Existence, Uniqueness, Stability, and Numerical Solution of Second‐Order Quantum Difference Equations: A General Approach via the Riccati Method and Chebyshev Collocation

In this paper, we investigate the stability and numerical solution of second‐order linear nonhomogeneous equations with the general quantum ‐difference operator. We prove Hyers–Ulam stability ( s ) and Hyers–Ulam–Rassias stability ( s ) for these equations using a Riccati equation approach and variation of parameters technique. Furthermore, we develop a robust numerical scheme based on the second‐kind Chebyshev collocation method with first‐kind Chebyshev nodes, combined with a suitable Chebyshev–Gauss quadrature formula for approximating the right‐hand side of the differential equation. Convergence theorems with rigorous proofs are established, and a detailed algorithm is provided. Three comprehensive numerical examples demonstrate the efficiency and accuracy of the proposed method, with tables reporting absolute errors, logarithmic errors, and CPU times. This work generalizes prior results on Hahn difference equations and provides new computational insights into quantum difference equations.