Approximate solutions for fractional partial differential equations using new operational matrices and collocation technique

Abstract This article presents a novel computational framework for developing operational matrices (matrix representations of fractional derivatives) constructed using generalised clique polynomials. Unlike conventional approaches that offer only approximate representations, the developed matrices exactly characterise the Liouville–Caputo and Caputo–Fabrizio fractional derivatives. The proposed matrix-based formulations significantly enhance the precision of approximate solutions while reducing computational complexity compared to traditional numerical techniques. The developed methodology is applied to a class of space and time fractional partial differential equations of orders within the interval (0, 2], encompassing well-known models such as the Klein–Gordon, Burgers, and diffusion–wave equations, among others. The method produces approximate solutions that show excellent agreement with exact solutions, particularly for polynomial-type problems, where it achieves notably higher accuracy than existing methods. Furthermore, it remains robust for non-polynomial cases, demonstrating superior precision and numerical stability over previously established techniques. By transforming the fractional differential equations into a system of algebraic equations through the use of operational matrices, the approach employs a collocation-based numerical strategy to compute solutions with high efficiency and minimal computational overhead. Additionally, the study demonstrates that increasing the computational precision and the degree of the generalised clique polynomials further improves the accuracy of the obtained results. Extensive numerical experiments validate the proposed framework, confirming its accuracy, stability, and computational efficiency, thereby establishing it as a powerful and reliable alternative to conventional numerical methods.