I've been able to solve cubic spline functions given values of the function using the code below.
F1[t_] = k1 + k2*t + k3*t^2 + k4*t^3;
F2[t_] = k5 + k6*t + k7*t^2 + k8*t^3;
F3[t_] = k9 + k10*t + k11*t^2 + k12*t^3;
F4[t_] = k13 + k14*t + k15*t^2 + k16*t^3;
F5[t_] = k17 + k18*t + k19*t^2 + k20*t^3;
Solve[{F1[0] == 0, F1[5] == 3, F2[5] == 3, F1'[5] == F2'[5],
F1''[5] == F2''[5],
F2[15] == 11, F3[15] == 11, F2'[15] == F3'[15], F2''[15] == F3''[15],
F3[30] == 16, F4[30] == 16, F3'[30] == F4'[30], F3''[30] == F4''[30],
F4[45] == 19, F5[45] == 19, F4'[45] == F5'[45], F4''[45] == F5''[45],
F5[72] == 20, F1'[0] == 0, F1''[0] == 0}]
What I would like to be able to do is to solve these symbolically like:
Solve[{F1[t0] == Fi1, F1[t1] == Fi2, F2[t1] == Fi2,
F1'[t1] == F2'[t1], F1''[t1] == F2''[t1],
F2[t2] == Fi3, F3[t2] == Fi3, F2'[t2] == F3'[t2],
F2''[t2] == F3''[t2],
F3[t3] == Fi4, F4[t3] == Fi4, F3'[t3] == F4'[t3],
F3''[t3] == F4''[t3],
F4[t4] == Fi5, F5[t4] == Fi5, F4'[t4] == F5'[t4],
F4''[t4] == F5''[t4],
F5[t5] == Fi6, F1'[t0] == 0, F1''[t0] == 0}]
for just the k values i.e. I don't want it to think of the Fi's and t's as unknowns but rather as values that I can then plug in.