LdeApprox[ToRatCoeffs]  - gives rational approximation of LDE coefficients over given interval

Calling Sequence:

     ToRatCoeffs(lde, x = a..b, [m, n])

Parameters:

     lde       - linear differential equation

      x          - independent variable

      a, b     - numerical values specifying the interval of approximation    

      m        - integer specifying the desired degree of the numerator

      n        - integer specifying the desired degree of the denominator

Description:

• If the given LDE has non-polynomial coefficients one can use the package procedure ToRatCoeffs  which gives a rational approximation of the coefficients. This procedure applies numapprox[minimax]  procedure to each non-polynomial coefficient of the LDE.
• It should be pointed out that the coefficients can not involve indeterminate variables except independent one. This is a restriction of numapprox  package methods as pure numerical ones.

Examples:

This loads the package.
restart:
with(LdeApprox):

This is a simple LDE with some non-polynomial coefficients.
lde:=diff(y(x),x)+erf(x)*y(x)=0;

Trying to find approximate solution of the IVP:  .
apr = ApproxSol({lde, y(0) = 1}, y(x), x=-1..1, 9);

Using ToRatCoeffs to convert the LDE into approximate one with polynomial coefficients on the given interval.
lde1 := ToRatCoeffs(lde, y(x), x=-1..1, 9);

Trying ApproxSol for the new approximate LDE.
apr := ApproxSol({lde1, y(0) = 1}, y(x), x=-1..1, 9);

The exact solution of the IVP is as follows:
sol:=dsolve({lde, y(0) = 1},y(x));

Comparing the results.
plot(subs(sol,y(x))-subs(apr,y(x)),x=-1..1);

See Also:

LdeApprox