Construction an Implicit Block Multi-Steps Approach for Solving Sixth-Order Fractional Differential Equations

In this paper, we focus on deriving an efficient method for solving ordinary differential equations (ODEs) of sixth order, and then, we modify the proposed method for solving fractional differential equations (FDEs).

The methodology of this paper used the approach of derivation of implicit numerical method. However, the purpose of this study is to derive a direct implicit block and multi-step strategy for solving sixth-order ODEs.

The main novelty of this work lies in the investigation of constructing a novel method for solving FDEs. This paper presents a general block two-point approach, known as GIBM2P, for solving sixth-order initial-value problems (IVPs) using Hermite-interpolating polynomials. Consequently, the FDE problem has been transformed into ODE using the α-fractional-derivative transform, and then, the sixth-order IVP is solved using the proposed GIBM2P method.

In this paper, we offered a method for solving sixth-order FDEs. The proposed method has been shown to be an accurate and efficient method; also, one can obtain the approximate efficient solutions of FDEs problems. The numerical implementations are used to show the high accuracy and efficacy of the proposed GIBM2P approach. However, the GIBM2P method is effective in terms of accuracy, processing time, and the number of function calls.