Diagonally implicit block backward differentiation formula with off step points for solving stiff ordinary differential equations - Abd Rasid, Norshakila (2021)
This research demonstrates an alternative method for solving stiff ordinary differential equations (ODEs) using a diagonally implicit block backward differentiation formula with off-step points (DOBBDF). The off-step points are the optimal points between two equidistant grid points that help provided stable and high-accuracy solutions. The diagonally implicit form optimized the computational cost since fewer differential coefficients caused reducing the execution times. The thesis is divided into two significant parts. The first part showed the derivation and implementation of the two-point DOBBDF using constant and variable step-size strategies for solving the first-order stiff ODEs. The methods satisfied the convergence properties and A-stable conditions and yielded the region which contains the whole negative real axis in the complex plane. Numerical results revealed that the derived method excels than the other same kind methods. The second part described the formulation of DOBBDF for solving second-order ODEs directly. The direct method is the best feature to replace the previously expensive approach. The costly technique involved reducing the higher-order ODEs to first-order ODEs and solve using the first-order method. The new direct methods emphasized approximation at two solution points and two off-step points simultaneously in a block using constant and variable step-size strategies. The methods satisfied the properties of consistency and zero-stable, guaranteed convergent method for directly solving second-order Initial value problems of ODEs. Last, the DOBBDF is validated with several application models, including cancer, gene regulations, Prothero-Robinson system, and oscillation problems. In conclusion, DOBBDF is a significant alternative solver for the stiff ODEs model in science and engineering.