A Comparative Analysis of Wavelet-Based Collocation Algorithms for Solving Systems of Fractional Order Differential Equations

Wavelet-based techniques have attracted the attention of researchers in solving systems of fractional order differential equations (FODEs) since they can detect singularities, are simple, have compact support, and are highly accurate with less computational cost. In this paper, we seek to survey some wavelet-based techniques, which have been applied to solve systems of linear and nonlinear FODEs. In view of this, we outlined or demonstrated how the selected wavelets are utilized to solve systems of FODEs. Also, the advantages and disadvantages of the selected wavelets are stated. By employing these wavelet approaches, the systems of FODEs are transformed to a system of algebraic equations by their operational matrices. These systems created can then be solved using any known technique to determine the unknown wavelet coefficients. Despite their advantages over other numerical methods, however, to apply wavelets for solving systems of FODEs, it is necessary to select a particular method that is most beneficial for the specific application. Research has indicated that for smooth problems, it is better to use wavelets, which possess a higher number of vanishing moments, while for nonsmooth problems, wavelets with fewer vanishing moments tend to be more beneficial. Wavelet-based methods are widely applicable and robust.