Numerical solutions of fractional differential equation with multiple delays via block boundary value method

The aim of this paper is to present a new numerical scheme, namely the fractional multi-delay block boundary value method (FMDBBVM), which is proposed to solve fractional differential equations (FDEs) with multiple delays. The FMDBBVM is proposed via a combination of the block boundary value method (BBVM) and the Lagrange interpolating polynomial. In the methodology of the FMDBBVM, the fractional derivative is approximated through a qth-order Lagrange interpolating polynomial with mth-order BBVM. The largest delay is dealt with a modification in the BBVM, and the remaining delays are approximated in the mesh concerning to the largest delay via the Lagrange interpolating polynomial of (w(1)+ w (2)+ 1)th order. The proposed scheme is theoretically showed to be globally stable and convergent with order min{m, q- alpha+ 1, w(1)+ w(2) + 1}. Further, a reduced scheme is obtained corresponding to the FMDBBVMfor the fractional delay differential equations (FDDEs) in whichmultiple delays can be accommodated in a singlemesh. This scheme is also convergent with order min{m, q - alpha+ 1}, and it is also globally stable. The numerical efficiency of both the proposed schemes is demonstrated with the help of some examples.