An Efficient Fractional‐Order Shifted Legendre Function for Solving Multiterm High‐order Fractional‐Differential Equations of Caputo‐Type With Convergence Analysis

ABSTRACT This study introduces a novel numerical technique employing fractional‐order shifted Legendre functions (FSLFs) defined on the interval to address multiterm linear fractional‐differential equations (LFDEs) of high fractional‐integer order of the Caputo type, subject to mixed boundary conditions. The methodology leverages FSLFs in conjunction with the Gauss‐Lobatto point technique. The core objective of this approach is to transform the LFDEs and their associated mixed boundary conditions into a system of matrix equations. The proposed procedure's effectiveness and reliability are thoroughly evaluated via absolute error and error reduction analyses, juxtaposing the computed solutions with exact solutions and established findings from the literature. Furthermore, a residual error correction strategy is integrated to significantly improve the accuracy of the numerical results. The convergence analysis for the proposed method demonstrates that as the number of basis functions increases, the approximate solution monotonically approaches the exact solution. The behavior of the solutions is visually examined through comprehensive graphical representations. All computational tasks and graphical visualizations were executed using Python.