information, see numeric,4"#)$ 0or

differential-algebraic !"# problems (DA), a modification of the r%f&' method is

used (0or more information, see daeetension)$

All .&Pmethods can be used for comple-valued !"#s with a real-valued

independent variable (though for the default stiff and nonstiff !"# methods, it may benecessary to specify that the problem is comple via the comple"option)$ one of

the &Por DAEmethods can currently be used for comple-valued problems,requiring the system be converted to a real system before calling dsolve$

The .&Pand DAEmethods have additional capabilities for the returned solutionprocedure$ +pecifically, it is possible to query the last computed solution value (useful

for problems with singularities), query the initial data, and change the initial data forsome solution procedures (DAEmethods support a subset of this)$ 0or more

information, see numeric,!"#and numeric,DA$

9hen the set of dependent variables is specified as a list, then this specifies the orderof the dependent variables as they appear in the output$ The order is otherwisealphabetical in the dependent variable name$

0ptions

The high-level options, most common to !"#s, 4"#s, and DAs are as follows$'output' = keyword or array

'stiff' = boolean

'events' = list




'abserr' = numeric or list

'relerr' = numeric



/


11/36

'output'= keyword or array

Geyword that can ta%e the values procedurelist, listprocedure, operator, or

piecewise, or an arrayor Arraythat gives the values of the independent variable at

which solution values are desired$The default %eyword is procedurelist, which gives the output from dsolveas aprocedure$ This procedure accepts the value of the independent variable as an

argument, and it returns a list of the solution values of the form variale=value,

where the left-hand sides are the names of the independent variable, the dependentvariable(s) and their derivatives (for higher order equations), and the right-hand sides

are the corresponding computed solution values$

The listprocedure%eyword gives the output as a list of equations of the formvariale=procedure, where the left-hand sides are the names of the independentvariable, the dependent variable(s) and derivatives, and the right-hand sides are

procedures that can be used to compute the corresponding solution components$ This

output form is most useful when the returned procedure is to be used later with, foreample, fsolve$

The operator%eyword gives the output as a list of equations of the form

operator=procedure, where the left-hand sides are operators that can be applied to

an independent variable value to give the function or derivative evaluated at a point,and the right-hand sides are procedures that can be used to compute the

corresponding solution components$ This output form is most useful for shortcut

evaluation forms where the returned list is to be evaluated at a point (for eample,1result2(,*)$

The piecewise%eyword is available only for non-stiff and stiff default !"# and DA

methods (r%f&', rosenbroc%, r%f&'dae, and rosenbroc%dae) and the taylorseries

method$ !t provides output as a list of equations of the form variale = pw!unc,where the left-hand sides are the names of the independent variable, the dependent

variable(s) and derivatives, and the right-hand sides arepiecewisepolynomial

functions describing the corresponding solution components$ Thesepiecewisefunctions are obtained from the method interpolants for each step of the computation$

!n addition, the form of the piecewise polynomials can be specified using an inde on

the piecewise output request$ +pecification of output=piecewise1horner2providesoutput in terms of horner-form polynomials (the default for taylorseries), while

specification of output=piecewise1polynomial2provides output in terms of standard

polynomial form (the default for all other methods)$ Hse of this output form requiresthat the 'range'argument be used to specify to dsolve/numericthe desired range of

the piecewise function$

!f an arrayis used, it must be a vector of floats that defines the values of the

independent variable at which solution values are desired$ 9hen this option is given,a 6 / matri"is returned (instead of a solution procedure)$ The 1,3 ,2entry is an

array containing the name of the independent variable and the names of the dependent

variable(s) and derivatives$ The 1-3 ,2entry is a matri$ The first column of thismatri is a copy of the outputvector, that is, the values of the independent variable$

//


12/36

The other columns are values of the dependent variable(s) and derivatives

corresponding to the vector of names in 1,3 ,2$ Fow iof this matri is the vector

containing both the value of the independent variable and the values of the dependentvariables evaluated at element iof the outputarray$

