An Iterative Technique for Solving Time Fractional Systems with Variable Coefficients

This paper presents a comprehensive study on constructing exact and approximate solutions to Cauchy problems for time-fractional systems with variable coefficients. An innovative iterative approach is developed for solving functional equations with initial conditions, combining rigorous mathematical foundations with practical computational efficiency. The proposed technique effectively handles the nonlocal nature of fractional operators through a carefully designed iterative scheme that maintains simplicity while achieving high accuracy. It demonstrates particular strength in solving nonlinear systems with well-defined conditions and variable coefficients, where traditional methods often fail. Through systematic theoretical analysis and numerical validation, we establish the method’s convergence properties and computational advantages, showing its capability to generate both exact closed-form solutions, when available, and high-precision approximations otherwise. The approach remains computationally tractable even for complex cases where variable coefficients and memory effects of fractional systems present significant challenges to conventional solution approaches.