rtaylor - obtain the Taylor series for an ODE or PDE system

rtaylor( solved , options )

rtaylor( solved , vars , options )

Parameters

solved - system in solved form

vars - (optional) solving variables of the system

options - (optional) sequence of options to specify the ranking for the solved form, initial data, and the order of the Taylor series

• The rtaylor function uses an output rifsimp form to obtain local Taylor series expansions for all dependent variables in the ODE or PDE system simultaneously. The Taylor series output is a list containing equations of the form depvar(indepvars)=Taylor series .
• The ranking related options that are accepted by rtaylor include the specification of the vars as a ranking, and the ranking and indep options described in rifsimp[ranking] , rifsimp[options] , and rifsimp[adv_options] .

• Note : specification of different vars than those used to obtain the result from rifsimp can give incomplete results.

• The order=n option specifies the order that the Taylor series should be computed to, and must be a non-negative integer. The default value is 2 .
• The point=[ivar1=val1,...] option specified an expansion point for the series. When this option is used, every independent variable must be given a value.
• The table resulting from a call to initialdata can be given as an option to rtaylor , in which case the Taylor series will be given in terms of the functions present in the initial data.

• In addition, the arbitrary functions and constants on the right hand sides of the specified initial data can be given specific values, and the expansion can be computed for these values.

> with(Rif):

A simple ODE

> rtaylor([diff(f(x),x,x)=-f(x)],order=4);

A PDE system with a single dependent variable

> rtaylor([diff(f(x,y),y,y)=diff(f(x,y),x)*f(x,y),
diff(f(x,y),x,x)=2*f(x,y)], order=3);

A PDE system with two dependent variables

> rtaylor([diff(f(x,y),x,x)=diff(g(x,y),y),
diff(f(x,y),y,y)=diff(g(x,y),x),
diff(g(x,y),x)=diff(g(x,y),y)]);

An example using initial data

> sys := {diff(f(x,y),x,x)=0,diff(f(x,y),x,y)=0};

> id := initialdata(sys);

> rtaylor(sys, id, order=3);

An example using specified initial data and an expansion point

> ids := eval(eval(id),{_F1(y)=sin(y),_C1=1});

> rtaylor(sys, ids, order=3, point=[x=1,y=Pi]);

caseplot , rifsimp , rifsimp[nonlinear]