4ote5!f an Arrayis used, then the output is the same as for an arrayecept the

newer Arrayand 6atri"datatypes are used in the output$ 0or a large number ofoutput points andor solution components, the output data may not be directly visible

because there is a default si8e at which the representation of an Arrayor 6atri"is

displayed instead of an inline display of the data itself$ 0or more information, see

interface(rtablesi8e)$

'sti!!'= oolean

4oolean that is used only for !"# and DA$ +etting sti!!=true(thoutselecting a

method specifies that the default stiff methods (rosenbroc%or rosenbroc%dae) must

be used instead of the default non-stiff methods (r%f&'or r%f&'dae)$ 9hen themethod is also specified, a consistency chec% is performed to verify that the method

matches with the 'sti!!'value$ 0or more information on the phenomena of stiffness,

see dsolve+tiffness.$

'events'= list

'event#pre'= keyword

'event#ma"iter'= integer

Arguments that are available only for real-valued !"#s or DAs using the non-stiff

and stiff default methods, r%f&', rosenbroc%, r%f&'dae, or rosenbroc%dae, and areused to specify and control the behavior of events$ These options are discussed in

detail in dsolvevents.$

'range'= numeric777numeric

"alues that specify the range of the independent variable over which solution valuesare desired$

0or !"#s and DAs this option is used only by the non-stiff and stiff default methods

(r%f&', rosenbroc%, r%f&'dae, rosenbroc%dae) and the taylorseriesmethod$ !t hastwo purposes for procedure-type output$ !f 'range'is used, then the call to dsolve

computes the solution over the desired range before returning, storing that solution

for later calls to the returned procedure, which then compute the return values through

interpolation$

4otes5

- 9hen computing a numerical solution for a problem that has large regions where

the solution is changing slowly, and small regions where the solution is changing

rapidly, use of 'range'combined with the re!ineoption of odeplot(rk!89:androsenrock:only) allows better plotting of the details of the solution$ !f 'range'is

not used, then the call to dsolvereturns a procedure that does notstore the computed

solution, but rather computes the solution whenever a point is requested$

- 0or long-time integration problems it is suggested that 'range'not be used, as the

storage of the entire solution can consume a fair bit of memory$

/6


13/36

0or the 4"#method, this option is only needed when the 4"# method is to be used

to solve an !"# with a global error bound, as for 4"# the information can be inferred

from the boundary conditions in deqns$ 0or more information, see numeric,4"#$

'aserr'= numeric or list

umeric value that gives a limit on the absolute error tolerance for a successful step

in the !"# and DA cases, and a measure of the maimum error between thecomputed solution and the eact solution in the 4"# case (in all but eceptionalsystems)$ !t is supported by all methods ecept the classicalmethods (as all classical

methods are implemented with a fied step si8e and no error control)$ The list form of

abserr is available for the rk!89:, rosenrock:, and med!i!"# and DA methods,and allows specification of absolute error tolerances that are different for each

dependent variable, or for each solution component$ 3ore detail on this can be found

in the dsolvenumeric,!"#.page and the dsolverror*ontrol.page$

'relerr'= numeric

umeric value that gives a limit on the relative error tolerance for a successful step

for an !"# or DA$ This option wor%s in conEunction with 'aserr'$ !t is supported by

all methods ecept classical, taylorseries, andbvp$

Defaults for 'aserr'and 'relerr'are specific to each method, but it should be notedthat the default error tolerance for the default non-stiff r%f&'method is now fied

(rather than controlled by the value of Digits)$

0or a detailed discussion about error control, see dsolverror*ontrol.$

'known'= name or list o! names

#rovides a list of functions that should be considered as kno(nwhen eamining the

system$ This allows for specification of a system containing user-defined functions$

ach user defined function must always return numerical values when given numericinput, and must return unevaluated when given name input$ 0or an illustration of the

use of this option, see the E"amplessection below$

4ote5!n most cases, use of a system with a user-defined function prevents the use of

evalhfwithin dsolve, so obtaining a solution using this facility will run noticeablyslower than specification of the system in terms of %nown functions (when it is

possible to epress the system in this way)$

All methods ecept taylorseries, rosenbroc%, and rosenbroc%daesupport this option

directly, but for these two methods additional information is required$ The %eyproblem is that these two methods require information on the derivatives of the input

