TY  - JOUR
AU  - Adamu, Abdulrahman
AU  - Alhassan, Buhari
PY  - 2026
DA  - 2026/09/04
TI  - A Modified Third-Order Block Numerical Integration Technique for First Order Stiff and Oscillatory Differential Equations
JO  - ICCK Journal of Applied Mathematics
T2  - ICCK Journal of Applied Mathematics
JF  - ICCK Journal of Applied Mathematics
VL  - 2
IS  - 3
SP  - 242
EP  - 250
DO  - 10.62762/JAM.2026.326643
UR  - https://www.icck.org/article/abs/JAM.2026.326643
KW  - a-stability
KW  - order
KW  - convergence
KW  - block backward differentiation formula
KW  - stiff
AB  - This paper proposes a modified two-point block backward differentiation formula (MBBDF) for the numerical solution of highly stiff and oscillatory ordinary differential equations. The method achieves third-order accuracy with a reduced error constant and computes two solution values simultaneously at each step. For the selected parameter value $\rho=-\dfrac{7}{8}$, stability analysis shows that the resulting scheme is A-stable, making it suitable for the numerical integration of stiff initial value problems. The method is also consistent and zero-stable, and hence convergent. Numerical experiments on benchmark stiff problems show that the relative accuracy of the considered methods depends on the problem and step size. While 2SBBDF generally produces the smallest maximum errors, the proposed MBBDF method maintains competitive accuracy and often requires substantially less computational time than 2SBBDF and 2ISBBDF. The results indicate that MBBDF provides a favorable compromise between numerical accuracy, stability, and computational efficiency for the solution of stiff and oscillatory ordinary differential equations.
SN  - 3068-5656
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Adamu2026A,
  author = {Abdulrahman Adamu and Buhari Alhassan},
  title = {A Modified Third-Order Block Numerical Integration Technique for First Order Stiff and Oscillatory Differential Equations},
  journal = {ICCK Journal of Applied Mathematics},
  year = {2026},
  volume = {2},
  number = {3},
  pages = {242-250},
  doi = {10.62762/JAM.2026.326643},
  url = {https://www.icck.org/article/abs/JAM.2026.326643},
  abstract = {This paper proposes a modified two-point block backward differentiation formula (MBBDF) for the numerical solution of highly stiff and oscillatory ordinary differential equations. The method achieves third-order accuracy with a reduced error constant and computes two solution values simultaneously at each step. For the selected parameter value \$\rho=-\dfrac{7}{8}\$, stability analysis shows that the resulting scheme is A-stable, making it suitable for the numerical integration of stiff initial value problems. The method is also consistent and zero-stable, and hence convergent. Numerical experiments on benchmark stiff problems show that the relative accuracy of the considered methods depends on the problem and step size. While 2SBBDF generally produces the smallest maximum errors, the proposed MBBDF method maintains competitive accuracy and often requires substantially less computational time than 2SBBDF and 2ISBBDF. The results indicate that MBBDF provides a favorable compromise between numerical accuracy, stability, and computational efficiency for the solution of stiff and oscillatory ordinary differential equations.},
  keywords = {a-stability, order, convergence, block backward differentiation formula, stiff},
  issn = {3068-5656},
  publisher = {Institute of Central Computation and Knowledge}
}