An Efficient Numerical Approach for Fractional PDEs With Proportional Delays

ABSTRACT In this paper, we present a novel, highly accurate, and efficient numerical method for solving fractional partial differential equations with proportional delays. These equations pose significant challenges in both numerical and analytical analysis due to their inherent nonlinearity and delay parameters. The proposed approach addresses these difficulties by transforming this equation into an equivalent Volterra integral equation using a collocation method and the matrix representation of the fractional integral operator. This process creates a nonlinear system of algebraic equations, and solving it results in an approximate solution to the original problem. The error analysis is investigated, and the results confirm that the method is convergent. Numerical experiments validate the theoretical results and confirm the accuracy and computational efficiency of the proposed approach. Furthermore, comparative studies show that the method outperforms existing numerical techniques in terms of accuracy.