D system, which is not directly available when functions are procedure defined$ !t

is still possible to use these methods with 'known', as long as the derivatives of the

function are defined via use of one or more ;di!!/;rules$ The rosenrockmethodrequires only a single derivative, but the taylorseriesmethod requires derivatives to

arbitrary order, so with taylorseriesit is useful only if some finite derivative of the'known'functions can be represented as a function instead of a procedure, in which

case, all derivatives up to that function derivative must have a di!!rule defined$ The

rosenrock#daemethod may require more than a single derivative, as the DApreprocessing may need to differentiate the input system$

/2


14/36

0or taylorseries, rosenbroc%, and rosenbroc%daeuse of multiple argument functions,

or functions that also depend on the dependent variables or derivatives of the problem

are tric%y, and have some limitations$ The details of this are deferred to the respectivehelp pages$

'optimi


15/36

: dsn0(0);

., ,=t /$ =( ) t A$'&A2A622/BB'>'6=C6 =( )y t -A$>&/&B//A//'BC/=C=

: [t = 05, x(t) = 35393$939$?9@0, y(t) =

"3560?009@3$49406]; ., ,=t /$ =( ) t A$'&A2A62A&2&6B'2C/& =( )y t -A$>&/&B//2CA6'=2=/&

: dsn$ := dsolve(dsys,numeric,[y(t),x(t)]): > s&/&B//A//'BC/=C=& =( ) t A$'&A2A622/BB'>'6=C

: [t = 05, y(t) = "3560?009@3$49406, x(t) =35393$939$?9@0];

., ,=t /$ =( )y t -A$>&/&B//2CA6'=2=/&> =( ) t A$'&A2A62A&2&6B'2C/

+traightforward second order !"#5

: deA0 := (t0)#$'diff(y(t),t,t) (t0)'diff(y(t),t)

((t0)#$"35$)'y(t) = 3;

57de)! =+ +( )+t / 6

d

d6

t6

( )yt ( )+t /

d

d

t

( )yt ( )( )+t / 6 $6' ( )yt

: ic0 := y(3) = 35@?094@?3?006393,

(y)(3) = 3534330??@:

: dsol0 := dsolve(1deA0,ic02, numeric, rane=3550);

57dsol! proc( ) $$$ end prox"rk#$%

: dsol0(3);

, ,=t $ =( )yt $CB/2=CBB/&/>2 =d

d

t( )yt $='&'/&&&B&B&

: dsol0(35);

, ,=t $' =( )yt $C&=>2>/2B2C>>>=>C =d

d

t( )yt -$/B'6='C6>/6>/2=

/'


16/36

: dsol0(0);

, ,=t /$ =( )yt $'/2/C=/&>/6B' =d

d

t( )yt -$2C22=>626&&2B6C

+ame second order !"# with operator output5

: dsol0+ := dsolve(1deA0,ic02, numeric, out2>A/2B2C>>>=>C =( )( )D y A$' -A$/BA'6='C6>/6>/2=6

: dsol0+(0);

., ,=t /$ =( )y / A$'/2A/CA=A/A&>/6B'A =( )( )D y / -A$2C2A2=>6A26&&2B/'

0irst order !"#5

: dsol$ := dsolve(1diff(y(x),x) = y(x)'cos(x), y(3) = 02,

numeric, out2&&C//A/AA$======B>'>&=AC6=

0irst order !"# with piecewise output5

: dsol$< := dsolve(1diff(y(x),x) = y(x)'cos(x), y(3) = 02,

numeric, out


17/36

dsol&* =x x ( )yx ,unde#ned '>'=6C26/$6BB'/6CC6/C/>x $&==&C=&&6=2=B( )x $6BBC6/62/'B/6>6+ +

$/&'2&''2&=2&( )x $6BBC6/62/'B/6>&2

$/22>2'6=''6'/6C( )x $6BBC6/62/'B/6>&& x $'''6&6&C2/&6'C>',

$=>=//''6/22/6/$/&B/=>'6/6/C&x $&>/=///BBB/B'( )x $/&=/26C6>&B=>C+ +

$=&=/>B&=BC&22( )x $/&=/26C6>&B=>C2

$/B62/26C'C'C&2( )x $/&=/26C6>&B=>C& x $6&6'66B=2>/B/&=6',

$=&>2=B&CC/&B>>/$2/'=B&66BC6=/x $2>B&6>C>2CB6C>2( )x $2&'B2>>62'B26+ +

$6&&'B/='6C=/B/=( )x $2&'B2>>62'B2

$66B/2B6&B//C/=( )x $2&'B2>>62'B2& x $&2BC/6&=CCC'&6=CB2,

$>=B&6'6/C>/''26/$&2&B'C6==B>6C'x $/>26>>6/&/2&//'( )x $'&/2CB'22&B='B6+ +

$&26C2&B&C&/&6/B( )x $'&/2CB'22&B='B2

$66'>>C='>>66=>( )x $'&/2CB'22&B='B& x $C&'/66'B6=2B/&'/,

$>=>6B6==2C2&C/$&&>2=6&26&Cx $/226''=B6&>C&>>( )x $B&2C6>'&>B=/BC6+

$'=>B&'CC>&CB'CB( )x $B&2C6>'&>B=/BC2

$/''/=''2C=>/( )x $B&2C6>'&>B=/BC&

x $>&6/2&'//&C&C&/C>=,

$=>/B&&&26>==/$2&/22B>66B&Bx $&BB6&6/C=>'66( )x $=6/CB6''B2626/6+

$CB2'22>B&==>=/( )x $=6/CB6''B2626/2

$/&2C''B6CB==B>( )x $=6/CB6''B2626/& x /$,,unde#ned other(se.

+tiff nonlinear problem5

: dsys9 := diff(x(t),t) = x(t)#$ " t#@, x(3)="350;

57dsys' ,=d

d

t( )t ( )t 6 tC =( ) -$/

: dsol9 := dsolve([dsys9], ty


18/36

57dsol'

.,t ( )t

$ -$/

$6'-$=B'C=B&B'//CB'2

$'-$=C2&B26'26B22=6&

$B'-$///B&&B=//62==2B>

/$ -$66=6B=6C&2'=2>22

=B'' =ddx

( )yx $C26CBB>'C>&C&&C

: dsol($);

, ,=x 6$ =( )yx /$6/6= =d

d

x( )yx /$=B&//=6&2C'6&>

onlinear 4"#5

: dsys := 1diff(y(x),x,x) a+s(y(x)) = 3, (y)(3) = 0,

y(0) = "02;

57dsys% ? @, ,=+d

d6

x6

( )yx ( )yx =( )y / -/ =( )( )Dy /

: dsol := dsolve(dsys, numeric,out


19/36

57dsol%

, ,x ( )yx d

d

x( )yx

$ -/$&=C&>&6='>B'6&62/$&&

$6' -/$6/2/B'2=C>/''=/=$CB'2/>'&&/BB/&C&6

$' -/$C>&C>C''/C2C=$2=2C&B>C//62=>C&'

$B' -/$6B2/'6B/2/&=2'$/2'''B=&>=C=>2

/$ -$===============>=-$//2'2=>>62C/'6'>B

DA5

: dsys@:= 1

diff(x(t),t,t) = "$'lam+da(t)'x(t),

diff(y(t),t,t) = "$'lam+da(t)'y(t)"!i#$,

x(t)#$y(t)#$ = 0, x(3)=3, (x)(3)=0&03, y(3)="0, (y)(3)=32;

dsys+ =+( )t 6 ( )yt 6 / =d

d

t6( )t 6 ( )t ( )t =

d

d

t6( )yt 6 ( )t ( )yt 6 =( ) =( )y -/, , , , ,?57

=( )( )Dx /

/=( )( )Dy , @

: dsol@ := dsolve(dsys@, numeric);

57dsol+ proc( ) $$$ end prox"rk#$%"dae

: dsol@(0);

=t /$ =( )t &$=2=>66B2=>'BC =( )t $C22CB22>&2226CC6//-', , ,

=d

d

t( )t -$/C&=B'B2>/B =( )yt -$==========B>>B/, ,

=d

d

t( )yt -$C22CB2B'==/C=&2/

-C

Hse with !solve5

: dsys? := 1diff(y(x),x)=$'y(x), y(3)=02;

57dsys, ? @,=d

d

x( )yx 6 ( )yx =( )y /

/=


20/36

: iits := $3;

57Dgts 6A

: dsol? := dsolve(dsys?, numeric, met-od=ear, a+serr=0e"0@,

relerr=0e"0@, out


21/36

end /&=6=&B==C/C&/

: dsol6($);

.,=x 6$ =( )y x &$/=/CB==2///CA='=

A method that requires derivatives,

: dsolve(dsys6, numeric, met-od=rosen+roc7, 7non=f);

Error !in dsolve"numeric"ceckknown% use of 'rosenbrock'metod wit procedure defined function f re&uires a

differentiation rule ,diff"f,which failed because a differentiation rule is required$ 9e then clear the remember table

of diff and define this as5

: foret(diff);

: Ediff&fE := C-ain rule

6/


22/36

diff(ars[0],ars[$])'ex


23/36

4oyce, 9$$, and Di#rima, F$*$ Elementary Di!!erential Equations and oundary

&alue Prolems$ ew Nor%5 Oohn 9iley P +ons, /==B$

*ash, O$F$ MThe !ntegration of +tiff !"# in D Hsing 3odified tended 4D0$MComputers and 6athematics with Applications$ "ol$ ?$ (/=>2)5 C&'-C'B$

Qear, *$9$ 4umerical .nitial &alue Prolems in 0rdinary Di!!erential Equations$

#rentice-Lall, /=B/$

Lindmarsh, Alan *$; +tepleman, F$+$; et al, eds$ 0depack3 a Systemi2$

Lubbard O$L$, and 9est, 4$L$ Di!!erential Equations5 A Dynamical Systems

Approach$ ew Nor%5 +pringer, /==$ #art .$ 0ne Dimensional Equations$

Lull, T$$; nright, 9$L$; 0ellen, 4$3$; and +edgwic%, A$$ M*omparing umerical3ethods for rdinary Differential quations$M S.A6 @7 4umer7 Anal7"ol$ ?$ (/=B6)5

C2-C2B$

+hampine,


24/36

Description

The dsolvecommand with the options numericand method=classicalormethod=classical1choice2finds a numerical solution by using one of the classcalnumerical methods as described in the following$

These methods use a fied step si8e, provide no error estimation or correction, and are

provided primarily for educational purposes$ !f practical numerical solution of D

initial value problems (!"#) is required, it is suggested that any of the other methodsmentioned in dsolvenumeric,!"#.help page be used instead$

A number of classical methods are available through use of the choiceoption in themethod=classical1choice2selection$ The default is !oreuler, the forward uler

method$

The available choices of the classicalmethod are described for the problem y'=!(t3y*,where 1i2is the estimated value of the solution at the time t1i2, his the fied step

si8e t1i2+t1i+,2, and for each formula the value of the solution 1nB,2at time t1nB,2is

being computed$

!oreuleris the#or(ard Eulermethod specified by the equation5 /n0+ = /n 0 1f!tn /n%

heun!ormis the.eunformula (also %nown as the improved uler method)$ !t usesthe#or(ard Eulermethod to predict the solution, and then applies the tra*e/od rule

as a corrector$ !t is specified by the pair of equations5

/p = /n 0 1f!tn /n%

/n0+ = /n 0 !"(%1!f!tn/n% 0 f!tn0+/p%%

impolyis the m*roved *olygonmethod (also %nown as the modified uler method),

as specified by the equation5 /n0+ = /n 0 1!f!tn0"( /n0!"(%1f!tn/n%%%

rk-is thesecond-order classcal Runge-Kuttamethod, as specified by5 k+ = f!tn /n%

k( = f!tn0 /n01k+%

/n0+ = /n 0 !"(%1!k+0k(%

4ote5This is the same as the.eunformula$

rk>is the thrd-order classcal Runge-Kuttamethod, as specified by5 k+ = f!tn /n%

k( = f!tn0!"(% /n0!"(%1k+%

k2 = f!tn0 /n01!-k+0(1k(%%

/n0+ = /n 0 !"3%1!k+041k(0k2%

6&


25/36

rk8is the#ourth-order classcal Runge-Kuttamethod, as specified by5 k+ = f!tn /n%

k( = f!tn0"( /n0!"(%1k+%

k2 = f!tn0"( /n0!"(%1k(%

k4 = f!tn0 /n01k2%

/n0+ = /n 0 !"3%1!k+0(1k(0(1k20k4%

This is not to be confused with method=rk!89, which uses aFehlberg #ourth-##thorder Runge-Kutta method$

adamashis the0dams-1ash#orthmethod (a predictor method), as specified by5 /n0+ = /n 0 !"(4% 1 !551f!tn/n% - 561f!tn-+/n-+%

0 271f!tn-(/n-(% - 61f!tn-2/n-2%%

amoultonis the0dams-1ash#orth-Moultonmethod (a predictor-corrector method),as specified by5

/n0+ = /n 0 !"(4% 1 !61f!tn0+/n0+% 0 +61f!tn/n%

- 51f!tn-+/n-+% 0 f!tn-(/n-(%%

where !(t1nB,231nB,2*is found by first applying the Adams-4ashforth method (the

predictor), then using the above Adams-4ashforth-3oulton method (the corrector)$

The following options are available for the classicalmethod5 'output' = keyword or array


'maxfun' = integer

'number' = integer

'procedure' = procedure

'start' = numeric

'initial' = array

'procvars' = list




'stepsize' = numeric

6'


26/36

'corrections' = integer

'output'

+pecifies the desired output from dsolve, and the knownoption specifies user-

defined %nown functions$ 0or more information, see dsolvenumeric.$

'ma"!un'

+pecifies a maimum on the number of evaluations of the right-hand side of the firstorder D system$ This option can be disabled by specifying ma"!un=)$ The default

value for classicalmethods is 9))))$ 0or more information, see dsolvemafun.$

'numer', 'procedure', 'start', 'initial', and 'procvars'

These options are used to specify the !"# using procedures$ 0or more information,

see dsolvenumeric,!"#.$

'startinit','implicit', and 'optimi


27/36

: init0 := y(3) = 0, (y)(3) = $, (GG$)(y)(3) = "0,

x(3) = , (x)(3) = 59;

57nt! , , , ,=( )y / =( )( )Dy 6 =( )( )( )D( )6

y -/ =( ) & =( )( )Dx &$2

: ans0 := dsolve(1dsys0,init02, numeric,

met-od=classical[a+moulton], corrections=$);

57ans! proc( ) $$$ end prox"classcal

: ans0(053);

=t /$ =( )t /2$/6&C=2B=6B/'C>2 =d

d

t( )t />$B>BC=/'2'/&6/', , ,

=( )yt 2$BC6>2&/262C/6C =ddt

( )yt '$2/BC2=BC>6/C2C6 =dd

6

t6

( )yt /$6'&C>2C62'=C2, ,

: deA$ := diff(y(x), x$) = y(x);

57de)& =d

d6

x6

( )yx ( )yx

: init$ := y(3) = 0, (y)(3) = 0;

57nt& ,=( )y A / =( )( )D y A /

: iits := $3:

ans$ := dsolve(1deA$, init$2, numeric,

met-od=classical[-eunform],

out


28/36

57ans&

, ,x ( )yx d

d

x( )yx

$/ /$/'/C'&6/2/B/'/$/'/C'&6/2/B/

$/ /$/'/B>==CC==&/'>>/$/'/B>==CC==&/'>>

/$ 6$B/>6>/2B'B'/BC>2==6$B/>6>/2B'B'/BC>2=

: iits := 03:

deA9 := 1 diff(y(t),t) " diff(y(t),t9) = y(t)'t#$ 2;

57de)' ? @=d

d&

t&

( )yt

dd

2

t2

( )yt ( )yt t6

: init9 := 1 y(3) = 95@, (y)(3) = 0$, (GG$)(y)(3) = ",

(GG9)(y)(3) = @5 2;

57nt' ? @, , ,=( )y 2$'C =( )( )Dy /6 =( )( )( )D( )6

y -& =( )( )( )D( )2

y C$'&&

: ans9 := dsolve(deA9 union init9, numeric,

met-od=classical[r79], outB'A&B=6&>AC=&=

See Also

dsolvedver%B>., dsolvegear., dsolvelsode., dsolvemafun., dsolvenumeric.,

dsolvenumeric,!"#.,plotsodeplot., dsolver%f&'., dsolverosenbroc%.,

dsolvetaylorseries.

