We propose as a model a sixth degree polynomial. Why a sixth degree polynomial? Because polynomials have continuous first, second and third derivatives. And because a sixth degree polynomial has seven coefficients to be determined and we have 7 constraint equations - see below.
Sixth degree polynomial -- proposed solution
Input :=
y1[q_] = Sum[ a[i] q^i, {i,0,6} ]
Output =
2 3 4 5
a[0] + q a[1] + q a[2] + q a[3] + q a[4] + q a[5] +
6
q a[6]
Constraint equations
Input :=
e0 = y1[0] == 0;
e1 = y1'[0] == 0;
e2 = y1''[0] == 0;
e3 = y1[Pi/2] == .1;
e4 = y1''[Pi] == 0;
e5 = y1'[Pi] == 0;
e6 = y1[Pi] == 0;
Solution for undetermined coefficients
Input :=
sol1 = Solve[ {e0, e1,e2,e3,e4,e5,e6},
{a[0],a[1],a[2],a[3],a[4],a[5],a[6]}]
Output =
-16
{{a[0] -> 0., a[1] -> 1.49403 10 , a[3] -> 0.20641,
a[4] -> -0.197107, a[5] -> 0.0627411,
a[6] -> -0.00665703, a[2] -> 0.}}
Substitution to create sixth degree polynomial solution
Input :=
y1[q_] = y1[q] /. sol1[[1]] //Chop
Output =
3 4 5 6
0.20641 q - 0.197107 q + 0.0627411 q - 0.00665703 q