I try to solve numerically a boundary VP for ODE with different order of discontinuity of right part.
Say, the following BVP is given:
Let's use piecewise right part
F := piecewise(x<=1, -x, x>1, 2*x+(x-1)^2)
piecewise(x<=1, 1-x, x>1, (x-1)^2)
as obviuos, satisfies the BVP exclung the point x=1, where its 1st and 2nd derivatives are discontinuos.
As:=dsolve([diff(y(x), x$2)+diff(y(x), x)+y(x)=F, y(0)=1, y(2)=1], y(x), type=numeric, output = Array([seq(2.0*k/N0, k=0..N0)]), 'maxmesh'=500, 'abserr'=1e-3):
provides the solution essentially different to exact one described above:
But if to use the right part
F := piecewise(x<=1, x^2+x+2, x>1, -x^2+x)
for which the function
piecewise(x<=1, 1-x+x^2, x>1, -1+3*x-x^2)
satisfies the BVP excluding x=1, where this function has discontinuity of 2nd derivative only, the corresponding numerical solution is very similar to this exact solution:
This reason of the difference between these two cases is clear. In the first case both 1st and 2nd derivatives are discontiuos, while in the second one -- 1st derivative is contiuos.
I wonder, if there are numerical methods, implemeted in Maple, for numerical solution of the first type BVP with non-smooth right part?