TY  - JOUR
AU  - Abubakar, Ahmad Umar
PY  - 2026
DA  - 2026/08/31
TI  - On the Convergence and Order of An Extended 2-Point Super Class BBDF with Two Intermediate Points for Solving Stiff IVPs
JO  - ICCK Journal of Applied Mathematics
T2  - ICCK Journal of Applied Mathematics
JF  - ICCK Journal of Applied Mathematics
VL  - 2
IS  - 3
SP  - 233
EP  - 241
DO  - 10.62762/JAM.2026.599481
UR  - https://www.icck.org/article/abs/JAM.2026.599481
KW  - BBDF
KW  - convergence
KW  - order
KW  - stiff IVPs
KW  - superclass
AB  - This study investigates the order and convergence properties of an extended two-point superclass Block Backward Differentiation Formula (BBDF) incorporating two intermediate (off-step) points for the numerical solution of stiff initial value problems (IVPs). Stiff systems frequently arise in scientific and engineering applications and require numerical methods that combine stability, accuracy, and computational efficiency. While classical Backward Differentiation Formula (BDF) methods are well known for their stability, their sequential nature limits efficiency, particularly for large-scale problems. To address these limitations, the proposed framework extends the BBDF approach by introducing off-step points within a two-point superclass structure, allowing multiple solution values to be computed simultaneously. This block formulation enhances parallelism and reduces the number of integration steps required. The method is formulated in matrix form, and its theoretical properties are rigorously analyzed. The order of the method is established using a Taylor series expansion of the associated linear difference operator, and the scheme is shown to achieve fifth-order accuracy. The method's convergence is examined using the standard consistency and zero-stability criteria. The proposed scheme satisfies the necessary and sufficient conditions for the convergence of linear multistep methods. The results confirm that the proposed method provides a reliable and efficient framework for solving stiff systems.
SN  - 3068-5656
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Abubakar2026On,
  author = {Ahmad Umar Abubakar},
  title = {On the Convergence and Order of An Extended 2-Point Super Class BBDF with Two Intermediate Points for Solving Stiff IVPs},
  journal = {ICCK Journal of Applied Mathematics},
  year = {2026},
  volume = {2},
  number = {3},
  pages = {233-241},
  doi = {10.62762/JAM.2026.599481},
  url = {https://www.icck.org/article/abs/JAM.2026.599481},
  abstract = {This study investigates the order and convergence properties of an extended two-point superclass Block Backward Differentiation Formula (BBDF) incorporating two intermediate (off-step) points for the numerical solution of stiff initial value problems (IVPs). Stiff systems frequently arise in scientific and engineering applications and require numerical methods that combine stability, accuracy, and computational efficiency. While classical Backward Differentiation Formula (BDF) methods are well known for their stability, their sequential nature limits efficiency, particularly for large-scale problems. To address these limitations, the proposed framework extends the BBDF approach by introducing off-step points within a two-point superclass structure, allowing multiple solution values to be computed simultaneously. This block formulation enhances parallelism and reduces the number of integration steps required. The method is formulated in matrix form, and its theoretical properties are rigorously analyzed. The order of the method is established using a Taylor series expansion of the associated linear difference operator, and the scheme is shown to achieve fifth-order accuracy. The method's convergence is examined using the standard consistency and zero-stability criteria. The proposed scheme satisfies the necessary and sufficient conditions for the convergence of linear multistep methods. The results confirm that the proposed method provides a reliable and efficient framework for solving stiff systems.},
  keywords = {BBDF, convergence, order, stiff IVPs, superclass},
  issn = {3068-5656},
  publisher = {Institute of Central Computation and Knowledge}
}