e!erences

4oyce, 9$$, and Di#rima, F$*$ Elementary Di!!erential Equations and oundary&alue Prolems$ 'th ed$ ew Nor%5 9iley, /==6$

*onte, +$D$, and *$ de 4oor$ Elementary 4umerical Analysis3 An Algorithmic

Approach$ 3cQraw-Lill, /=>$

6>


29/36

0o, B$


30/36

This method can be used in combination with the re!ineoption of odeplotto produce

an adaptive plot (that is, odeplotuses the precomputed points to produce the plot

when re!ineis specified$)

!t is not recommended that this method be used for problems in which the solution

can become singular, as each step is stored, and many steps may be ta%en when near a

singularity, so memory usage can become a significant issue$The storage of the interpolant in use by this method can be disabled by using theinterpolation=!alseoption described below$ This is recommended for high accuracy

solutions where storage of the interpolant (in addition to the discrete solution)

requires too much memory$ Disabling the interpolant is not generally recommendedbecause the solution values are obtained from an interpolation of the ' closest points,

and does not necessarily provide an interpolant with order & error$

9ithout the rangeoption9hen used without the rangeoption, the !"# solution values are not stored, butrather computed when requested$

4ecause not all solution values are stored, computation must restart at the initialvalues whenever a point is requested between the initial point and the most recently

computed point (to avoid reversal of the integration direction), so it is advisable tocollect solution values moving away from the initial value$

0ptions

The following options are available for the rk!89method$'output' = keyword or array


'abserr' = numeric

'relerr' = numeric

'initstep' = numeric

'interr' = boolean

'maxfun' = integer

'number' = integer

'procedure' = procedure

'start' = numeric

'initial' = array

'procvars' = list




2


31/36

'compile' = boolean or 'auto'


'events' = list



'event_iterate' = keyword

'event_initial' = boolean

'complex' = boolean

output

+pecifies the desired output from dsolve$ The %eywords procedurelist,

listprocedure, or operatorprovide procedure-type output, the %eyword piecewiseprovides output in the form ofpiecewisefunctions over a specified range of

independent variable values, and a /-D array or Array provide output at fied values

of the independent variable$ 0or more information, see dsolvenumeric.$

known

+pecifies user-defined %nown functions$ 0or more information, see dsolvenumeric.$

aserr, relerr, and initstep

+pecify the desired accuracy of the solution, and the starting step si8e for the method$

0or more information, see dsolverror*ontrol.$ The default values for rk!89areaserr = 7,e+and relerr = 7,e+9$ The value for initstep, if not specified, is

determined by the method, ta%ing into account the local behavior of the D system$interr

4y default this is set to true, and controls whether the solution interpolant error(including the interpolant on inde-/ variables for DA problems) is integrated into

the error control$ 9hen set to !alse, areas where the solutions is varying rapidly (e$g$

a discontinuity in a derivative due to a piecewise) may have a much larger solution

error than dictated by the specified error tolerances$ 9hen set to true, the step si8e isreduced to minimi8e error in these regions, but for problems where there is a Eump

discontinuity in the variables, the integration may fail with an error indicating that a

singularity is present$ !n the latter case where an error is thrown, it may beadvantageous to model the discontinuities using events (see dsolvevents.)$

ma"!un

+pecifies a maimum on the number of evaluations of the right-hand side of the first

order D system$ This option is disabled by specifying ma"!un = )$ The defaultvalue for rk!89is >))))$

numer, procedure, start, initial, and procvars

These options are used to specify the !"# using procedures$ 0or more information,

see dsolvenumeric,!"#.$

2/


32/36

startinit, implicit, and optimior gearmethodsbe used instead$

Fesults can be plotted by using the function odeplotin the plotspac%age$

E"amples

+olution with a range - solution over ) 77 ,is stored5

: dsys0 := 1diff(x(t),t)=y(t), diff(y(t),t)=x(t)y(t),

x(3)=$, y(3)=02;

26


33/36

57dsys! ? @, , ,=d

d

t( )t ( )yt =

d

d

t( )yt +( )t ( )yt =( ) 6 =( )y /

: dsol0 := dsolve(dsys0, numeric, out6C>>=6C&===&/'B

+olution without a range - each point is computed when requested5

: dsol$ := dsolve(dsys0, numeric, met-od=r7f,

out


34/36

: dsol9 := dsolve(dsys9 union init9, numeric, met-od=r7f,

out$=C2&/6//=2C'C&66>-$/>&>C&&>/6/BB>/ , , ,

$C2//6>2>BB6/'''''/$/&>6/>&'6B&>B&/B=, .

/$/=$BCC=/&BB=62/C/&&-$C'&2'=>'/>=266C= , , ,

/$6C==C'>/=B2&&66/2/$&6/6B&C66>/C2>BB6, .

/$'$2>BB&>&6>62BC&=&2&-/$6>6B&22C'B/&62C= , , ,

/$>>>&22>/CC'B'C///$CB&'6BC&>=B6'B, .

6$6==============>6-/$2=&'BCC/=='CB2B=-2$2B6B62'=C2/26'&= , , ,

2$2B=2==22=2&/B'&/$=B>/&C==&2/B'>=, .

6$'-6$/2=2&/CC//'=2&BC'-&$>>'>'C62/B/'6/ , , ,

2$BB2CBC//=B&6&>6/$=&>B/C=//'B>C', ..

procedurelistoutput5

: deAn := 1 diff(y(t),t9) " $'diff(y(t),t$) $'y(t) 2;

57de)n$ ? @ +d

d2

t2( )yt 6

d

d6

t6( )yt 6 ( )yt

: init := 1 y(3)=0, (y)(3)=0, (GG$)(y)(3)=0 2;

57nt$ ? @, ,=( )y / =( )( )Dy / =( )( )( )D( )6

y /

: dsol := dsolve(deAn union init, numeric, met-od=r7f,

relerr=Float(0,"6), a+serr=Float(0,"6),

maxfun=3, out


35/36

: dsol(353);

=t $' =( )yt /$'/6&=&C2'22>6C2 =d

d

t( )yt /$&=='CB2>CB>&2, , ,

=dd

6

t6 ( )yt $==B2B/>6/'2=>2C6&

: dsol("35);

=t -$&' =( )yt $C&>C2B>6>'>B/C' =d

d

t( )yt $'B/B'B'2'B>'&'6, , ,

=d

d6

t6

( )yt $>B>=B/2CB'2'>&/

#iecewise output for comple-valued problem5: deAn := 1 diff(y(t),t) = D'y(t) 2;

57de)n% ? @=d

d

t( )yt ( )yt 2

: init := 1 y(3)=0 2;

57nt% ? @=( )y A /

: dsol := dsolve(deAn union init, numeric, met-od=r7f,

out=B= +-/$C&2 $62>>'&&2&/- 2

See Also

2'


36/36

dsolverror*ontrol, dsolveclassical., dsolvedver%B>., dsolvevents., dsolvegear.,

dsolvelsode., dsolvemafun., dsolvenumeric,!"#., dsolvenumeric.,

dsolverosenbroc%., dsolve+tiffness., dsolvetaylorseries., infolevel,plotsodeplot.

e!erences

nright, 9$L$; Oac%son, G$F$; orsett, +$#$; and Thomsen, #$Q$ M!nterpolants forFunge-Gutta 0ormulas$M AC6 %06S, "ol$ ,-$ (/=>C)5 /=2-6/>$

0ehlberg, $ MGlassische Funge-Gutta-0ormeln vierter und niedrigerer rdnung mit

+chrittweiten-Gontrolle und ihre Anwendung auf 9aermeleitungsproblemeM$

Computing$ "ol$ $ (/=B)5 C/-B/$

0orsythe, Q$$; 3alcolm, 3$A$; and 3oler, *$4$ Computer 6ethods !or

6athematical Computations$ ew Oersey5 #rentice Lall, /=BB$

+hampine,

Date post:	17-Feb-2018
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Maple Dsolve Numeric

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