An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative

The vast applicability of fractional calculus to model physical phenomena in the form of fractional differential equations and their complexity has created a massive demand for efficient analytic and semi-analytic techniques to solve fractional differential equations. This paper has derived a new operational matrix using the independence polynomial of a complete bipartite graph to solve multi-order fractional differential equations. While deriving the operational matrix, the Caputo sense fractional derivatives have been considered. Series solutions are found by using the collocation matrix method. The main characteristic of this approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The convergence analysis and the time complexity analysis of the proposed scheme are also presented in this paper. Six examples have been considered to illustrate the relevance and applicability of the method described. The results obtained are compared with the exact solutions. We have also compared our results with the ones obtained by other methods available in the literature. © 2021, The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada.