Fractional Caputo-type simultaneous scheme for finding all polynomial roots

A wide range of scientific fields, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and in particular biomedical engineering, can adequately describe a number of real-world scenarios using fractional calculus. Numerous fractional calculus problems can be resolved using various derivative types. In this chapter, we propose a modified Caputo-type fractional Weierstrass method for simultaneously finding all polynomial roots. The ordered convergence of the proposed family of methods is 2σ+1, as shown by convergence analysis. The numerical results of the test examples illustrate that the newly proposed method performs better than the other classical fractional iterative scheme previously used in the literature in terms of residual error, computing time, computational order of convergence, basins of attraction, efficiency, and absolute error.