Hybrid Spectral-Adomian Method for Nonlinear Stochastic Fractional Integro-Differential Equations on Time Scales

This study explored the challenges of solving nonlinear stochastic fractional integro-differential equations on time scales, which are critical in various applied mathematics and engineering problems. The complexity of these equations and the lack of efficient analytical methods highlighted the need for an improved computational approach.The aim of the research was to develop a hybrid spectral-Adomian decomposition method to solve these types of equations with enhanced accuracy and efficiency. The focus was on bridging existing methodological gaps and providing a robust tool for researchers dealing with stochastic fractional systems. The method involved integrating the spectral method with the Adomian decomposition technique to create a hybrid algorithm. Numerical experiments were carried out on benchmark problems to assess the performance of the proposed method, and computational tests were implemented using MATLAB.The results indicated that the hybrid spectral-Adomian method achieved superior accuracy and faster convergence compared to conventional methods. The numerical solutions closely matched exact results where available, confirming the method’s validity and reliability.The study concluded that the hybrid method is a highly effective approach for solving nonlinear stochastic fractional integro-differential equations on time scales. It demonstrated significant improvements over traditional numerical schemes, making it a valuable addition to computational mathematics.This research contributed a new hybrid numerical method that combines the strengths of spectral accuracy with the flexibility of the Adomian decomposition method. The findings are expected to benefit the field of applied mathematics, particularly in advancing solution techniques for complex stochastic systems, aligning with the journal’s scope.