Application of (q, τ)-Bernoulli Interpolation to the Spectral Solution of Quantum Differential Equations

In order to solve fractional differential equations on quantum domains, this work provides a spectral approach based on higher-order (q, τ)-Bernoulli functions and polynomials. We build a robust basis for approximation in (q, τ)-weighted Hilbert spaces by using the orthogonality properties of these extended polynomials and the Sheffer-type generating function. Prototype equations of the form D_q,τu(x) = f(x) are numerically solved using the (q, τ)-Lagrange interpolation approach modified to represent arbitrary functions in terms of Bernoulli bases. Spectral expansion is used to recreate the solution, and a thorough example is given. The technique shows spectral convergence and shows how well higher-order (q, τ)-Bernoulli systems capture the global structure and local behavior of fractional quantum calculus solutions.