LdeApprox[ApproxSol]  - find polynomial approximation for solutions of an LDE with polynomial coefficients

Calling Sequence:

     ApproxSol( {lde, cond}, y(x), x = a..b, n, opt)

Parameters:

     lde - an LDE with polynomial coefficients

      y(x) - any indeterminate function of one variable

     n - degree of the polynomial approximation
     x = a..b - interval of the approximation (a, b can be symbolic or numeric)

     cond - (optional)  initial or boundary conditions

      opt - (optional) options

Description:

• ApproxSol  procedure find polynomial approximation for solutions of an LDE with polynomial coefficients using a-method .
• The approximation problem can be treated as initial value problem (IVP) or boundary value problem (BVP) with regular singular points (RSP) or without them. The treatment is as follows.
• If no condition (cond) presented the problem treated as IVP and the initial point of the approximation is the middle point of the giving interval: x0=(a+b)/2 (see Example 1 ).
• If condition (cond) is one point condition (for instance cond is y(x0)=0, D(y)(x0)=1, ... with one point x0) then  the problem treated as IVP and x0 must be the middle point or the left point of the approximation interval, thus  x0 = (a+b)/2 or x0 = a (see Example 2 ).
•  If condition (cond) is two points condition (for instance cond is y(x0)=0, D(y)(x1)=1, ... with two points x0 and x1) then the problem is treated as BVP and it must be x0 = a and x1 = b or x0 = b and x1 = a (see Example 3 ).
• If an LDE has an RSP then it must be the left point of the interval of approximation (see RSP examples ).
• The options available are (with default values):
  method = classic  - method of approximation (see Example 4 );
  exact = true - mode of calculation (makes effect for homogeneous BVP only; see also LdeApprox[Normalize] ) , if exact = true then the procedure uses exact arithmetic to calculate corresponding eigenvalues (see Example 5 );
  eigenvalue = NULL  - eigenvalue variable name (makes effect for homogeneous BVP only; see also LdeApprox[Normalize] ), if not specified the procedure assumes that any indeterminate in lde  and cond  can be taken as an eigenvalue variable and  makes calculation for each indeterminate in turn (see Example 5 ) .
• The a-method  is fast saturated and gives uniform approximation (see Example 6 ).

Examples:

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

Example 1

This declares an LDE without any condition:
ivp:={diff(y(x),x,x,x)*(1+x^2) = 1};

As no condition presented  the approximation has been done in the middle of the given interval: x=0..1, so x0=1/2
apr:=ApproxSol(ivp,y(x),x=0..1,3);

Example 2

This declares an IVP with one point condition:
ivp:={diff(y(x),x,x,x)*(1+x^2) = 1, y(0)=A, D(y)(0)=B};

x0 = 0 is the middle point of the interval x = -1..1
apr:=ApproxSol(ivp,y(x),x=-1..1,3);

x0 = 0 is the left point of the interval x = 0..1
apr:=ApproxSol(ivp,y(x),x=0..1,3);

error otherwise
apr:=ApproxSol(ivp,y(x),x=-1..2,3);

Example 3

This declares a BVP with two point condition:
bvp:={diff(y(x),x,x,x)*(1+x^2) = 1, y(0)=A, y(1)+D(y)(0)=B,`@@`(D,2)(y)(0)=C};

x0 = 0 and x1 = 1 then x = 0..1
apr:=ApproxSol(bvp,y(x),x=0..1,3);

error otherwise
apr:=ApproxSol(bvp,y(x),x=-1..1,3);

Example 4

This declares a simple IVP:
ivp:={diff(y(x),x)*(1+x^2) = 1,y(0)=0};

Now one can use ApproxSol function to find polynomial approximation of solution of the IVP by classic or modified a-method

Classic a-method:
apr0:=ApproxSol(ivp,y(x),x=-1..1,5,method = classic);

Modified a-method:
apr1:=ApproxSol(ivp,y(x),x=-1..1,5,method = modified(2));

Exact solution:
sol:=dsolve(ivp,y(x));

Comparing results for classic a-method:
plot(subs(sol,y(x))-subs(apr0,y(x)),x=-1..1);

Comparing results for modified a-method (more uniform and precise approximation):
plot(subs(sol,y(x))-subs(apr1,y(x)),x=-1..1);

Example 5

This declares a homogeneous BVP:
bvp:={diff(y(x),x,x) + lambda*y(x) = 0, y(0)=0, y(Pi)=0};

exact = true - slow mode of calculation (egenvalue is not specified)
apr:=ApproxSol(bvp,y(x),x=0..Pi,3);

exact = false - fast mode of calculation (egenvalue is not specified)
apr:=ApproxSol(bvp,y(x),x=0..Pi,3,exact=false);

declare egenvalue = lambda and exact = false mode of calculation
apr:=ApproxSol(bvp,y(x),x=0..Pi,3,exact=false,eigenvalue=lambda);

Example 6

This declares a simple LDE with polynomial coefficients:
lde := diff(y(x),`$`(x,2))-x*y(x)=0;

Now one can use ApproxSol function to find polynomial approximation of solution of the following IVP:

apr:=ApproxSol({lde,y(0)=0,D(y)(0)=1}, y(x), x=-1..1, 5);

Using Maple™ function dsolve  to get the exact solution of the IVP.
sol:=dsolve({lde,y(0)=0,D(y)(0)=1},y(x));

Now one can compare the results using Maple™ function plot .
plot(subs(sol,y(x))-subs(apr,y(x)),x=-1..1);

The a-method  is fast saturated. To see this one can try to calculate the approximation polynomial of 9-degree or higher.
apr1:=ApproxSol({lde,y(0)=0,D(y)(0)=1}, y(x), x=-1..1, 9);

Comparing the results:
plot(subs(sol,y(x))-subs(apr1,y(x)),x=-1..1);

See Also:

LdeApprox