Computing integer- and fractional-order $ q $-$ h $ boundary-value problems

We study integer- and fractional-order Dirichlet problems generated by the affine map $ \beta_a(x) = a+q(x-a)+qh $, where $ 0 < q < 1 $ and $ h > 0 $. On the finite orbit $ x_k = \beta_a^k(a) $ with $ N+1 $ nodes, the second-order problem is written entirely on consecutive orbit nodes. For arbitrary prescribed data we obtain a unique solution through an explicit nonnegative $ (N+1)\times(N-1) $ Green matrix. Its row sum is exactly $ (x_k-a)(b-x_k)/(1+q) $, giving the computable bound $ M_N\le (b-a)^2/[4(1+q)] $. Consequently, the nonlinear problem has a nonnegative solution in $ [0, R]^{N-1} $ when $ F_RM_N\le R $, and that solution is unique when $ L_fM_N < 1 $. For constant loading the closed form is $ y(x) = (x-a)(b-x)/(1+q) $, and, with $ N $ and $ h $ fixed, the scheme converges to the forward $ h $-difference problem as $ q\to1 $. For $ 1 < \alpha\le2 $, we also record a branchwise Riemann–Liouville affine integral obtained by exact conjugacy with the Jackson $ q $-integral. The conjugacy yields the corresponding commutation formula, including its orientation signs and boundary terms. The fractional Dirichlet problem is posed on one fixed geometric orbit, and there the Green representation is an identity of orbit sums, and a compact cone operator gives a positive solution. Finally, we prove the sampled-kernel integer-order consistency between the two constructions: as $ \alpha\to2^- $, samples of the fractional kernel converge entrywise to the finite Green matrix after translation to the orbit coordinates.