Numerical Computation for Nabla Fractional Order Systems via Time-Frequency Domain Joint Technique

This paper proposes an algorithm that combines time and frequency domain methods to compute the time-domain response of Nabla fractional order systems. The algorithm first uses frequency-domain techniques to estimate the system's initial values and then refines them in the time domain to improve accuracy. From a frequency-domain perspective, the problem is reformulated as an operator approximation task. Inspired by the Bode plot, the method approximates fractional difference and sum operators during the transition from continuous to discrete domains to obtain the initial values. In the time domain, the system state at each step is expressed as an equation based on the properties of Nabla calculus. By enforcing the convergence of the equation's solution to zero, the problem is transformed into an optimization task, allowing iterative refinement of the time-domain response.