Efficient families of higher-order Caputo-type numerical schemes for solving fractional order differential equations

Fractional calculus has gained widespread application in various scientific and engineering disciplines due to its ability to model systems with memory effects, long-range dependencies, and non-local behaviors. However, traditional analytical methods often prove insufficient for solving the complex nonlinear fractional problems encountered in these fields. In this paper, we introduce two novel families of numerical methods designed to approximate solutions for such nonlinear fractional problems. Theoretical and numerical analyses confirm the effectiveness of the proposed schemes, which achieve convergence orders of 2ß+1 and 3ß+1. Compared to existing methods, the newly developed approaches demonstrate improved convergence behavior, stability, and computational efficiency. By leveraging fractional calculus, these methods provide deeper insights and practical solutions for tackling complex nonlinear problems in science and engineering.