Diagonally implicit block backward differentiation formula with off step points for solving stiff ordinary differential equations
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 prov...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2021
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/92775/1/FS%202021%2038%20-%20IR.pdf |
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| Summary: | 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. |
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