Malaysian Journal of Mathematical Sciences, March 2024, Vol. 18, No. 1

Fractional Block Method for the Solution of Fractional Order Differential Equations

Noor, N. M., Yatim, S. A. M., and Ibrahim, Z. B.

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Received date: 13 October 2023
Accepted date: 23 January 2024

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be \textit{A}-stable for different values of $\alpha$ within the interval $\left( 0,1\right) $. Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order $\alpha$ lies within the range of $0$ and $1$. Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods.

Keywords: linear multistep method; fractional block method; fractional order; single order FDEs; stability



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